This document discusses methods for analyzing relationships between categorical variables using two-way tables. It explains how to calculate marginal distributions by finding row totals, column totals, and grand totals from two-way tables. Conditional distributions are also discussed, which show the proportion of each category within each row or column total. An example two-way table is provided analyzing the relationship between age group and gender using percentages to describe the marginal and conditional distributions.
The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.
The document defines and provides examples of various types of functions, including:
- Linear functions, which can be represented by a straight line and have the equation y = mx + b.
- Quadratic functions, whose graphs are parabolas with the equation y = ax2 + bx + c.
- Inversely proportional functions with the equation y = k/x.
- Radical functions containing a variable inside a root, like y = √x.
- Exponential functions where the variable is an exponent, like y = ax.
It also discusses how to graph and analyze the key features of each type of function.
This document discusses simplifying radical expressions using the product, quotient, and power rules for radicals. It also covers adding, subtracting, multiplying, and dividing radicals. Rationalizing denominators is explained as well as solving radical equations. Key steps include isolating the radical term, squaring both sides to remove the radical, and checking solutions in the original equation.
Linearization involves developing a linear approximation of a nonlinear system around an operating point. This allows tools from linear systems theory to be applied to analyze and design controllers for nonlinear systems. Specifically, Taylor's theorem is used to expand the nonlinear functions as a linear combination of deviations from the operating point. The resulting linearized model is only valid locally but provides an approximate way to analyze system behavior if well-controlled near the operating point. Examples show how to derive linearized models for common nonlinear systems like tanks and chemical reactors.
The document provides instructions for students on how to simplify radicals. It defines what it means to simplify a radical as having no perfect square factors in the radicand. The objective is for students to be able to simplify at least 9 out of 10 radicals correctly. Students are taught to rewrite the radicand using prime factorization, replace any perfect square factors with their numeric equivalent, and check if the radicand is simplified by having no remaining perfect square factors. Examples are worked through and students complete guided practice problems to demonstrate their understanding.
Equilibrium point analysis linearization techniqueTarun Gehlot
The document discusses the linearization technique for analyzing the behavior of solutions near equilibrium points of nonlinear systems of differential equations. It explains that nonlinear systems can be approximated by linearizing around equilibrium points using a Jacobian matrix. The eigenvalues of the Jacobian matrix then allow classifying the equilibrium point and predicting whether solutions will converge or diverge from it. This technique is demonstrated on examples, including the Van der Pol oscillator and pendulum equations.
This chapter discusses examining relationships between two quantitative variables using scatterplots and correlation. Scatterplots show the relationship between an explanatory and response variable by plotting each data point. Correlation measures the strength and direction of the linear relationship between two variables on a scale of -1 to 1. The least squares regression line fits a straight line to minimize the residuals, or vertical distances between the data points and the line. It is used to model and make predictions about the relationship between two variables.
The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.
The document defines and provides examples of various types of functions, including:
- Linear functions, which can be represented by a straight line and have the equation y = mx + b.
- Quadratic functions, whose graphs are parabolas with the equation y = ax2 + bx + c.
- Inversely proportional functions with the equation y = k/x.
- Radical functions containing a variable inside a root, like y = √x.
- Exponential functions where the variable is an exponent, like y = ax.
It also discusses how to graph and analyze the key features of each type of function.
This document discusses simplifying radical expressions using the product, quotient, and power rules for radicals. It also covers adding, subtracting, multiplying, and dividing radicals. Rationalizing denominators is explained as well as solving radical equations. Key steps include isolating the radical term, squaring both sides to remove the radical, and checking solutions in the original equation.
Linearization involves developing a linear approximation of a nonlinear system around an operating point. This allows tools from linear systems theory to be applied to analyze and design controllers for nonlinear systems. Specifically, Taylor's theorem is used to expand the nonlinear functions as a linear combination of deviations from the operating point. The resulting linearized model is only valid locally but provides an approximate way to analyze system behavior if well-controlled near the operating point. Examples show how to derive linearized models for common nonlinear systems like tanks and chemical reactors.
The document provides instructions for students on how to simplify radicals. It defines what it means to simplify a radical as having no perfect square factors in the radicand. The objective is for students to be able to simplify at least 9 out of 10 radicals correctly. Students are taught to rewrite the radicand using prime factorization, replace any perfect square factors with their numeric equivalent, and check if the radicand is simplified by having no remaining perfect square factors. Examples are worked through and students complete guided practice problems to demonstrate their understanding.
Equilibrium point analysis linearization techniqueTarun Gehlot
The document discusses the linearization technique for analyzing the behavior of solutions near equilibrium points of nonlinear systems of differential equations. It explains that nonlinear systems can be approximated by linearizing around equilibrium points using a Jacobian matrix. The eigenvalues of the Jacobian matrix then allow classifying the equilibrium point and predicting whether solutions will converge or diverge from it. This technique is demonstrated on examples, including the Van der Pol oscillator and pendulum equations.
This chapter discusses examining relationships between two quantitative variables using scatterplots and correlation. Scatterplots show the relationship between an explanatory and response variable by plotting each data point. Correlation measures the strength and direction of the linear relationship between two variables on a scale of -1 to 1. The least squares regression line fits a straight line to minimize the residuals, or vertical distances between the data points and the line. It is used to model and make predictions about the relationship between two variables.
Lagrange's theorem states that for any finite group G and subgroup H of G, the order of H divides the order of G. The document provides the proof of Lagrange's theorem and several examples. It also discusses corollaries, including that every group of prime order is cyclic, every group of order less than 6 is abelian, and the order of an element must divide the group order. However, the converse of Lagrange's theorem is false - there can exist groups where not every divisor of the group order is a possible subgroup order.
This document provides information on radicals and root expressions. It defines the key parts of a radical expression, including the index and radicand. It explains that roots and radicals are inverse operations of exponents, and how to "undo" a power or radical. Examples are given of various nth roots and their relationships to exponents. The document outlines the rules for principal roots, noting even roots have two solutions while odd roots only have one. It cautions the reader to check the index when evaluating roots. Finally, it discusses how to simplify nth roots involving variables by dividing the exponent by the index.
Radical equations contain radicals or rational exponents. To solve them, isolate the radical term and raise both sides of the equation to the same power to cancel out the radical. This may result in extraneous solutions that do not satisfy the original equation, so all solutions must be checked. Solving equations with rational exponents involves isolating the power and raising both sides to the reciprocal power.
This document provides an introduction to regression and correlation analysis. It discusses simple and multiple linear regression models, how to interpret regression coefficients, and how to check the assumptions and adequacy of regression models. Key aspects covered include computing the regression line using the least squares method, interpreting the slope and intercept, checking the normality of residuals, and examining residual plots to validate the model. The goal of regression analysis is to model the relationship between a dependent variable and one or more independent variables.
This document provides an introduction to calculus and functions. It discusses that calculus originated in the 17th century through the work of Newton and Leibniz. A function is defined as a relationship between variables where each value of one variable corresponds to a unique value of the other. Functions can relate multiple variables as well. Examples of functions include the area of a circle as a function of its radius and the perimeter of a rectangle as a function of its length. The document provides exercises on evaluating functions and relating composite functions.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
This document discusses rational exponents and nth roots. It explains that an nth root of a number a is a number whose nth power is equal to a. It also notes that if the index n is even, then the radicand a must be non-negative. The document provides rules for simplifying expressions with rational exponents, such as eliminating the root and then the power. It also discusses strategies for solving equations with exponents and radicals.
This presentation covered the following topics:
1. Definition of Correlation and Regression
2. Meaning of Correlation and Regression
3. Types of Correlation and Regression
4. Karl Pearson's methods of correlation
5. Bivariate Grouped data method
6. Spearman's Rank correlation Method
7. Scattered diagram method
8. Interpretation of correlation coefficient
9. Lines of Regression
10. regression Equations
11. Difference between correlation and regression
12. Related examples
The document discusses correlation analysis and different types of correlation:
1. It defines correlation as measuring the relationship between two variables, and describes correlation as examining how two variables co-vary rather than being causative.
2. Positive correlation occurs when both variables increase or decrease together, while negative correlation is when one variable increases as the other decreases.
3. Zero correlation means no relationship between the variables as change in one does not correspond to change in the other.
This document discusses correlation and linear regression analysis. It begins by outlining the learning objectives which are to describe relationships between variables using correlation, estimate effects of independent variables on dependents with regression, and perform and interpret different types of regression analyses. It then provides examples of how correlation calculates the strength and direction of relationships between interval variables and how regression finds the best fitting linear equation to estimate relationships between variables. It emphasizes that regression minimizes the sum of squared errors to find the line of best fit for the data.
1) The document provides an overview of key concepts in probability and statistics, including random variables, probability distributions, and characteristics of distributions such as expected value and variance.
2) It defines key probability terms such as population, sample, mutually exclusive events, independent events, and exhaustive events. It also covers how to calculate the probability of single and multiple events.
3) The document distinguishes between discrete and continuous random variables and probability distributions. It explains how probability distributions associate probabilities with individual outcomes for discrete variables but use probability density functions to provide probabilities over intervals for continuous variables.
This document discusses correlation coefficient and different types of correlation. It defines correlation coefficient as the measure of the degree of relationship between two variables. It explains different types of correlation such as perfect positive correlation, perfect negative correlation, moderately positive correlation, moderately negative correlation, and no correlation. It also discusses different methods to study correlation including scatter diagram method, graphic method, Karl Pearson's coefficient of correlation method, and Spearman's rank correlation method. It provides examples and steps to calculate correlation coefficient using these different methods.
This document discusses correlation and linear regression. It defines correlation as the association between two variables, which can be positive, negative, or non-existent. Linear correlation exists when plotted points approximate a straight line. The correlation coefficient r measures the strength of a linear relationship between -1 and 1. Linear regression finds the linear relationship that best fits the data using a regression equation to predict y values from x. Multiple linear regression extends this to use multiple explanatory variables.
1. The document discusses the concept of correlation, including the different types and methods of measuring correlation.
2. It provides background on the history of correlation, beginning with its proposal by French scientist A. Bravis, and development of graphical representation by Sir Francis Galton.
3. Key aspects covered include Karl Pearson's coefficient of correlation (r), which measures the strength and direction of linear relationships between variables ranging from -1 to 1. Examples are provided to illustrate different degrees of positive and negative correlation.
The document discusses rules for simplifying radical expressions. It states that if the index of radicals in a division or multiplication expression are the same, the division can be rewritten as a division of the radicals and the multiplication can be rewritten as a multiplication of the radicals. Examples are provided to demonstrate simplifying expressions using these rules together with rules for rational exponents and reducing the index of fractional exponents.
1. The document discusses logarithmic transformations, which can be used to transform non-linear data into a linear format to better model exponential relationships.
2. There are two options for logarithmic transformations: taking the log of just the response variable y, or taking the log of both the explanatory variable x and the response variable y.
3. Graphing the transformed data allows one to determine which option produces a more linear relationship and thus the better transformation to use.
This document discusses correlation and linear regression. It defines correlation as a measure of the linear association between two variables. The strength of the correlation is quantified from 0 (no association) to 1 (perfect association). Regression analysis predicts the value of a dependent variable based on independent variables. Simple linear regression fits a linear equation to the data of the form Y=β0 + β1X + ε, where β0 is the Y-intercept and β1 is the slope of the regression line. The coefficient of determination, R-squared, indicates how much of the variation in the dependent variable is explained by the independent variable.
The document discusses the parties, settings, purposes, and reference sources for psychological testing and assessment. It identifies the main parties as test developers/publishers, test users, test takers, and society. Assessments are commonly conducted in educational, geriatric, counseling, clinical, business, military, and other settings to evaluate individuals' abilities, diagnose issues, inform treatment, and make organizational decisions. Authoritative information on specific tests can be found in test manuals, reference books, journal articles, online databases, and test publishers' catalogs.
The document discusses psychological research methods. It begins by defining research and its goals, which include describing behavior, establishing relationships between causes and effects, and developing theories about human behavior. It then describes the empirical research cycle and different research methods, both primary like experiments and secondary like meta-analyses. It discusses variables, research designs, qualitative and quantitative data collection and analysis, and drawing conclusions. Finally, it covers ethical issues in research and challenges in determining causality.
Lagrange's theorem states that for any finite group G and subgroup H of G, the order of H divides the order of G. The document provides the proof of Lagrange's theorem and several examples. It also discusses corollaries, including that every group of prime order is cyclic, every group of order less than 6 is abelian, and the order of an element must divide the group order. However, the converse of Lagrange's theorem is false - there can exist groups where not every divisor of the group order is a possible subgroup order.
This document provides information on radicals and root expressions. It defines the key parts of a radical expression, including the index and radicand. It explains that roots and radicals are inverse operations of exponents, and how to "undo" a power or radical. Examples are given of various nth roots and their relationships to exponents. The document outlines the rules for principal roots, noting even roots have two solutions while odd roots only have one. It cautions the reader to check the index when evaluating roots. Finally, it discusses how to simplify nth roots involving variables by dividing the exponent by the index.
Radical equations contain radicals or rational exponents. To solve them, isolate the radical term and raise both sides of the equation to the same power to cancel out the radical. This may result in extraneous solutions that do not satisfy the original equation, so all solutions must be checked. Solving equations with rational exponents involves isolating the power and raising both sides to the reciprocal power.
This document provides an introduction to regression and correlation analysis. It discusses simple and multiple linear regression models, how to interpret regression coefficients, and how to check the assumptions and adequacy of regression models. Key aspects covered include computing the regression line using the least squares method, interpreting the slope and intercept, checking the normality of residuals, and examining residual plots to validate the model. The goal of regression analysis is to model the relationship between a dependent variable and one or more independent variables.
This document provides an introduction to calculus and functions. It discusses that calculus originated in the 17th century through the work of Newton and Leibniz. A function is defined as a relationship between variables where each value of one variable corresponds to a unique value of the other. Functions can relate multiple variables as well. Examples of functions include the area of a circle as a function of its radius and the perimeter of a rectangle as a function of its length. The document provides exercises on evaluating functions and relating composite functions.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
This document discusses rational exponents and nth roots. It explains that an nth root of a number a is a number whose nth power is equal to a. It also notes that if the index n is even, then the radicand a must be non-negative. The document provides rules for simplifying expressions with rational exponents, such as eliminating the root and then the power. It also discusses strategies for solving equations with exponents and radicals.
This presentation covered the following topics:
1. Definition of Correlation and Regression
2. Meaning of Correlation and Regression
3. Types of Correlation and Regression
4. Karl Pearson's methods of correlation
5. Bivariate Grouped data method
6. Spearman's Rank correlation Method
7. Scattered diagram method
8. Interpretation of correlation coefficient
9. Lines of Regression
10. regression Equations
11. Difference between correlation and regression
12. Related examples
The document discusses correlation analysis and different types of correlation:
1. It defines correlation as measuring the relationship between two variables, and describes correlation as examining how two variables co-vary rather than being causative.
2. Positive correlation occurs when both variables increase or decrease together, while negative correlation is when one variable increases as the other decreases.
3. Zero correlation means no relationship between the variables as change in one does not correspond to change in the other.
This document discusses correlation and linear regression analysis. It begins by outlining the learning objectives which are to describe relationships between variables using correlation, estimate effects of independent variables on dependents with regression, and perform and interpret different types of regression analyses. It then provides examples of how correlation calculates the strength and direction of relationships between interval variables and how regression finds the best fitting linear equation to estimate relationships between variables. It emphasizes that regression minimizes the sum of squared errors to find the line of best fit for the data.
1) The document provides an overview of key concepts in probability and statistics, including random variables, probability distributions, and characteristics of distributions such as expected value and variance.
2) It defines key probability terms such as population, sample, mutually exclusive events, independent events, and exhaustive events. It also covers how to calculate the probability of single and multiple events.
3) The document distinguishes between discrete and continuous random variables and probability distributions. It explains how probability distributions associate probabilities with individual outcomes for discrete variables but use probability density functions to provide probabilities over intervals for continuous variables.
This document discusses correlation coefficient and different types of correlation. It defines correlation coefficient as the measure of the degree of relationship between two variables. It explains different types of correlation such as perfect positive correlation, perfect negative correlation, moderately positive correlation, moderately negative correlation, and no correlation. It also discusses different methods to study correlation including scatter diagram method, graphic method, Karl Pearson's coefficient of correlation method, and Spearman's rank correlation method. It provides examples and steps to calculate correlation coefficient using these different methods.
This document discusses correlation and linear regression. It defines correlation as the association between two variables, which can be positive, negative, or non-existent. Linear correlation exists when plotted points approximate a straight line. The correlation coefficient r measures the strength of a linear relationship between -1 and 1. Linear regression finds the linear relationship that best fits the data using a regression equation to predict y values from x. Multiple linear regression extends this to use multiple explanatory variables.
1. The document discusses the concept of correlation, including the different types and methods of measuring correlation.
2. It provides background on the history of correlation, beginning with its proposal by French scientist A. Bravis, and development of graphical representation by Sir Francis Galton.
3. Key aspects covered include Karl Pearson's coefficient of correlation (r), which measures the strength and direction of linear relationships between variables ranging from -1 to 1. Examples are provided to illustrate different degrees of positive and negative correlation.
The document discusses rules for simplifying radical expressions. It states that if the index of radicals in a division or multiplication expression are the same, the division can be rewritten as a division of the radicals and the multiplication can be rewritten as a multiplication of the radicals. Examples are provided to demonstrate simplifying expressions using these rules together with rules for rational exponents and reducing the index of fractional exponents.
1. The document discusses logarithmic transformations, which can be used to transform non-linear data into a linear format to better model exponential relationships.
2. There are two options for logarithmic transformations: taking the log of just the response variable y, or taking the log of both the explanatory variable x and the response variable y.
3. Graphing the transformed data allows one to determine which option produces a more linear relationship and thus the better transformation to use.
This document discusses correlation and linear regression. It defines correlation as a measure of the linear association between two variables. The strength of the correlation is quantified from 0 (no association) to 1 (perfect association). Regression analysis predicts the value of a dependent variable based on independent variables. Simple linear regression fits a linear equation to the data of the form Y=β0 + β1X + ε, where β0 is the Y-intercept and β1 is the slope of the regression line. The coefficient of determination, R-squared, indicates how much of the variation in the dependent variable is explained by the independent variable.
The document discusses the parties, settings, purposes, and reference sources for psychological testing and assessment. It identifies the main parties as test developers/publishers, test users, test takers, and society. Assessments are commonly conducted in educational, geriatric, counseling, clinical, business, military, and other settings to evaluate individuals' abilities, diagnose issues, inform treatment, and make organizational decisions. Authoritative information on specific tests can be found in test manuals, reference books, journal articles, online databases, and test publishers' catalogs.
The document discusses psychological research methods. It begins by defining research and its goals, which include describing behavior, establishing relationships between causes and effects, and developing theories about human behavior. It then describes the empirical research cycle and different research methods, both primary like experiments and secondary like meta-analyses. It discusses variables, research designs, qualitative and quantitative data collection and analysis, and drawing conclusions. Finally, it covers ethical issues in research and challenges in determining causality.
This document provides an introduction to psychological testing and assessment. It defines key terms like tests, testing, and assessment. It describes the different types of constructs that can be measured by psychological tests, like traits, conditions, intelligence, and attitudes. It also discusses the differences between testing and assessment and contrasts the two processes. The document outlines the various tools used in psychological assessment, including tests, interviews, observations, and more. It discusses important topics like test administration and interpretation, the parties involved in testing, and the various settings where assessment is conducted. Finally, it covers issues like culture, ethics, and laws related to psychological assessment.
This module introduces key concepts in statistics. It will cover defining statistics and related terms, the history and importance of statistics, summation rules, sampling techniques, organizing data in tables, constructing frequency distributions, and measures of central tendency for ungrouped data. The goal is for students to understand how statistics is used in daily life and to learn techniques for collecting, organizing, and analyzing data.
Psychological testing is a field characterized by the use of samples of performance in order to assess psychological construct, such as cognitive and emotional implementation, about a given individual.
Psychological tests are formal tools used to measure mental functioning and behaviors. They can be administered in various settings like schools, hospitals, and workplaces to assess abilities, personality, and neurological status. Common uses of tests include education placement, career counseling, diagnosing disorders, and selecting job applicants. Tests vary in their administration method, targeted behaviors, and purpose between ability, personality, and clinical domains. Proper tests are standardized, objective, use norms, and are reliable and valid measures of their intended construct.
Correlation is a statistical technique used to determine the degree of relationship between two variables. Correlational research aims to identify and describe relationships but does not imply causation. Positive correlation indicates high scores on one variable are associated with high scores on the other, while negative correlation means high scores on one variable are associated with low scores on the other. Correlational research can be used for explanatory or predictive purposes. More complex techniques like multiple regression allow prediction using combinations of variables. Threats to internal validity like subject characteristics must be controlled.
This document discusses correlation analysis and its various types. Correlation is the degree of relationship between two or more variables. There are three stages to solve correlation problems: determining the relationship, measuring significance, and establishing causation. Correlation can be positive, negative, simple, partial, or multiple depending on the direction and number of variables. It is used to understand relationships, reduce uncertainty in predictions, and present average relationships. Conditions like probable error and coefficient of determination help interpret correlation values.
This document discusses psychological assessment and tests. It describes the development and types of psychological tests, including intelligence tests like the Stanford-Binet and Wechsler scales, achievement tests, aptitude tests, personality tests like the MMPI, and projective tests like the Rorschach inkblot test. It also outlines the nurse's role in psychological assessment, which includes educating patients, observing behaviors, and documenting changes.
Psychological tests were developed to assist in understanding human behavior and making important decisions in an objective manner. Tests provide standardized samples of behavior that can be used to infer underlying traits and make comparisons to norms. This allows for decisions to be made with less bias than relying solely on subjective human judgment. Tests quantify results to precisely describe behaviors and allow for clearer communication than qualitative descriptions alone.
This document discusses rank correlation and Spearman's rank correlation coefficient. It defines correlation as a relationship between two variables where a change in one variable corresponds to a change in the other. Rank correlation involves ranking observations from highest to lowest rather than using the original values, which avoids assumptions about the population distribution. Spearman's rank correlation coefficient measures the correspondence between two rankings and is calculated based on the differences between ranks of paired items. It provides a distribution-free measure of correlation.
Correlation of subjects in school (b.ed notes)Namrata Saxena
This document discusses the concept of correlation in education. It defines correlation as the mutual relationship between different subjects or variables in a curriculum. The document outlines the importance of correlation, including that it helps students perceive knowledge as a whole, strengthens retention of knowledge, and promotes well-rounded development. It discusses different types of correlation, including vertical/internal correlation between topics within a subject and horizontal/external correlation between different subjects. Examples are provided of how mathematics can be correlated with other subjects like science, geography, and economics.
Psychological test meaning, concept, need & importancejd singh
This document discusses psychological testing. It defines psychological testing as a standardized measure of a person's behavior that is used to observe differences among individuals. It notes that tests measure constructs like abilities, functioning, and personality. The document outlines the objectives, need, importance and types of psychological tests. It describes the major characteristics of tests including standardization, norms, reliability and validity. Finally, it provides examples of commonly used psychological tests.
The document discusses correlation analysis and different types of correlation. It defines correlation as the linear association between two random variables. There are three main types of correlation:
1) Positive vs negative vs no correlation based on the relationship between two variables as one increases or decreases.
2) Linear vs non-linear correlation based on the shape of the relationship when plotted on a graph.
3) Simple vs multiple vs partial correlation based on the number of variables.
The document also discusses methods for studying correlation including scatter plots, Karl Pearson's coefficient of correlation r, and Spearman's rank correlation coefficient. It provides interpretations of the correlation coefficient r and coefficient of determination r2.
Correlation describes the relationship between two or more variables. A positive correlation means that as one variable increases, the other also increases, while a negative correlation means that as one variable increases, the other decreases. Correlation is measured numerically using coefficients like the Pearson correlation coefficient r, which ranges from -1 to 1, with values farther from 0 indicating stronger linear relationships and the direction indicating positive or negative correlation. Correlation is used in business and economics to study relationships between variables like price and demand.
This document provides an introduction to correlation and regression analysis. It defines correlation as a measure of the association between two variables and regression as using one variable to predict another. The key aspects covered are:
- Calculating correlation using Pearson's correlation coefficient r to measure the strength and direction of association between variables.
- Performing simple linear regression to find the "line of best fit" to predict a dependent variable from an independent variable.
- Using a TI-83 calculator to graphically display scatter plots of data and calculate the regression equation and correlation coefficient.
This document summarizes Steve Hoang's presentation on linear models in R. The presentation introduced linear models, demonstrated how to fit and interpret them in R, and discussed transforming data and hypothesis testing. It also briefly covered mixed effects models. The goal was to provide basic tools for interpreting linear models and pursuing more advanced topics.
This document provides information about assumptions of linear regression and checking those assumptions. It discusses the key assumptions of linear regression as normal distribution of residuals, linear relationship between variables, equal variance of residuals, and independence of observations. It outlines steps to check the assumptions, including plotting residuals against fitted and explanatory values and a normal probability plot of residuals. Specific things to look for that may indicate violations of assumptions are also described.
Data pre-processing for neural networks
NNs learn faster and give better performance if the input variables are pre-processed before being used to train the network
Mathematics in the Modern World (Patterns and Sequences).pptxReignAnntonetteYayai
Patterns exist in different variety of forms. The petals of a flower, arrangement of leaves reveals a sequential pattern. Natures are bounded by different colors and shapes – the rainbow mosaic of a butterfly’s wings, the undulating ripples of a desert dune. But these miraculous creations not only delight the imagination, they also challenge our understanding. How do these patterns develop? What sorts of rules and guidelines, shape the patterns in the world around us? Some patterns are molded with a strict regularity. At least superficially, the origin of regular patterns often seems easy to explain. Thousands of times over, the cells of a honeycomb repeat their hexagonal symmetry. The honeybee is a skilled and tireless artisan with an innate ability to measure the width and to gauge the thickness of the honeycomb it builds. Although the workings of an insect's mind may baffle biologists, the regularity of the honeycomb attests to the honey bee's remarkable architectural abilities. A pattern is something which helps us anticipate what we might see or expect to happen next. It may also help us know what may have come before or what we are seeing currently. There are four types of patterns; (1) logic patterns, (2) number patterns, (3) geometric patterns and (4) word patterns. 4. Rippled pattern observed on the desert sand. 5. Honeycomb structure A pattern has symmetry. Isometry of the plane that preserves the pattern. It is a way of transforming the plane that preserves geometrical properties such as length. There are four types of isometries according to Euclidian isometry of plane transformation (1) Translation (2) Reflection (3) Rotation (4) Dilation. Moreover, we have to consider sometimes the combination of Reflection, translation and rotations makes another isometry called rigid transformation which leave the dimensions of the object and its image unchanged. A propositional variable represented by a lowercase or capital letter in the English alphabet denotes an arbitrary proposition with an unspecified truth value. An assertion which contains at least one proposition variable is called a propositional form. In the preposition: “if I study the lesson, Then I will pass the test” In a condition hypothetical proposition, the truth does not rest on the truth of every statement taken singly. Rather, it depends on valid sequence between members of the proposition. In the example given, we don’t assert, “I study the lesson” nor we assert, “I will be able to pass the test”. We ae going to simply declare the fact that the statement “I study the lesson” is dependent on the other statement which is “I will pass the test” and vice versa. Note: The word “then’ as part of the consequent maybe omitted. “IF I study the lesson, I will pass the test” The consequent may also be written ahead of the antecedent and the word “then” is omitted. “I will pass the test, IF I study t
This document discusses multiple linear regression analysis. It begins by defining a multiple regression equation that describes the relationship between a response variable and two or more explanatory variables. It notes that multiple regression allows prediction of a response using more than one predictor variable. The document outlines key elements of multiple regression including visualization of relationships, statistical significance testing, and evaluating model fit. It provides examples of interpreting multiple regression output and using the technique to predict outcomes.
The document discusses linear inequalities in one variable. It defines a linear inequality in one variable as an inequality that can be written in the form ax + b > c, where a, b, and c are real numbers. It notes that the > symbol can be replaced by ≥, <, or ≤. The document provides examples and steps for transforming linear inequalities into standard form where the leading coefficient a is positive and the inequality is written as ax + b > 0. It emphasizes using properties of inequalities and multiplying by -1 when a is negative.
The "Great Lakes" data set is an example of a non-seasonal, non-stationary time series that
experiences a slight upward linear trend. The series is differenced and transformed using
"Box-Cox" in order to stabilize the mean and variance, correcting for stationarity. The best
model fitted for the data was an ARIMA(4,1,0) found by observing the partial and auto
correlation functions. The fit suggested the best estimates for the coefficients via the AIC.
Verified as independent random variables, the residuals of the fitted model were tested for
normality using the McLeod-Li, Ljung-Box, and Shapiro-Wilk test. The model proved to be
an adequate representation of the data providing reasonable predictions for precipitation.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
Lecture 4 - Linear Regression, a lecture in subject module Statistical & Mach...Maninda Edirisooriya
Simplest Machine Learning algorithm or one of the most fundamental Statistical Learning technique is Linear Regression. This was one of the lectures of a full course I taught in University of Moratuwa, Sri Lanka on 2023 second half of the year.
This document discusses various methods for transforming variables to achieve normality, stabilize variance, and ensure linearity. It describes common transformations like logarithmic, square root, power, inverse, reciprocal, and cube root. The objectives of transformation are outlined. Guidelines are provided for choosing a transformation method and assessing whether it successfully achieves linearity. Precautions for using transformations and ensuring assumptions still hold are also noted.
This document provides guidelines for carrying out statistical analyses in SPSS and R using various datasets. It discusses how to replicate analyses from 2x2 tables using individual level data, and how to perform tests such as the Kappa test, McNemar's test, chi-square tests, tests for independent proportions, Fisher's exact test, Levene's test, Wilcoxon signed-rank tests, Mann-Whitney U tests, t-tests, and Q-Q plots in both SPSS and R. Instructions are provided for reading in SPSS data files into R and accessing variable values.
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
This document provides an introduction and overview of linear equations. It defines key terms like equations, variables, and solutions. It explains that the goal in solving equations is to find the value of the unknown that makes the statement true. The document outlines various properties of equality that can be used to solve equations, such as distributing the same operation to both sides. It also distinguishes between linear and nonlinear equations. Several examples are provided to demonstrate how to solve different types of linear equations, including those with fractions and those that simplify to linear form. The document also briefly introduces solving power equations, which involve variables raised to powers, as well as equations with fractional exponents.
Probability theory provides a framework for quantifying and manipulating uncertainty. It allows optimal predictions given incomplete information. The document outlines key probability concepts like sample spaces, events, axioms of probability, joint/conditional probabilities, and Bayes' rule. It also covers important probability distributions like binomial, Gaussian, and multivariate Gaussian. Finally, it discusses optimization concepts for machine learning like functions, derivatives, and using derivatives to find optima like maxima and minima.
The document discusses various sorting algorithms in Java including bubble sort, insertion sort, selection sort, merge sort, heapsort, and quicksort. It provides explanations of how each algorithm works and comparisons of the time performance of each algorithm based on testing multiple runs. Quicksort and heapsort generally had the best performance while bubble sort consistently had the worst performance.
Slides for "Do Deep Generative Models Know What They Don't know?"Julius Hietala
My slides that discuss different deep generative models, mainly normalizing flows for density estimation at a deep learning seminar at Aalto University fall 2019.
The document discusses bivariate and multivariate linear regression analysis, explaining how to estimate regression coefficients using software like SPSS and interpret their results. It covers topics such as estimating and interpreting intercept and slope coefficients, measuring predictive power using R-squared, and testing the significance of individual regression coefficients and the overall regression model through techniques like t-tests and F-tests.
Stuck with your Regression Assignment? Get 24/7 help from tutors with Phd in the subject. Email us at support@helpwithassignment.com
Reach us at http://www.HelpWithAssignment.com
Changing the subject of a formula (Simple Formulae)Alona Hall
This document provides instructions for changing the subject of a formula. It begins by explaining the concept of changing the subject, which means taking a formula with one subject and making a different term the subject. It then demonstrates the procedure through examples of changing the subject in formulas involving addition, subtraction, multiplication, division and combinations of operations. The key steps are to identify what mathematical operation was applied to the original subject and then apply the inverse operation to isolate the new subject. A shorter method is also introduced which involves flipping the sides of the formula and applying the inverse operation to the other side. Worked examples demonstrating both methods are provided.
The document provides an overview of the structure and content covered on the AP Calculus AB exam, including:
- The exam is 3 hours 15 minutes long and divided into multiple choice and free response sections testing limits, derivatives, integrals, and applications of calculus.
- Content topics covered include limits of functions, continuity, derivatives and their applications (related rates, max/min problems), integrals, and differential equations.
- Formulas and strategies are provided for evaluating limits, finding derivatives using various rules, applying derivatives to sketch curves, solve optimization problems, and solve motion problems using related rates.
1. The document discusses probability and chance experiments. It provides examples to illustrate key concepts such as sample space, events, and how to calculate probabilities.
2. One example examines student food preferences in a cafeteria, with the sample space consisting of all possible combinations of student gender and food line choice.
3. The document also covers conditional probability, explaining how to calculate the probability of an event given that another event has occurred. An example calculates the probability of nausea given being seated in the front of a bus.
This document discusses key concepts for collecting data and conducting research studies. It defines variables, data sets, and types of bias that can occur in data collection. Common sampling methods like simple random sampling, stratified sampling, and cluster sampling are described. The document also distinguishes between observational studies and experiments, noting that experiments allow researchers to control variables and determine causal effects. Key aspects of experimental design like treatments, placebos, and control groups are also explained.
This document provides examples and explanations of various graphical methods for describing data, including frequency distributions, bar charts, pie charts, stem-and-leaf diagrams, histograms, and cumulative relative frequency plots. It demonstrates how to construct these graphs using sample data on student weights, grades, ages, and other examples. The goal is to help readers understand different ways to visually represent data distributions and patterns.
This document discusses sampling distributions and the central limit theorem. It defines key terms like population, statistic, and sampling distribution. It shows examples of how sampling distributions become more normal and less variable as the sample size increases. The central limit theorem states that for large sample sizes, the sampling distribution of the sample mean will be approximately normally distributed even if the population is not. It provides properties and rules for the sampling distributions of the sample mean and sample proportion.
This document discusses the importance of statistics and introduces key concepts. It explains that statistics involves collecting, analyzing, and drawing conclusions from data. It also defines important statistical terms like population, sample, variable, and different types of data. Frequency distributions are introduced as a way to organize categorical data by displaying the categories and associated frequencies or relative frequencies. An example frequency distribution is provided using vision correction data from a classroom example.
This document discusses various numerical methods for describing data, including measures of central tendency (mean, median), variability (range, variance, standard deviation), and graphical representations (boxplots). It provides examples and formulas for calculating the mean, median, quartiles, interquartile range, variance, standard deviation, and constructing boxplots. Outliers are defined as observations more than 1.5 times the interquartile range from the quartiles.
This document provides an overview of random variables and probability distributions. It defines discrete and continuous random variables and gives examples of each. Discrete random variables have probabilities associated with each possible value, while continuous random variables are defined by probability density functions where the area under the curve equals the probability. The document discusses how to calculate the mean, variance and standard deviation of discrete random variables from their probability distributions. It also covers how the mean and variance are affected for linear transformations of random variables.
This document summarizes bivariate data and linear regression analysis. It introduces scatterplots and the Pearson correlation coefficient as ways to examine relationships between two variables. A positive correlation indicates that as one variable increases, so does the other, while a negative correlation means one variable increases as the other decreases. The least squares line provides the best fit linear relationship between two variables by minimizing the sum of squared residuals. Calculating the slope and y-intercept of this line allows predicting y-values from x-values. Examples using bus fare and distance data demonstrate these concepts.
This document provides information on estimating population characteristics from sample data, including:
- Point estimates are single numbers based on sample data that represent plausible values of population characteristics.
- Confidence intervals provide a range of plausible values for population characteristics with a specified degree of confidence.
- Formulas are given for constructing confidence intervals for population proportions and means using large sample approximations or t-distributions.
- Guidelines for determining necessary sample sizes to estimate population values within a specified margin of error are also outlined.
The document describes multiple regression models and their applications. It begins by defining a general multiple regression model that relates a dependent variable to multiple predictor variables. It then discusses key aspects of multiple regression models like regression coefficients, the regression function, polynomial regression models, and qualitative predictor variables. The document provides examples of applying multiple regression to model lung capacity based on variables like height, age, gender, and activity level. It describes building different regression models and evaluating their fit and significance.
1. A study examined survival times of patients with advanced cancers in different organs (stomach, bronchus, colon, ovary, or breast) treated with ascorbate.
2. An analysis of variance (ANOVA) was used to determine if survival times differed based on the affected organ. ANOVA compares the means of multiple groups and tests if they are equal.
3. The ANOVA test statistic, F, compares the variation between groups (mean square for treatments) to the variation within groups (mean square for error). If F exceeds a critical value, then at least one group mean is significantly different from the others.
This document discusses methods for comparing two population or treatment means, including notation, hypothesis tests, and confidence intervals. Key points covered include:
1) Notation for comparing two means includes the sample size, mean, variance, and standard deviation for each population or treatment.
2) Hypothesis tests for comparing two means can use a z-test if the population standard deviations are known, or a two-sample t-test if the standard deviations are unknown.
3) Confidence intervals can be constructed for the difference between two population means using a t-distribution, assuming independent random samples of sufficient size or approximately normal populations.
1. The document discusses categorical data analysis and goodness-of-fit tests. It introduces concepts such as univariate categorical data, expected counts, the chi-square test statistic, and assumptions of the chi-square test.
2. An example analyzes faculty status data from a university using a goodness-of-fit test to determine if the proportions are equal across categories. The test fails to reject the null hypothesis that the proportions are equal.
3. Tests for homogeneity and independence in two-way tables are described. Examples calculate expected counts and perform chi-square tests to compare populations' category proportions.
1. The document discusses hypothesis testing using a single sample. It outlines the formal structure of hypothesis tests including the null and alternative hypotheses.
2. Common hypothesis tests are presented including tests of a population proportion, mean, and variability. Examples of hypotheses and solutions are provided.
3. The key steps in a hypothesis testing analysis are defined including stating the hypotheses, selecting the significance level, computing the test statistic and p-value, and making a conclusion. Large sample and small sample tests are described.
This document provides an overview of simple linear regression and correlation. It defines key concepts such as the population regression line, the simple linear regression model equation, and assumptions of the model. Examples are provided to demonstrate calculating the least squares regression line, interpreting the slope and intercept, and evaluating goodness of fit using r-squared. Formulas are given for computing sums of squares, estimating the standard deviation of residuals, and constructing confidence intervals for the slope of the population regression line.
This document provides instructions for adding grades to a Google Site by hosting the grades on Dropbox. It explains how to sign up for a Dropbox account, export grades from EasyGrade Pro and save them to the Dropbox public folder, and embed the grades on a new "Grades" page on the Google Site using an iframe. The process is then tested and instructions are provided for uploading updated grades by exporting new reports from EasyGrade Pro and allowing Dropbox to automatically sync the changes.
This document provides an overview of how to perform chi-square tests for goodness of fit and tests of homogeneity using categorical data. It explains how to set up and carry out chi-square tests through defining hypotheses, calculating test statistics, determining p-values, and making decisions. Examples are provided for chi-square goodness of fit tests to determine if observed count data fits an expected distribution, as well as chi-square tests of homogeneity to assess if the distribution of one categorical variable is the same across categories of another variable. Calculator instructions are also given for performing the relevant calculations and statistical analyses on a TI-83/84 graphing calculator.
The document discusses regression analysis and constructing confidence intervals and conducting significance tests for the slope (β) of the regression line. It provides guidance on checking the assumptions of the regression model, outlines the steps for constructing a confidence interval for β which involves calculating the standard error of the slope (SEb) and the appropriate t-statistic. It also outlines the steps for a significance test on β, which involves defining the null and alternative hypotheses, checking assumptions, and determining whether to reject or fail to reject the null based on the calculated p-value. An example problem is presented to demonstrate applying these procedures.
The document discusses comparing two population parameters using sample data. It covers comparing two means using a two-sample t-test or z-test, and comparing two proportions using a two-sample z-test. Key assumptions for these tests include independent random samples from each population and sample sizes large enough for the sampling distributions to be approximately normal. An example compares systolic blood pressure for two groups, one taking calcium and one placebo, and finds no significant difference. A second example finds preschool significantly reduces the proportion needing later social services.
This document discusses significance tests for population means and proportions using Student's t-distribution and the normal distribution. It provides examples of hypothesis testing for a population mean using a paired t-test and for a population proportion using a single-sample z-test. It also discusses the assumptions, test statistics, and interpretations for these tests. Confidence intervals are presented as complementary to significance tests for estimating population parameters.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Make a Field Mandatory in Odoo 17Celine George
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
3. What if the scatterplot is not
linear?
• Of course not all data is linear!
• Our method in statistics will involving
mathematically operating on one or both
of the explanatory and response variables
• An inverse transformation will be used to
create a non-linear regression model
• This will be a little “mathy”
4. Transformations
• Before we begin transformations,
remember that some well known
phenomenon act in predictable ways
– I.e. when working with time and gravity, you
should know that there is a square
relationship between distance and time!
5. The Basics
• The data from measurements (raw data)
must be operated on.
• Apply the same mathematical
transformation on the raw data
– Ex. “Square every response”
• Use methods from the previous chapter to
find the LSRL for the transformed data
• Analyze your regression to ensure the LSRL is
appropriate
• Apply an inverse transformation on the LSRL
to find the regression for the raw data.
6. Example
Please refer to p 265 exercise 4.2
Length (cm) Period (s)
16.5 0.777
17.5 0.839
19.5 0.912
22.5 0.878
28.5 1.004
31.5 1.087
34.5 1.129
37.5 1.111
43.5 1.290
46.5 1.371
106.5 2.115
9. Example
• L3 = L2^.5 (square
root)
• LinReg L1, L3
• Note that the value
of r2 has increased
• Note that the value
of the residual of the
last point has
decreased
10. Exponential Models
• Many natural phenomenon are explained
by an exponential model.
• Exponential models are marked by sharp
increases in growth and decay.
• Basic model: y = A·Bx
• For this transformation, you need to take
the logarithm of the response data.
• You may use “log10” or “ln” your choice.
– I prefer “ln” (of course)
11. Exponential Models
After the transformation, we have the
following linear model: ln(y) = a + b·x
1. ln(y) = a + b·x
2. eln(y) = e(a + b·x) exponentiate
3. y = ea · ebx property of logarithms
4. Let ‘A’ = ea redefine variables
‘B’ = eb
5. y = A·Bx this is our model
12. Exponential Models
• Since this is an ‘applied math’ course,
you need not remember how to apply the
inverse transformation
• Whew
• BUT you do need to memorize:
when ln(y) = a + bx
y = A·Bx
where ‘A’ = ea and ‘B’ = eb
27. Power Models
• These models are used when the rate of
increase is less severe than an
exponential model, or if you suspect a
‘root’ model
• For this model, you will find the
logarithms of both the expl var and the
resp var
28. Power models
LSRL on transformed data yields:
ln(y) = a + b·ln(x)
1. ln(y) = a + b·ln(x)
2. e ln(y) = e(a + b·ln(x))
3. y = ea·eln(x^b)
4. y = ea ·xb
5. Let ‘A’ = ea
6. y = A · xb
46. Power models
• Much like the exponential model, you
only need to know how the transformed
model becomes the model for the raw
data.
• When ln(y) = a + b·ln(x),
y = A · xb
where ‘A’ = ea
47. Transformation thoughts
• Although this is not a major topic for the
course, you still need to be able to apply
these two transformations (exp and power)
• Be sure to check the residuals for the LSRL
on transformed data! You may have picked
the wrong model :/
• If one model doesn’t work, try the other. I
would start with the exponential model.
• Don’t transform into a cockroach. Ask Kafka!
50. Marginal Distributions
• Tables that relate two categorical variables
are called “Two-Way Tables”
– Ex 4.11 pg 292
• Marginal Distribution
– Very fancy term for “row totals and column
totals”
– Named because the totals appear in the margins
of the table. Wow.
• Often, the percentage of the row or column
table is very informative
56. Marginal Distributions “Age Group”
Age
Group
Female Male Total Marg. Dist.
15-17 89 61 150
18-24 5668 4697 10365
25-34 1904 1589 3494
35 or
older
1660 970 2630
Totals 9321 7317 16639
57. Marginal Distributions “Age Group”
Age
Group
Female Male Total Marg. Dist.
15-17 89 61 150
18-24 5668 4697 10365
25-34 1904 1589 3494
35 or
older
1660 970 2630
Totals 9321 7317 16639
Row total / grand total
150/16639=0.009
58. Marginal Distributions “Age Group”
Age
Group
Female Male Total Marg. Dist.
15-17 89 61 150 0.9%
18-24 5668 4697 10365
25-34 1904 1589 3494
35 or
older
1660 970 2630
Totals 9321 7317 16639
Row total / grand total
150/16639=0.009
59. Marginal Distributions “Age Group”
Age
Group
Female Male Total Marg. Dist.
15-17 89 61 150 0.9%
18-24 5668 4697 10365 62.3%
25-34 1904 1589 3494 21.0%
35 or
older
1660 970 2630 15.8%
Totals 9321 7317 16639 100%
Adds to 100%
60. Marginal Distributions “Gender”
Age
Group
Female Male Total
15-17 89 61 150
18-24 5668 4697 10365
25-34 1904 1589 3494
35 &up 1660 970 2630
Totals 9321 7317 16639
Margin
dist.
56% 44% 100%
Similarly for columns
61. Describing Relationships
• Some relationships are easier to see when
we look at the proportions within each
group
• These distributions are called “Conditional
Distributions”
• To find a conditional distribution, find each
percentage of the row or column total.
• Let’s look at the same table, and find the
conditional distribution of gender, given
each age group
62. Conditional Distributions
Age
Group
Female Male Total
15-17 89 61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
63. Conditional Distributions
Age
Group
Female Male Total
15-17 89 61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
We will look at the
conditional
distribution for this
row
64. Conditional Distributions
Age
Group
Female Male Total
15-17 89 61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
This cell is 89/150
(cell total /row total)
=53.9%
65. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
This cell is 89/150
(cell total /row total)
=59.3%
66. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
This cell is 61/150
(cell total /row total)
=40.7%
67. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
This cell is 61/150
(cell total /row total)
=40.7%
68. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
69. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
The table with
complete conditional
distributions for each
row
70. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
For an analysis of the
effect of age groups,
compare a row’s
conditional
distribution…
71. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
With the marginal
distribution for the
columns…
72. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
They should be close
…
73. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
… unless there is an
effect caused by the
age group (?)
74. Conditional Distributions
Age
Group
Female Male Total
15-17 89
(59.3%)
61
(40.7%)
150
(100%)
18-24 5668
(54.7%)
4697
(45.3%)
10365
(100%)
25-34 1904
(54.5%)
1589
(45.5%)
3494
(100%)
35 or
older
1660
(63.1%)
970
(36.9%)
2630
(100%)
Totals 9321
(56%)
7317
(44%)
16639
(100%)
… and these are not
close to the marginal
distribution!
75. Conditional Distributions
• Based on the previous table, the
distribution of “gender given age group”
are not that different.
• We can see that the “35 and older” group
seems to differ slightly from the overall
trend.
76. Conditional Distributions
“age group given gender”
Age
Group
Female Male Total
15-17 89
(1%)
61
(0.8%)
150
(0.9%)
18-24 5668
(60.8%)
4697
(64.2%)
10365
(62.3%)
25-34 1904
(20.4%)
1589
(21.7%)
3494
(21.0%)
35 or
older
1660
(17.8%)
970
(13.3%)
2630
(15.8%)
Totals 9321
(100%)
7317
(100%)
16639
(100%)
77. Conditional Distributions
“age group given gender”
Age
Group
Female Male Total
15-17 89
(1%)
61
(0.8%)
150
(0.9%)
18-24 5668
(60.8%)
4697
(64.2%)
10365
(62.3%)
25-34 1904
(20.4%)
1589
(21.7%)
3494
(21.0%)
35 or
older
1660
(17.8%)
970
(13.3%)
2630
(15.8%)
Totals 9321
(100%)
7317
(100%)
16639
(100%)
Here is the same chart
with the conditional
distributions by
gender…
78. Conditional Distributions
“age group given gender”
Age
Group
Female Male Total
15-17 89
(1%)
61
(0.8%)
150
(0.9%)
18-24 5668
(60.8%)
4697
(64.2%)
10365
(62.3%)
25-34 1904
(20.4%)
1589
(21.7%)
3494
(21.0%)
35 or
older
1660
(17.8%)
970
(13.3%)
2630
(15.8%)
Totals 9321
(100%)
7317
(100%)
16639
(100%)
Is there a gender
effect noticeable from
this table?
79. Conditional Distributions
“age group given gender”
Age
Group
Female Male Total
15-17 89
(1%)
61
(0.8%)
150
(0.9%)
18-24 5668
(60.8%)
4697
(64.2%)
10365
(62.3%)
25-34 1904
(20.4%)
1589
(21.7%)
3494
(21.0%)
35 or
older
1660
(17.8%)
970
(13.3%)
2630
(15.8%)
Totals 9321
(100%)
7317
(100%)
16639
(100%)
80. Conditional Distribution
Conclusions from the previous chart
• Females are more likely to be in the “35 and
older group” and less likely to be in the “18
to 24” group
• Males are more likely to be in the “18 to 24”
group and less likely to be in the “35 and
older” group
• These differences appear slight. Are
actually “significant” with respect to the
overall distribution?
81. Conditional Distribution
• No single graph portrays the form of the
relationship between categorical
variables.
• No single numerical measure (such as
correlation) summarizes the strength of
the association.
82. Simpson’s Paradox
• Associations that hold true for all of
several groups can reverse direction
when teh data is combined to form a
single group.
• EX 4.15 pg 299
• This phenomenon is often the result of an
“unaccounted” variable.
85. Different Relationships
• Suppose two variables (X and Y) have
some correlation
– i.e. when X increases in value, Y increases as
well
– One of the following relationships may hold.
86. Different Relationships
Causation
• In this relationship, the explanatory
variable is somehow affecting the
response variable.
• In most instances, we are looking to find
evidence of a causation relationship
88. Different Relationships
Common Response
• In this relationship, both X and Y are
correlated to a third (unknown) variable
(Z).
• EX, When Z increases X increases and Y
increases.
• Unless we known about Z, it appears as
though X and Y have a causation
relationship.
90. Different Relationships
Confounding
• X and Y have correlation,
• An (often unknown) third variable ‘Z”
also has correlation with Y
• Is X the explanatory variable, or is Z the
explanatory variable, or are the both
explanatory variables?
92. Causation
• The best way to establish causation is
with a carefully designed experiment
– Possible ‘lurking variables’ are controlled
• Experiments cannot always be conducted
– Many times, they are costly or even unethical
• Some guidelines need to be established in
cases where an observational study is the
only method to measure variables.
93. Causation- some criteria
• Association is strong
• Association is consistent (among different
studies)
• Large values of the response variable are
associated with stronger responses
(typo?)
• The alleged cause precedes the effect in
time
• The alleged cause is probable