Mean- Mean is an essential concept in mathematics and statistics. The mean is the average or the most common value in a collection of numbers
Types of Mean
A. Arithmetic Mean
a. Simple Arithmetic Mean
b. Weighted Arithmetic Mean
B. Geometric Mean
C. Harmonic Mean
1.Calculation of Simple Arithmetic Mean
a) Direct Method
b) Shortcut Method
c) Step Deviation Method
2. Calculation of Weighted Arithmetic Mean
a) Direct Method
b) Shortcut Method
Merits and Demerits of Different types of Mean.
This document discusses measures of central tendency, specifically the arithmetic mean. It provides examples and step-by-step solutions for calculating the arithmetic mean of individual data sets, discrete series with frequencies, and continuous series grouped into class intervals. For continuous series, the formula uses the mid-point of each class interval. The document also includes one problem that requires solving for a missing frequency given the calculated arithmetic mean.
Measures of dispersion
Absolute measure, relative measures
Range of Coe. of Range
Mean deviation and coe. of mean deviation
Quartile deviation IQR, coefficient of QD
Standard deviation and coefficient of variation
The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population.
The document defines and provides examples for calculating the coefficient of variation, which is a measure used to compare the dispersion of data sets. It gives the formula for coefficient of variation as the standard deviation divided by the mean, expressed as a percentage. Two examples are shown comparing the stability of prices between two cities and production between two manufacturing plants, with the data set having the lower coefficient of variation considered more consistent or stable.
The mean deviation is a measure of how spread out values are from the average. It is calculated by:
1) Finding the mean of all values.
2) Calculating the distance between each value and the mean.
3) Taking the average of those distances. This provides the mean deviation, which tells us how far on average values are from the central mean. Examples show calculating mean deviation for both grouped and ungrouped data sets.
Frequency distribution is a method to organize and summarize data by grouping it into intervals called classes. It displays how often observations from a sample fall into each class. To create a frequency distribution, one first determines the class intervals and then counts the frequency of observations in each interval. This information can be presented in a table or graphically as a line graph, bar graph, pie chart, or other visualizations. Frequency distributions provide a simplified view of the overall patterns in large data sets.
This document defines mean or average deviation and provides information about calculating it. Mean deviation is the average of the absolute differences between each value in a data set and the mean. It is easier to understand and calculate than other measures of variation. Mean deviation is less affected by extreme observations than other measures. It is frequently used to analyze the distribution of personal wealth by considering both the rich and poor. The document outlines the steps to calculate mean deviation for individual, discrete, and continuous data series.
This document provides information about various measures of central tendency including arithmetic mean, median, mode, and quartiles. It defines each measure and provides formulas and examples for calculating them for different types of data series, including individual, discrete, frequency distribution, and cumulative frequency series. Formulas are given for calculating the arithmetic mean, median, quartiles, and mode of a data set, along with examples worked out step-by-step. Advantages and disadvantages of each measure are also discussed.
This document discusses measures of central tendency, specifically the arithmetic mean. It provides examples and step-by-step solutions for calculating the arithmetic mean of individual data sets, discrete series with frequencies, and continuous series grouped into class intervals. For continuous series, the formula uses the mid-point of each class interval. The document also includes one problem that requires solving for a missing frequency given the calculated arithmetic mean.
Measures of dispersion
Absolute measure, relative measures
Range of Coe. of Range
Mean deviation and coe. of mean deviation
Quartile deviation IQR, coefficient of QD
Standard deviation and coefficient of variation
The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population.
The document defines and provides examples for calculating the coefficient of variation, which is a measure used to compare the dispersion of data sets. It gives the formula for coefficient of variation as the standard deviation divided by the mean, expressed as a percentage. Two examples are shown comparing the stability of prices between two cities and production between two manufacturing plants, with the data set having the lower coefficient of variation considered more consistent or stable.
The mean deviation is a measure of how spread out values are from the average. It is calculated by:
1) Finding the mean of all values.
2) Calculating the distance between each value and the mean.
3) Taking the average of those distances. This provides the mean deviation, which tells us how far on average values are from the central mean. Examples show calculating mean deviation for both grouped and ungrouped data sets.
Frequency distribution is a method to organize and summarize data by grouping it into intervals called classes. It displays how often observations from a sample fall into each class. To create a frequency distribution, one first determines the class intervals and then counts the frequency of observations in each interval. This information can be presented in a table or graphically as a line graph, bar graph, pie chart, or other visualizations. Frequency distributions provide a simplified view of the overall patterns in large data sets.
This document defines mean or average deviation and provides information about calculating it. Mean deviation is the average of the absolute differences between each value in a data set and the mean. It is easier to understand and calculate than other measures of variation. Mean deviation is less affected by extreme observations than other measures. It is frequently used to analyze the distribution of personal wealth by considering both the rich and poor. The document outlines the steps to calculate mean deviation for individual, discrete, and continuous data series.
This document provides information about various measures of central tendency including arithmetic mean, median, mode, and quartiles. It defines each measure and provides formulas and examples for calculating them for different types of data series, including individual, discrete, frequency distribution, and cumulative frequency series. Formulas are given for calculating the arithmetic mean, median, quartiles, and mode of a data set, along with examples worked out step-by-step. Advantages and disadvantages of each measure are also discussed.
This document discusses skewness and kurtosis in a financial context. It defines skewness as a measure of asymmetry in a distribution, with positive skewness indicating a long right tail and negative skewness a long left tail. Kurtosis is defined as a measure of the "peakedness" of a probability distribution, with positive excess kurtosis indicating flatness/long fat tails and negative excess kurtosis indicating peakedness. Formulas are provided for calculating skewness and kurtosis from a data set. Examples of positively and negatively skewed distributions are given to illustrate these concepts.
Frequency distribution, types of frequency distribution.
Ungrouped frequency distribution
Grouped frequency distribution
Cumulative frequency distribution
Relative frequency distribution
Relative cumulative frequency distribution
Graphical representation of frequency distribution
I. Representation of Grouped data
1.Line graphs
2.Bar diagrams
a) Simple bar diagram
b)Multiple/Grouped bar diagram
c)Sub-divided bar diagram.
d) % bar diagram
3. Pie charts
4.Pictogram
II. Graphical representation of ungrouped data
1, Histogram
2.Frequency polygon
3.Cumulative change diagram
4. Proportional change diagram
5. Ratio diagram
It is most useful for the students of BBA for the subject of "Data Analysis and Modeling"/
It has covered the content of chapter- Data regression Model
Visit for more on www.ramkumarshah.com.np/
Multiple regression analysis allows researchers to examine the relationship between one dependent or outcome variable and two or more independent or predictor variables. It extends simple linear regression to model more complex relationships. Stepwise regression is a technique that automates the process of building regression models by sequentially adding or removing variables based on statistical criteria. It begins with no variables in the model and adds variables one at a time based on their contribution to the model until none improve it significantly.
This document provides information about various statistical measures of central tendency including the median, mode, and quartiles. It defines each measure and provides examples of how to calculate them from both grouped and ungrouped data sets. Formulas are given for calculating the median, quartiles, deciles, and percentiles for grouped data. The mode is defined as the value that occurs most frequently in a data set, and a formula is provided for calculating it from grouped frequency distributions.
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
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.
One of the three points that divide a data set into four equal parts. Or the values that divide data into quarters. Each group contains equal number of observations or data. Median acts as base for calculation of quartile.
This document discusses measures of dispersion in statistics. It defines dispersion as the extent of variation in a data set from the average value. There are two main types of dispersion - absolute and relative. Absolute measures express variation in units of the data and include range, variance, standard deviation, and quartile deviation. Relative measures allow comparison between data sets by being unit-free, such as the coefficient of variation. Key absolute measures are then explained in more detail, along with their merits and demerits.
Correlation and regression analysis are statistical tools used to analyze relationships between variables. Correlation measures the strength and direction of association between two variables on a scale from -1 to 1. Regression analysis uses one variable to predict the value of another variable and draws a best-fit line to represent their relationship. There are always two lines of regression - one showing the regression of x on y and the other showing the regression of y on x. Regression coefficients from these lines indicate the slope and intercept of the lines and can help estimate unknown variable values based on known values.
This document discusses skewness and kurtosis, which are statistical measures of the distribution of a variable. Skewness measures the asymmetry of a distribution and can be positive, negative, or zero. Kurtosis measures the peakedness of a distribution and can be platykurtic (flatter than normal), mesokurtic (normal), or leptokurtic (more peaked than normal). The document provides formulas for calculating skewness using Pearson's, Bowley's, and Kelly's coefficients as well as calculating kurtosis using the fourth standardized moment. Examples of applying skewness and kurtosis to determine if a variable's distribution or resource use is normal are also discussed.
Standard deviation measures how dispersed data values are from the average. It is the most reliable measure of dispersion and shows the average distance of each data point from the mean. While it is more difficult to calculate than other measures, standard deviation provides important information about how concentrated or spread out the data is. The presentation defines standard deviation, lists its merits and demerits, and shows how to calculate it for both populations and samples.
Introduction to statistics...ppt rahulRahul Dhaker
This document provides an introduction to statistics and biostatistics. It discusses key concepts including:
- The definitions and origins of statistics and biostatistics. Biostatistics applies statistical methods to biological and medical data.
- The four main scales of measurement: nominal, ordinal, interval, and ratio scales. Nominal scales classify data into categories while ratio scales allow for comparisons of magnitudes and ratios.
- Descriptive statistics which organize and summarize data through methods like frequency distributions, measures of central tendency, and graphs. Frequency distributions condense data into tables and charts. Measures of central tendency include the mean, median, and mode.
The sign test is a nonparametric test that uses the signs (positive or negative) of deviations from a measure of central tendency, rather than the magnitudes of the deviations. There are one-sample and paired-sample versions. For the one-sample sign test, the null hypothesis is that the probability of a positive sign is 0.5. Signs are counted and compared to a critical value to determine if the null can be rejected. The document then provides examples of applying the one-sample and paired-sample sign tests to various data sets involving numbers of late workers, golf scores, and accounts receivable.
The document discusses coefficient of variation (CV), which is the ratio of the standard deviation to the mean. It provides an example comparing the CV of two multiple choice tests with different conditions. Formulas for calculating CV by hand and in Excel are shown. Methods for finding quartiles in ungrouped and grouped data are explained. The document also demonstrates how to calculate quartile deviation and construct box and whisker plots, and provides references for further information.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
This document discusses measures of central tendency including the mean, median, and mode. It provides examples and definitions for each measure. The mean is the average and is calculated by summing all values and dividing by the total number. The median is the middle value when values are ranked in order. The mode is the most frequent value. The best measure depends on the scale of measurement and shape of the distribution, such as whether it is symmetrical or skewed.
This document defines variance and standard deviation and provides formulas and examples to calculate them. It states that variance is the average squared deviation from the mean and measures how far data points are from the average. Standard deviation tells how clustered data is around the mean and is the square root of the variance. It provides step-by-step instructions to find variance and standard deviation, including calculating the mean, deviations from the mean, summing the squared deviations, and taking the square root. Worked examples are shown to find the variance and standard deviation of students' test scores and people's heights in a room.
The mode is defined as the score that occurs most frequently in a data set. It is a measure of central tendency that indicates the most common value. For an ungrouped data set, the mode is simply the value that repeats most. For a grouped data set, the mode is calculated using a formula that finds the midpoint of the interval with the highest frequency while accounting for the differences in frequencies on either side of that interval. The mode is useful when wanting a quick approximation of central tendency or when trying to find the most typical value in a data set. Data sets can have one, two, or more modes depending on the number of most frequent values.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
The document provides information on measures of central tendency. It discusses five main measures - arithmetic mean, geometric mean, harmonic mean, mode, and median. For arithmetic mean, it provides formulas and examples for calculating the mean from ungrouped and grouped data using both the direct and assumed mean methods. It also discusses the merits and demerits of each measure.
This document discusses the calculation of arithmetic mean from data in various formats. It defines arithmetic mean as the sum of all values divided by the number of observations. It then provides examples of calculating the arithmetic mean using direct and shortcut methods for individual series, discrete series with frequencies, continuous series with class intervals, and series with open-ended classes. The different methods are demonstrated using example data sets. References on biostatistics are also included.
This document discusses skewness and kurtosis in a financial context. It defines skewness as a measure of asymmetry in a distribution, with positive skewness indicating a long right tail and negative skewness a long left tail. Kurtosis is defined as a measure of the "peakedness" of a probability distribution, with positive excess kurtosis indicating flatness/long fat tails and negative excess kurtosis indicating peakedness. Formulas are provided for calculating skewness and kurtosis from a data set. Examples of positively and negatively skewed distributions are given to illustrate these concepts.
Frequency distribution, types of frequency distribution.
Ungrouped frequency distribution
Grouped frequency distribution
Cumulative frequency distribution
Relative frequency distribution
Relative cumulative frequency distribution
Graphical representation of frequency distribution
I. Representation of Grouped data
1.Line graphs
2.Bar diagrams
a) Simple bar diagram
b)Multiple/Grouped bar diagram
c)Sub-divided bar diagram.
d) % bar diagram
3. Pie charts
4.Pictogram
II. Graphical representation of ungrouped data
1, Histogram
2.Frequency polygon
3.Cumulative change diagram
4. Proportional change diagram
5. Ratio diagram
It is most useful for the students of BBA for the subject of "Data Analysis and Modeling"/
It has covered the content of chapter- Data regression Model
Visit for more on www.ramkumarshah.com.np/
Multiple regression analysis allows researchers to examine the relationship between one dependent or outcome variable and two or more independent or predictor variables. It extends simple linear regression to model more complex relationships. Stepwise regression is a technique that automates the process of building regression models by sequentially adding or removing variables based on statistical criteria. It begins with no variables in the model and adds variables one at a time based on their contribution to the model until none improve it significantly.
This document provides information about various statistical measures of central tendency including the median, mode, and quartiles. It defines each measure and provides examples of how to calculate them from both grouped and ungrouped data sets. Formulas are given for calculating the median, quartiles, deciles, and percentiles for grouped data. The mode is defined as the value that occurs most frequently in a data set, and a formula is provided for calculating it from grouped frequency distributions.
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
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.
One of the three points that divide a data set into four equal parts. Or the values that divide data into quarters. Each group contains equal number of observations or data. Median acts as base for calculation of quartile.
This document discusses measures of dispersion in statistics. It defines dispersion as the extent of variation in a data set from the average value. There are two main types of dispersion - absolute and relative. Absolute measures express variation in units of the data and include range, variance, standard deviation, and quartile deviation. Relative measures allow comparison between data sets by being unit-free, such as the coefficient of variation. Key absolute measures are then explained in more detail, along with their merits and demerits.
Correlation and regression analysis are statistical tools used to analyze relationships between variables. Correlation measures the strength and direction of association between two variables on a scale from -1 to 1. Regression analysis uses one variable to predict the value of another variable and draws a best-fit line to represent their relationship. There are always two lines of regression - one showing the regression of x on y and the other showing the regression of y on x. Regression coefficients from these lines indicate the slope and intercept of the lines and can help estimate unknown variable values based on known values.
This document discusses skewness and kurtosis, which are statistical measures of the distribution of a variable. Skewness measures the asymmetry of a distribution and can be positive, negative, or zero. Kurtosis measures the peakedness of a distribution and can be platykurtic (flatter than normal), mesokurtic (normal), or leptokurtic (more peaked than normal). The document provides formulas for calculating skewness using Pearson's, Bowley's, and Kelly's coefficients as well as calculating kurtosis using the fourth standardized moment. Examples of applying skewness and kurtosis to determine if a variable's distribution or resource use is normal are also discussed.
Standard deviation measures how dispersed data values are from the average. It is the most reliable measure of dispersion and shows the average distance of each data point from the mean. While it is more difficult to calculate than other measures, standard deviation provides important information about how concentrated or spread out the data is. The presentation defines standard deviation, lists its merits and demerits, and shows how to calculate it for both populations and samples.
Introduction to statistics...ppt rahulRahul Dhaker
This document provides an introduction to statistics and biostatistics. It discusses key concepts including:
- The definitions and origins of statistics and biostatistics. Biostatistics applies statistical methods to biological and medical data.
- The four main scales of measurement: nominal, ordinal, interval, and ratio scales. Nominal scales classify data into categories while ratio scales allow for comparisons of magnitudes and ratios.
- Descriptive statistics which organize and summarize data through methods like frequency distributions, measures of central tendency, and graphs. Frequency distributions condense data into tables and charts. Measures of central tendency include the mean, median, and mode.
The sign test is a nonparametric test that uses the signs (positive or negative) of deviations from a measure of central tendency, rather than the magnitudes of the deviations. There are one-sample and paired-sample versions. For the one-sample sign test, the null hypothesis is that the probability of a positive sign is 0.5. Signs are counted and compared to a critical value to determine if the null can be rejected. The document then provides examples of applying the one-sample and paired-sample sign tests to various data sets involving numbers of late workers, golf scores, and accounts receivable.
The document discusses coefficient of variation (CV), which is the ratio of the standard deviation to the mean. It provides an example comparing the CV of two multiple choice tests with different conditions. Formulas for calculating CV by hand and in Excel are shown. Methods for finding quartiles in ungrouped and grouped data are explained. The document also demonstrates how to calculate quartile deviation and construct box and whisker plots, and provides references for further information.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
This document discusses measures of central tendency including the mean, median, and mode. It provides examples and definitions for each measure. The mean is the average and is calculated by summing all values and dividing by the total number. The median is the middle value when values are ranked in order. The mode is the most frequent value. The best measure depends on the scale of measurement and shape of the distribution, such as whether it is symmetrical or skewed.
This document defines variance and standard deviation and provides formulas and examples to calculate them. It states that variance is the average squared deviation from the mean and measures how far data points are from the average. Standard deviation tells how clustered data is around the mean and is the square root of the variance. It provides step-by-step instructions to find variance and standard deviation, including calculating the mean, deviations from the mean, summing the squared deviations, and taking the square root. Worked examples are shown to find the variance and standard deviation of students' test scores and people's heights in a room.
The mode is defined as the score that occurs most frequently in a data set. It is a measure of central tendency that indicates the most common value. For an ungrouped data set, the mode is simply the value that repeats most. For a grouped data set, the mode is calculated using a formula that finds the midpoint of the interval with the highest frequency while accounting for the differences in frequencies on either side of that interval. The mode is useful when wanting a quick approximation of central tendency or when trying to find the most typical value in a data set. Data sets can have one, two, or more modes depending on the number of most frequent values.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
The document provides information on measures of central tendency. It discusses five main measures - arithmetic mean, geometric mean, harmonic mean, mode, and median. For arithmetic mean, it provides formulas and examples for calculating the mean from ungrouped and grouped data using both the direct and assumed mean methods. It also discusses the merits and demerits of each measure.
This document discusses the calculation of arithmetic mean from data in various formats. It defines arithmetic mean as the sum of all values divided by the number of observations. It then provides examples of calculating the arithmetic mean using direct and shortcut methods for individual series, discrete series with frequencies, continuous series with class intervals, and series with open-ended classes. The different methods are demonstrated using example data sets. References on biostatistics are also included.
1. The document discusses different types of means or averages, including arithmetic mean, geometric mean, and harmonic mean.
2. It provides definitions and formulas for calculating simple arithmetic mean, combined arithmetic mean, and arithmetic mean of grouped data using both direct and shortcut methods.
3. Examples are given to demonstrate calculating the arithmetic mean from both ungrouped and grouped data using the frequency distribution method and the assumed mean method.
Dispersion- It is a statistical term that describes the size of the distribution of values expected for a particular variable and can be measured by several different statistics, such as Range, Variance and standard deviation.
Method of Dispersion-A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution.
Methods of Dispersion.
1.Relative Dispersion
a. Coefficient of Mean Deviation
b. Coefficient of Quartile Deviation
c. Coefficient of Range
d. Coefficient of Variation
2. Absolute Dispersion
a. Range
b. Quartile range
c. Standard deviation
d. Mean Deviation
Range- It is the difference between smallest & largest values in the dataset. Also the relative measure of range is known as Coefficient of Range.
Advantages and disadvantages of Range.
Calculation of Range by different Methods.
b. Quartile Range- The interquartile range of a group of observations is the interval between the values of upper quartile and the lower quartile for that group.
Advantages and Disadvantages of Quartile Range.
Calculation of Quartile Range by different Methods.
c. Standard Deviation- It measures the absolute dispersion (or) variability of a distribution. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity in the series.
Advantages and Disadvantages of Quartile Range.
Calculation of Standard Deviation using.
i) Direct Method
ii) Short-cut Method
iii) Step Deviation Method.
This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is defined as the sum of all values divided by the total number of observations. It can be calculated using direct or shortcut methods for discrete, continuous, and weighted data. The geometric and harmonic means are also defined. The median is the middle value when data is arranged in order. Formulas are provided for calculating the median from discrete and continuous distributions. Examples are included to demonstrate calculating each measure of central tendency.
Median
Middle value in a distribution is known as Median.
Calculation of median.
1. Calculation of median in a series of individual observations or Calculation of median for ungrouped data
2. Calculation of median for grouped data
a) Calculation of median in a discrete series.
b) Calculation of median in a continuous series.
c) Calculation of median in unequal class intervals.
d) Calculation of median in open-end classes.
Merits and Demerits of Median.
This document discusses measures of central tendency and variation for numerical data. It defines and provides formulas for the mean, median, mode, range, variance, standard deviation, and coefficient of variation. Quartiles and interquartile range are introduced as measures of spread less influenced by outliers. The relationship between these measures and the shape of a distribution are also covered at a high level.
Measure of central tendency provides a very convenient way of describing a set of scores with a single number that describes the PERFORMANCE of the group.
It is also defined as a single value that is used to describe the “center” of the data.
This document discusses measures of central tendency and different methods for calculating averages. It begins by defining central tendency as a single value that represents the characteristics of an entire data set. Three common measures of central tendency are introduced: the mean, median, and mode. The document then focuses on explaining how to calculate the arithmetic mean, or average, including the direct method, shortcut method, and how it applies to discrete and continuous data series. Weighted averages are also covered. In summary, the document provides an overview of key concepts in measures of central tendency and how to calculate various types of averages.
Arithmetic mean of a series is obtained by dividing total of values in series by the number of items i.e.
Arithmetic mean is denoted by .
Determination of Arithmetic Mean for Ungrouped Data
Let be the “n” values of variable , then the arithmetic mean can be obtained by using the following formula:
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Chapter 12: Analysis of Variance
12.1: One-Way ANOVA
BIOSTATISTICS MEAN MEDIAN MODE SEMESTER 8 AND M PHARMACY BIOSTATISTICS.pptxPayaamvohra1
1. The document provides information about biostatistics including measures of central tendency, dispersion, correlation, and regression. It defines terms like mean, median, mode, range, and standard deviation.
2. Examples of calculating mean, median, and mode from individual data sets, grouped frequency distributions, and continuous series are shown step-by-step.
3. Parametric tests like t-test, ANOVA, and tests of significance are also introduced. Overall, the document covers fundamental concepts in biostatistics through examples.
1. The document summarizes key concepts from a lecture on statistics for engineers, including the normal distribution, the central limit theorem, and normal approximations to the binomial and Poisson distributions.
2. It provides an example of using the normal approximation to the Poisson distribution to calculate how many pills should be ordered to ensure the probability of running out is less than 0.005.
3. The document cautions that normal approximations may provide inaccurate results if assumptions like independence are violated, as with infectious diseases. Simple approximations are not advisable if failure could have important consequences, as with estimating rare event probabilities.
This document introduces common measures of central tendency (mean, median, mode) and variation (range, variance, standard deviation, coefficient of variation) in biostatistics. It defines each measure and provides examples of calculating and interpreting them. The mean is the most common measure of central tendency but the median is more robust to outliers. The choice of central tendency measure depends on whether the data is skewed. The mode is used for measuring popularity. Measures of variation quantify how spread out the data values are.
C2 st lecture 13 revision for test b handoutfatima d
This document provides an outline for a lecture series revising key concepts for Test B, including:
- Pythagoras' theorem, trigonometry, sine and cosine rules, and calculating triangle areas.
- Probability, probability trees, and examples calculating probabilities of dice rolls.
- Descriptive statistics like mode, median, interquartile range, mean, absolute deviation, and standard deviation.
- Hypothesis testing using z-tests, t-tests, and chi-squared tests; including setting up hypotheses, finding critical values, calculating test statistics, and making conclusions.
The revision is in preparation for Standard Track Test B which will be held the week of April 21st.
This document discusses measures of central tendency, specifically the arithmetic mean. It provides formulas and examples for calculating the arithmetic mean using direct, short-cut, and step-deviation methods for both ungrouped and grouped data. It also discusses calculating the weighted mean and combined mean of two or more related groups. Key characteristics of the arithmetic mean are that the sum of deviations from the mean is zero and the sum of squared deviations is minimum.
The document discusses various methods for describing data distributions numerically, including measures of center (mean, median), measures of spread (standard deviation, interquartile range), and graphical representations (boxplots). It explains how to calculate and interpret the mean, median, quartiles, five-number summary, standard deviation, and identifies outliers. Choosing an appropriate measure of center and spread depends on the symmetry of the distribution and presence of outliers. Changing the measurement units affects the calculated values but not the underlying shape of the distribution.
This document provides an introduction to research methodology concepts including population, sample, sampling methods, hypothesis testing, and types of errors. It defines key terms like population, sample, probability and non-probability sampling, null and alternative hypotheses. It explains probability sampling methods like simple random sampling, stratified sampling and cluster sampling. It also summarizes non-probability methods like convenience and purposive sampling. The document concludes by describing type I and type II errors and their relationship to hypothesis testing.
Microsoft Excel is a spreadsheet program used to record and analyse numerical and statistical data. Microsoft Excel provides multiple features to perform various operations like calculations, pivot tables, graph tools, macro programming, etc.
An Excel spreadsheet can be understood as a collection of columns and rows that form a table. Alphabetical letters are usually assigned to columns, and numbers are usually assigned to rows. The point where a column and a row meet is called a cell.
SPSS (Statistical Package for the Social Sciences) is a versatile and responsive program designed to undertake a range of statistical procedures. SPSS software is widely used in a range of disciplines and is available from all computer pools within the University of South Australia.
DOE is an essential tool to ensure products and processes satisfy Quality by Design requirements imposed by regulatory agencies. Using a QbD approach to develop your testing process can help you reduce waste, meet compliance criteria and get to market faster.
DOE helps you create a reliable QbD process for assessing formula robustness, determining critical quality attributes and predicting shelf life by using a few months of historical data.
Minitab is a statistics package developed at the Pennsylvania State University by researchers Barbara F. Ryan, Thomas A. Ryan, Jr., and Brian L. Joiner in conjunction with Triola Statistics Company in 1972.
It began as a light version of OMNITAB 80, a statistical analysis program by NIST, which was conceived by Joseph Hilsenrath in years 1962-1964 as OMNITAB program for IBM 7090. The documentation for OMNITAB 80 was last published 1986, and there has been no significant development since then.
R is a language and environment for statistical computing and graphics."
"R provides a wide variety of statistical (linear and nonlinear modelling, classical statistical tests, time-series analysis, classification, clustering) and graphical techniques, and is highly extensible."
"One of R's strengths is the ease with which well-designed publication-quality plots can be produced, including mathematical symbols and formulae where needed.“
SAMPLE SIZE DETERMINATION
Sample size determination is the essential step of research methodology. It is an act of choosing the number of observers or replicates to include in a statistical sample.
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample.
Precision
A measure of how close an estimate is to the true value of a population parameter. Or it can be thought of as the amount of fluctuation from the population parameter that we can expect by chance alone in sample estimates.
Degree of Precision
This is presented in the form of a confidence interval (Range of values within which confidence lies).
RESEARCH REPORT
A research report is considered a major component of any research study as the research remains incomplete till the report has been presented or written. No matter how good a research study, and how meticulously the research study has been conducted, the findings of the research are of little value unless they are effectively documented and communicated to others.
TYPES OF RESEARCH REPORT
The research report is classified based on 2 things; Nature of research and Target audience.
COHORT STUDIES
A research study that compares a particular outcome in groups of individuals who are alike in many ways but differ by a certain characteristic is called as Cohort study.
Cohort studies are a type of research design that follow groups of people over time. Researchers use data from cohort studies to understand human health and the environmental and social factors that influence it.
CLINICAL TRIALS
A clinical trial, also known as a clinical research study, is a protocol to evaluate the effects and efficacy of experimental medical treatments or behavioral interventions on health outcomes. This type of study gathers data from volunteer human subjects and is typically funded by a medical institution, university or nonprofit group, or by pharmaceutical companies and government agencies.
Clinical trial vs. clinical study
A clinical study is research conducted with the intent of gaining medical knowledge. Observational and interventional are the two main types of clinical studies. A clinical trial is an interventional study.
DESIGN OF EXPERIMENTS (DOE)
DOE is invented by Sir Ronald Fisher in 1920’s and 1930’s.
The following designs of experiments will be usually followed:
Completely randomised design(CRD)
Randomised complete block design(RCBD)
Latin square design(LSD)
Factorial design or experiment
Confounding
Split and strip plot design
FACTORIAL DESIGN
When a several factors are investigated simultaneously in a single experiment such experiments are known as factorial experiments. Though it is not an experimental design, indeed any of the designs may be used for factorial experiments.
For example, the yield of a product depends on the particular type of synthetic substance used and also on the type of chemical used.
ADVANTAGES OF FACTORIAL DESIGN.
Factorial experiments are advantageous to study the combined effect of two or more factors simultaneously and analyze their interrelationships. Such factorial experiments are economic in nature and provide a lot of relevant information about the phenomenon under study. It also increases the efficiency of the experiment.
It is an advantageous because a wide range of factor combination are used. This will give us an idea to predict about what will happen when two or more factors are used in combination.
DISADVANTAGES
It is disadvantageous because the execution of the experiment and the statistical analysis becomes more complex when several treatments combinations or factors are involved simultaneously.
It is also disadvantageous in cases where may not be interested in certain treatment combinations but we are forced to include them in the experiment. This will lead to wastage of time and also the experimental material.
2(square) FACTORIAL EXPERIMENT
A special set of factorial experiment consist of experiments in which all factors have 2 levels such experiments are referred to generally as 2n factorials.
If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3 or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn factorial experiment.
The calculation of the sum of squares is as follows:
Correction factor (CF) = (𝐺𝑇)2/𝑛
GT = grand total
n = total no of observations
Total sum of squares = ∑▒〖𝑥2−𝐶𝐹〗
Replication sum of squares (RSS) = ((𝑅1)2+(𝑅2)2+…+(𝑅𝑛)2)/𝑛 - CF
Or
1/𝑛 ∑▒𝑅2−𝐶𝐹
2(Cube) FACTORIAL DESIGN
In this type of design, one independent variable has 2 levels, and the other independent variable has 3 levels.
Estimating the effect:
In a factorial design the main effect of an independent variable is its overall effect averaged across all other independent variable.
Effect of a factor A is the average of the runs where A is at the high level minus the average of the runs
NEED FOR RESEARCH
Research is a systemic process of collecting and analyzing information to increase the understanding of the phenomenon under study.
It strengthens pharmacist-provided services, builds the evidence base for developing and commissioning new services, improves patient care and contributes to health service knowledge.
Phase I studies: Are done on healthy volunteers who agree to take the study drug to help the doctors determine how safe the drug is and if there are any side effects. Usually a small number of subjects (20-100) participate in Phase I studies. Approximately 70% of new drugs will pass this phase.
Phase II studies: Measure the effect of the new drug in patients with the disease or disorder to be treated. The main purpose is to determine safety and effectiveness of the new drug. Usually several hundred patients participate. These studies are usually “Double-blinded, randomized and controlled”.
Phase III studies: also use patients with the disorder to be treated by the new drug. These studies are done to gain a more thorough understanding of the effectiveness, benefits and side effects of the study drug.
NEED FOR DESIGN OF EXPERIMENTS
Design of experiments (DOE) is defined as a branch of applied statistics that deals with planning, conducting, analyzing, and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters.
DOE is a powerful data collection and analysis tool that can be used in a variety of experimental situations.
1. PRE-EXPERIMENTAL DESIGN
In pre-experimental research design, either a group or various dependent groups are observed for the effect of the application of an independent variable which is presumed to cause change.
It is the simplest form of experimental research design and is treated with no control group
2. TRUE EXPERIMENTAL DESIGN
The true experimental research design relies on statistical analysis to approve or disprove a hypothesis. It is the most accurate type of experimental design and may be carried out with or without a pretest on at least 2 randomly assigned dependent subjects.
The true experimental research design must contain a control group, a variable that can be manipulated by the researcher, and the distribution must be random.
3. QUASI EXPERIMENTAL DESIGN
The word "quasi" means partial, half, or pseudo. Therefore, the quasi-experimental research bearing a resemblance to the true experimental research, but not the same. In quasi-experiments, the participants are not randomly assigned, and as such, they are used in settings where randomization is difficult or impossible.
This is very common in educational research, where administrators are unwilling to allow the random selection of students for experimental samples.
PLAGIARISM
The word Plagiarism is derived from the Latin word Plagiarius, which means abducting, kidnapping, seducing, or plundering.
Non Parametric Test
1. Wilcoxon Signed Rank Test: (WSRT)
In this test the difference in positive and negative value is taken into consideration without assigning any weightage to the magnitude of the differences as a result, the sign test is often used in practice.
The Wilcoxon Sign Rank test can be used to overcome this limitation.
2. Wilcoxon Rank Sum test: (WRST)
This is also called as Mann- Whitney U test.
WRST is used to compare two independent sample while WSRT compare two related or two dependent samples.
This test is applicable if the data are at least ordinal {i.e. the observation can be ordered}
3. MANN-WHITNEY U-TEST
It is a non-parametric method used to determine whether two independent samples have been drawn from populations with same distribution. This test is also known as U-Test.
This test enables us to test the null hypothesis that both population medians are equal(or that the two samples are drawn from a single population).
4. KRUSKAL WALLIS TEST
This test is employed when more then 2 population are involved where as Man Whitney test is used when there are 2 populations. The use of this test will enable us to determine weather independent samples have been drawn from the sample population (or) different populations have the same distribution.
5. FRIEDMAN TEST
It is a non-parametric test applied to a data i.e. at least ranked and it is in the form of a 2 way ANOVA design. This test which may be applied to ranked or Interval or Ratio type of data is used when more than 2 treatment, group are included in the experiment.
Correlation- If two variables are so inter-related in such a manner that change in one variable brings about change in the other variable, then this type of relation of variable is known as correlation.
Types of Correlation.
1.Based on the direction of change of variables
a. Positive
correlation
b. Negative
correlation
2. Based upon the number of variables studied
a. Simple
correlation
b. Partial correlation
c. Multiple correlation
3. Based upon the constancy of the ratio of change between the variables
a. Linear correlation
b. Non-linear correlation
METHODS OF STUDYING CORRELATION
1) GRAPHIC
METHODS
A) SCATTER DIAGRAM
B) CORRELATION
GRAPH
2). ALGEBRIC METHOD
A) KARL PEARSON COEFFICIENT OF CORRELATION
B) RANK CORRELATION METHOD
C) CONCURRENT DEVIATION METHOD
Uses of Correlation.
Merits of Correlation.
Demerits of Correlation.
Introduction to Mode.
Calculation of modes by different methods.
Merits and Demerits of Mode.
Mode is the value which occurs the maximum number of times in a series of observations and has the highest frequency.
Calculation of Mode
1. Calculation of mode in a series of individual observations (Ungrouped data)
2. Calculation of mode in a discrete series (Grouped data)
3. Calculation of mode in a continuous series (Grouped data)
4. Calculation of mode in a unequal class intervals (Grouped data)
Introduction to biostatistics and its application in various sectors.
Introduction to variables and variation.
Different types of variables and their introduction.
Use of biostatistics in various fields.
I am Mrs. G. Sreelatha, Assistant Professor, CMR College Of Pharmacy, Hyderabad.
I will be uploading notes on Biostatistics And Research Methodology (BRM) of B.Pharmacy, 4th year II sem based on PCI syllabus - JNTUH.
Topic included in this PPT are Origin and History of Statistics.
Hope it will be useful for your studies and will clear your all the doubts.
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.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Main Java[All of the Base Concepts}.docxadhitya5119
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
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How to Fix the Import Error in the Odoo 17Celine George
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Digital Artefact 1 - Tiny Home Environmental Design
MEAN.pptx
1. BIOSTATISTICS AND RESEARCH METHODOLOGY
Unit-1: mean
PRESENTED BY
Himanshu Rasyara
B. Pharmacy IV Year
UNDER THE GUIDANCE OF
Gangu Sreelatha M.Pharm., (Ph.D)
Assistant Professor
CMR College of Pharmacy, Hyderabad.
email: sreelatha1801@gmail.com
2. MEAN
• Mean is an essential concept in mathematics and statistics. The mean is the average or the most
common value in a collection of numbers.
• In statistics, it is a measure of central tendency of a probability distribution along median and
mode.
Mean
Arithmetic
Mean
Simple
Arithmetic
Mean
Weighted
Arithmetic
Mean
Geometric
Mean
Harmonic
Mean
3. A) ARITHMETIC MEAN
• The arithmetic mean is the sum of all observations divided by the total number of observations is known
as simple Arithmetic mean.
1. Calculation of Simple Arithmetic Mean
• Simple Arithmetic Mean can be calculated by 2 methods:
a) Direct Method
b) Shortcut Method
I. Calculation of simple arithmetic mean in a series of individual observations.
a) Direct Method: x
̄ =
𝐱𝟏+𝐱𝟐+𝐱𝟑−−−−−−−−−− +𝐱𝐧
𝐧
𝐨𝐫 x̄=Σx/n
x
̄ = Arithmetic Mean
Σx= Sum of all values of the variable, x
n= Number of observations.
• Example: Calculate the mean incubation period of 12 polio cases from the data given below.
16,20,21,19,17,24,23,20,22,18,18,22
Solution: x̄=Σx/n
where Σx= 240 ; n =12
x
̄ = 240/12= 20 So, the mean incubation period of polio cases was found to be 20.
4. b) Shortcut Method: x
̄ = A+
Σ𝒅
𝐧
where
x
̄ = Arithmetic Mean
A= assumed mean
d= deviation of items from the assumed mean (x-A)
Σd= sum of deviations.
Example: The following example shows the calculation of arithmetic mean from the observations.
Number of mean= 18,20,21,19,28,22,29,30
Solution: Assumed mean=24
x
̄ = A+
Σ𝒅
𝐧
A= 24; 𝛴d = −5; n = 8
x
̄ = 24+
−5
8
= 24-0.625=23.37
Number of Observations
x
Deviations from assumed mean
d= (x-A)
18 -6
20 -5
21 -4
19 -3
28 -2
22 4
29 5
30 6
n= 8 Σ𝑑 = −20 + 15 = −5
5. II. Calculation of Simple Arithmetic Mean in a Discrete Series.
a) Direct Method :x̄ =
𝜮𝐟𝐱
𝒏
or x
̄ =
𝜮𝐟𝐱
𝜮𝒇
Where
x
̄ = arithmetic mean
n= number of observations
Σf= sum of frequencies
Σfx= sum of values of the variable and their corresponding frequencies.
Example: Find the arithmetic mean of a sample of
reported cases of mumps in school children
of the following data.
Solution: n=Σf= 30 and Σfx= 1848
Mean = x
̄ =
𝜮𝐟𝐱
𝜮𝒇
=𝟏𝟖𝟒𝟖
𝟑𝟎
= 𝟔𝟏. 𝟔
Blood
LDL (x)
Frequenc
y (f)
f *x
52 7 364
58 5 290
60 4 240
65 6 390
68 3 204
70 3 210
75 2 150
Total n/Σf=30 Σfx=1848
Blood LDL No. of
Patients
52 7
58 5
60 4
65 6
68 3
70 3
75 2
6. b) Shortcut Method:
x
̄ = 𝐀 +
𝚺𝐟𝐝
𝚺𝐟
where
x
̄ = Arithmetic mean
A= Assumed mean
Σfd= Sum of the deviations from the assumed mean and the respective frequencies.
Σf= Sum of frequencies
• Example: Ten patients were examined for uric acid test. The operation was performed 1050 times and
the frequencies thus obtained for different number of patients (x) are shown in the following table.
Calculate the arithmetic mean by short-cut method.
Solution:
• Let 5 be the assumed mean. i.e., a=5. Let us prepare the following table in order to calculate the
arithmetic mean.
x 0 1 2 3 4 5 6 7 8 9 10
f 2 8 43 133 207 260 213 120 54 9 1
8. III) Calculation of Simple arithmetic Mean in a Continuous series.
a) Direct Method: x
̄ =
𝚺𝐟𝐦
𝚺𝐟
Where x
̄ = arithmetic mean
Σfm = sum of values of mid − points mutiplied by the respective frequency of each class.
Σf= sum of frequency
m= mid-point of various classes.
Mid point (m)=
lower limit+upper limit
2
Example: Compute the arithmetic mean from the following data:
Age
Group
0-10 10-20 20-30 30-40 40-50 50-60
No. of
Patients
5 10 25 30 20 10
Age group No. of Patients (f) Mid – points (m) fm
0-10 5 5 25
10-20 10 15 150
20-30 25 25 625
30-40 30 35 1050
40-50 20 45 900
50-60 10 55 550
Σf= 100 Σfm= 3300
Solution:
x
̄ =
𝚺𝐟𝐦
𝚺𝐟
Σfm= 3300 ;Σf= 100
x
̄ =
𝟑𝟑𝟎𝟎
𝟏𝟎𝟎
= 33
9. b) Shortcut Method: x
̄ =A+
𝚺𝐟𝐝
𝚺𝐟
Where x
̄ = Arithmetic mean
A= Assumed mean
Σf = Sum of frequency
d= Deviation of mid-points from assumed mean(m-a)
f= Frequency of each other
Example: Compute the arithmetic mean from the following data:
Age
Group
0-10 10-20 20-30 30-40 40-50 50-60
No. of
Patients
5 10 25 30 20 10
Age group No. of Patients (f) Mid – points (m) fm
0-10 5 5 25
10-20 10 15 150
20-30 25 25 625
30-40 30 35 1050
40-50 20 45 900
50-60 10 55 550
Σf= 100 Σfm= 3300
Solution:
x
̄ =A+
𝚺𝐟𝐝
𝚺𝐟
A= 35; Σfd= -200; Σf= 100
x
̄ = 35+
(−200)
100
= 35+ (-2)= 33
10. c. Step Deviation Method:
x
̄ = 𝑨 +
𝚺𝐟d1
𝚺𝐟
x i – when class-interval are unequal
x
̄ = 𝐀 +
𝚺𝐟d1
𝚺𝐟
𝐱 𝐜 − when class − interval are unequal
Where x
̄ = assumed mean
d1 = deviations of mid points from assumed mean (m-A)/I or c
Σf= sum of frequencies
i= class- interval
c= common factor
d= deviations from assumed mean
• Example: Calculate by step deviation method the arithmetic mean of the blood test for triglyceride of
384 patients.
Triglyceri
des
5 10 15 20 25 30 35 40 45 50
No. of
Patients
20 43 75 67 72 45 39 9 8 6
11. x f Dx= x-a D’= d/i f x d’
5 20 -25 -5 -100
10 43 -15 -4 -172
15 75 -10 -3 -225
20 67 -5 -2 -134
25 72 0 -1 -72
30 45 5 0 0
35 39 10 1 39
40 9 10 2 18
45 8 15 3 24
50 6 20 4 20
Σdx= 384 Σfds’x’= -598
Solution:
Let a=20 and I= 5
x
̄ = a +
Σfd′x′
Σf
x i
30 −
598
384
x 5 = 30 – 1.56 x 5 = 30 – 7.80= 22.2
12. • Merits of Arithmetic Mean.
• It is easy to understand and easy to calculate.
• It is rigidly defined.
• It is based on all the observations.
• It provides a good basis for comparison.
• It is not affected by fluctuations of sampling.
• Demerits of Arithmetic Mean.
• The mean is unduly affected by the extreme items.
• It cannot be accurately determined even if one of the values is not known.
13. WEIGHTED ARITHMETIC MEAN
• It may be defined as the average whose component items are being multiplied by certain value known
as “Weights” and the aggregate of the multiplied results are being divided by the total sum of their
weights instead of the sum of the items.
1. Direct Method:
x̄w=
𝐱𝟏𝐰𝟏
+
𝐱𝟐𝐰𝟐
+
𝐱𝟑𝐰𝟑
+⋯.+
𝐱𝐧𝐰𝐧
𝐰𝟏
+
𝐰𝟐
+
𝐰𝟑
+⋯…….+
𝐰𝐧
=
𝚺𝐱𝐰
𝚺𝐰
where
x̄w= Weighted arithmetic mean
x1,x2,x3……….. xn= variables
w1,w2,w3…….. wn= weights
Σ𝑥𝑤= sum of the products of the values and their respective weights
Σ𝑤= sum of the weights
14. Example: Suppose that a pharmaceutical company conducts a clinical trial on1,000 patients to
determine the average number of disorders in each patients. The data show a large number of patients has
two or three disorders and a smaller number with one or four. Find the mean number of disorders per
patients.
Number of
disorders per
patients
No. of patients
1 73
2 378
3 459
4 90
Solution: As many of the values in this data set are repeated multiple
times, you can easily compute the sample mean as a weighted mean.
Follow these steps to calculate the weighted arithmetic mean:
Step 1: Assign a weight to each value in the dataset:
Step 2: Compute the numerator of the weighted mean formula.
Multiply each sample by its weight and then add the products together:
= (1)(73)+(2)(378)+(3)(459)+(4)(90)
= 73 + 756 + 1377 +360 =2566
Step 3: Now, compute the denominator of the weighted mean formula by
adding the weights together.
= 73 + 378 + 459 + 90 =1000
Step 4: Divide the numerator by the denominator
The mean number of TVs per household in this sample is 2.566.
15. 2. Shortcut (or) Indirect Method:
x̄w = 𝐀𝐰 +
𝚺𝐰. 𝐝𝐱
𝚺𝐰
Where
x̄w= Weighted arithmetic mean
Aw= Assumed weighted mean
dx= Deviation items from assumed mean
w= Weights of various items.
Example: Suppose that a pharmaceutical company conducts a clinical trial on1,000 patients to
determine the average number of disorders in each patients. The data show a large number of patients
has two or three disorders and a smaller number with one or four. Find the mean number of disorders
per patients.
Number of disorders per patients No. of patients
1 73
2 378
3 459
4 90
16. • Solution:
x̄w = Aw +
Σw. dx
Σw
x̄w= 378+
−11006
1000
= 378- 11.006
x̄w= 366.994 = 367
• Merits of Weighted mean.
i. It is simple to calculate
ii. It is easy to understand.
iii. It is rigidly defined.
iv. It is the most representative measures of central tendency.
v. It is not affected by sampling fluctuations.
• Demerits of Weighted mean.
i. The mean can not be calculated when the frequency distribution has open end classes at both the
ends.
ii. It is simple to calculate and can not be located by inspection like median or mode.
iii. It is unduly affected by extremely high or low values.
X w Xw d(x-A) wd
1 73 73 -305 -22265
2 378 756 0 0
3 459 1377 81 37179
4 90 360 -288 -25920
Σw=
1000
Σwd= -
11006
17. B) GEOMETRIC MEAN
• It is defined as the nth root of the product of n items of a series.
• If the geometric mean of items 5 and 20 is to be calculated, we apply the square root.
• If the geometric mean of items 5, 10 and 20 is to be calculated, we apply the cube root.
1. Calculation of geometric mean in a series of individual observations.
i. The log are found out of the given values (x).
ii. Then the logs are added (Σlog x).
iii. The sum of the logs is divided by number of observations and the antilog of the value is found out.
• G.M. of items 5,10,12 and 15 would be fourth root of the product of these figures
G.M.= x1x2x3 −− − xn
4
5 ∗ 10 ∗ 12 ∗ 15 =
4
9000 = 9.74
Where
G.M.= geometric mean
n= number of items
x= values of the variable
• The value of geometric mean is always less than the value of the arithmetic mean.
2. Calculation of geometric mean in a discrete.
G.M. = Antilog{
𝚺𝐟 𝐥𝐨𝐠 𝐱
𝚺𝐟
}
i. Find logs of the variable x.
ii. Multiply these logs with the respective frequencies and obtain the total Σf log x.
18. Example: The number of Basophils in blood of 40 patients of a hospital and their frequency were
recorded as: 12,15,17,20,24 and frequencies 7,9,11,7,6. Find out the value of geometric mean.
Solution:
G.M. = Antilog{
Σf log x
Σf
}
= antilog{
49.0619
40
}
= antilog (1.2265)
= 16.85
No. of Basophils
x
Frequency
f
Log x f log x
12 7 1.0792 7.5544
15 9 1.1761 10.5849
17 11 1.2304 13.5344
20 7 1.3010 9.1070
24 6 1.3802 8.2812
Total n= 40 Σf log x = 49.0619
19. Merits of Geometric Mean.
• It is based on all the observations.
• It is rigidly defined.
• It is not much affected by the fluctuations.
• It is suitable for averaging ration, rates and percentages.
Demerits of Geometric Mean.
• It is difficult to understand and to calculate.
• It cannot be calculated when there are both negative and positive values.
• If one or more of the values are zero, the geometric mean would also be zero.
20. C) HARMONIC MEAN
• It is a type of statistical average which is a suitable measure of central tendency when the data pertains to speed,
rates and time.
• According to Downie and Heath, “ The harmonic mean is used in averaging rates when the time factor is variable
and the act being performed is constant”.
H.M.= n ÷ Σ
1
X
Calculations of Harmonic Mean
i. Calculation of Harmonic Mean in a series of Individual Observations.
H.M.=n÷ 𝛴
1
x
ii. Calculation of Harmonic Mean in a Discrete Series'
H.M.= Σf÷ 𝛴(fx
1
x
)
iii. Calculation of Harmonic Mean in a continuous series.
H.M.=f÷ 𝛴(
f
m
)
Where
n= number of observations
f= frequency
m= mid-points Relationship among Arithmetic Mean, Geometric Mean and Harmonic Mean
A.M.>G.M.>H.M.
21. Calculation of Harmonic Mean in a series of Individual Observations
Example: Haemoglobin percentage of six patients were recorded as 1,5,10,15,20 and 25. Find out the
value of harmonic mean.
Solution: HM = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]= n÷ 𝛴
1
x
H.M=
6
437
300
=
1800
437
= 4.19
Merits of Harmonic Mean.
• It is rigidly defined.
• It is based on all the observations of a series.
• It gives greater weightage to the smaller items.
• It is not much affected by sampling fluctuations.
Demerits of Harmonic Mean.
• It is not easy to calculate and understand.
• It cannot be calculated if the one value is zero.
• It gives larger weightage to the small items.
• It cannot be calculated if negative and positivity