The document discusses descriptive statistics such as the mean, median, variance and standard deviation. It uses examples of birth weight data from 5 newborn babies to illustrate how to calculate these statistics. The mean birth weight was 3.68 kg. The standard deviation describes how much the weights vary around the mean. But the sample mean itself has some error since it is based on a subset of data. This error is quantified by the standard error of the mean, which depends on the sample size and standard deviation.
This document provides an introduction to earth science concepts. It discusses that earth science is the study of Earth's systems, including meteorology, astronomy, geology, and oceanography. It also covers key concepts like making observations versus inferences, using metric measurements, scientific notation, calculating percent error, determining density, understanding different types of graphs, and calculating rates of change.
This document provides an overview of nonparametric statistical methods for analyzing ranked data. It discusses the Wilcoxon rank-sum test and sign test, which are nonparametric alternatives to the t-test that do not assume a normal distribution. The document explains how to rank data values and handle ties. It also provides examples of using the sign test to compare a sample mean to a hypothesized value and interpreting the results.
The document discusses key aspects of the scientific method including observations, hypotheses, experiments, analysis, and theories. It explains that the scientific method involves making observations, asking questions, developing hypotheses, testing hypotheses through experiments, analyzing data, and drawing conclusions to support or revise hypotheses. The document also covers measurements and units in the metric system, significant figures, and basic calculations involving conversions between units.
Please Subscribe to this Channel for more solutions and lectures
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
The document describes various statistical methods for describing and analyzing data, including measures of central tendency (mean, median), variability (range, standard deviation, interquartile range), and distribution (histograms, boxplots). It provides examples of calculating these statistics and interpreting them for real data sets. Comparisons are made between the sample mean and median, and between theoretical descriptions of data distributions (Chebyshev's Rule and the Empirical Rule) and actual data analyses.
The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases. It allows using a normal distribution for applications involving sample means. The mean of the sample means equals the population mean, and the standard deviation of sample means is the population standard deviation divided by the square root of the sample size. For samples larger than 30, the distribution of means can be approximated as normal, becoming closer for larger samples. If the population is already normal, the sample means will be normally distributed for any sample size.
This document discusses scientific notation and units of measurement. It introduces scientific notation as a way to represent very large or small numbers between 1 and 10 with an exponent. Accuracy refers to how close a measurement is to the true value, while precision refers to the agreement between repeated measurements. Significant figures indicate the precision or number of digits known in a measurement. Calculations are only as precise as the least precise input measurement. Proper rounding of results is also covered. Standard International (SI) units for length, mass, temperature, time, and amount are defined. Common metric units for volume, mass, and temperature are also discussed.
- Ryosuke Ishii is a researcher from Tokyo, Japan who is studying the Central Limit Theorem through online courses including MITx and HarvardX.
- The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, even if the population is not normally distributed.
- Ishii provides an example where samples are taken from a population following a normal distribution N(27.6, 28.3) and calculates the mean and standard deviation of the sample means, finding them to be consistent with the theoretical values predicted by the Central Limit Theorem.
This document provides an introduction to earth science concepts. It discusses that earth science is the study of Earth's systems, including meteorology, astronomy, geology, and oceanography. It also covers key concepts like making observations versus inferences, using metric measurements, scientific notation, calculating percent error, determining density, understanding different types of graphs, and calculating rates of change.
This document provides an overview of nonparametric statistical methods for analyzing ranked data. It discusses the Wilcoxon rank-sum test and sign test, which are nonparametric alternatives to the t-test that do not assume a normal distribution. The document explains how to rank data values and handle ties. It also provides examples of using the sign test to compare a sample mean to a hypothesized value and interpreting the results.
The document discusses key aspects of the scientific method including observations, hypotheses, experiments, analysis, and theories. It explains that the scientific method involves making observations, asking questions, developing hypotheses, testing hypotheses through experiments, analyzing data, and drawing conclusions to support or revise hypotheses. The document also covers measurements and units in the metric system, significant figures, and basic calculations involving conversions between units.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
The document describes various statistical methods for describing and analyzing data, including measures of central tendency (mean, median), variability (range, standard deviation, interquartile range), and distribution (histograms, boxplots). It provides examples of calculating these statistics and interpreting them for real data sets. Comparisons are made between the sample mean and median, and between theoretical descriptions of data distributions (Chebyshev's Rule and the Empirical Rule) and actual data analyses.
The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases. It allows using a normal distribution for applications involving sample means. The mean of the sample means equals the population mean, and the standard deviation of sample means is the population standard deviation divided by the square root of the sample size. For samples larger than 30, the distribution of means can be approximated as normal, becoming closer for larger samples. If the population is already normal, the sample means will be normally distributed for any sample size.
This document discusses scientific notation and units of measurement. It introduces scientific notation as a way to represent very large or small numbers between 1 and 10 with an exponent. Accuracy refers to how close a measurement is to the true value, while precision refers to the agreement between repeated measurements. Significant figures indicate the precision or number of digits known in a measurement. Calculations are only as precise as the least precise input measurement. Proper rounding of results is also covered. Standard International (SI) units for length, mass, temperature, time, and amount are defined. Common metric units for volume, mass, and temperature are also discussed.
- Ryosuke Ishii is a researcher from Tokyo, Japan who is studying the Central Limit Theorem through online courses including MITx and HarvardX.
- The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, even if the population is not normally distributed.
- Ishii provides an example where samples are taken from a population following a normal distribution N(27.6, 28.3) and calculates the mean and standard deviation of the sample means, finding them to be consistent with the theoretical values predicted by the Central Limit Theorem.
This document discusses accuracy, precision, and uncertainty in measurement. It defines systematic and random uncertainties and explains how to determine uncertainty through estimation or by taking multiple measurements and calculating the average deviation. The key ways to express uncertainty are through stating a range of possible values (e.g. 5'6" ± 1/2") or through calculating percentage uncertainty. The document provides examples of calculating uncertainty in addition, subtraction, and multiplication operations. Expressing uncertainty is important for determining if two measured values are the same or different.
This lecture discusses scientific measurements and units. It covers the metric system and SI units, dimensional analysis, unit conversions, and significant figures. Key points include:
1. The metric system uses meters, grams, and seconds as fundamental units. There are seven base SI units including the meter for length and gram for mass.
2. Dimensional analysis uses conversion factors to change between units while maintaining the correct dimensions. It is useful for solving chemistry problems.
3. Significant figures reflect the precision of a measurement and determine how many digits are reported in calculations. Rules for significant figures depend on the operation being used.
This document provides an overview of key concepts from the first chapter of a physics textbook. It introduces why physics is studied, important terminology in physics, use of mathematics in physics, scientific notation and significant figures, units and dimensional analysis, problem-solving techniques, and graphing. Examples are provided for many topics to illustrate physics concepts and calculations involving units, proportions, percentages, and graphing patient temperature data.
The document provides an outline of topics covered in Chapter 6 of The Pharmacy Technician 4E including basic pharmaceutical measurements, calculations, and conversions. Key areas discussed include numbers, fractions, decimals, ratios, proportions, percents, and metric and household conversions. Examples are provided for calculating common denominators, multiplying fractions, setting up and solving proportions, and converting between ratios, percents, and fractions.
This document provides an overview of scientific measurement and units. It discusses qualitative vs quantitative measurements, scientific notation, accuracy and precision, significant figures, and the International System of Units (SI). Some key points covered include:
- Quantitative measurements provide numeric results with defined units, while qualitative measurements use descriptive terms.
- Scientific notation expresses numbers as a coefficient and exponent of 10.
- Accuracy refers to how close a measurement is to the accepted value, while precision describes how consistent repeated measurements are.
- Significant figures determine the precision expressed in a measurement based on the precision of the measuring tool.
- The SI system standardizes units of length, mass, volume, temperature and more based on powers of 10.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
2nd Law of Motion and Free Body DiagramsJan Parker
The document provides instructions for a lab on Newton's Second Law of Motion. Students are told to get their lab books and materials ready. They will review a previous quiz, read about an investigation, make predictions, take notes, and conduct the investigation. The investigation involves measuring force, mass and acceleration of carts. Students will use an equation relating these variables and take quantitative measurements.
1) The document discusses various measures used to analyze grouped data, including mean, median, mode, range, and standard deviation.
2) It provides examples of calculating these measures from frequency distribution tables showing heights of students and exam marks of math students.
3) The document also discusses histograms and how they can visually display the distribution of data in grouped intervals.
The summary discusses criteria for evaluating qualitative research, including credibility/trustworthiness and consistency/dependability. It explains how these criteria relate to philosophical assumptions about knowledge (epistemology) and the nature of reality (ontology). It also identifies potential ethical issues that could influence research design, such as protecting participants' privacy, and discusses how some topics are well-suited to qualitative study by allowing exploration of meanings and experiences.
This document discusses the central limit theorem through simulations in R. It shows how drawing multiple samples from a normal distribution with mean 100 and standard deviation 10 results in the distribution of sample means being normal, even for small sample sizes (n=5). The distribution of sample means becomes narrower as the sample size increases (n=10). Key ideas are that the distribution of sample means will be normal and have the same mean as the original population, and increasing the sample size narrows the spread of this distribution. Homework exercises are suggested to further experiment with these concepts.
The document discusses ratios, proportions, and percentages. It provides examples of calculating ratios, proportions, and percentages from word problems. It also explains the key differences between ratios and proportions. Ratios compare quantities using units while proportions express the relationship between two ratios as an equation. Percentages express a number or amount as a fraction of 100.
The document outlines objectives and concepts related to scientific measurement in chemistry, including defining units in the International System of Units (SI) such as meters, kilograms, and kelvins; distinguishing between accuracy and precision in measurements; and explaining the proper handling of significant figures in measurements and calculations.
The document is a set of teaching slides that introduces key concepts related to taking measurements, including variation, range, mean, accuracy, and precision. It uses examples like timing a falling parachute and weighing objects to explain these terms and illustrate the difference between accuracy and precision. The goal is for students to understand these measurement concepts as explained in a physics textbook.
This document discusses scientific measurement and units. It covers accuracy, precision, and error in measurements. It introduces the International System of Units (SI) including the base units for length, volume, mass, temperature, and energy. It discusses significant figures and proper handling of calculations and conversions between units using dimensional analysis and conversion factors.
This document discusses units of measurement, scientific notation, significant figures, and percent error in chemistry. It provides examples for converting between standard and scientific notation. The prefixes used in units are explained. Rules are given for determining the number of significant figures in measurements and calculations. An example problem demonstrates calculating percent error by comparing a measured value to the accepted value.
This document discusses measurement and units in physics. It begins by explaining that all measurements relate to fundamental quantities like length, mass and time. The International System of Units (SI units) was developed to standardize measurement across countries. There are two types of physical quantities - base quantities that cannot be defined in terms of other quantities, and derived quantities that can be expressed in terms of base quantities using mathematical equations. The document then discusses various base and derived quantities like length, area, volume, time and mass, and defines their respective SI units.
Rencana Pelaksanaan Pembelajaran (RPP) ini membahas pembelajaran bahasa Inggris untuk siswa kelas VII tentang mengenalkan dan menyebutkan nama benda di lingkungan rumah. RPP ini mencakup tujuan pembelajaran, materi, metode, dan langkah-langkah kegiatan selama 4 pertemuan untuk mengidentifikasi, menyebutkan, dan mempraktikkan ungkapan tentang berbagai benda di rumah.
I Love things arounds me : Things in the Classroomdeedewi
There are six chairs in the classroom but no cupboards or erasers. There is one map and three pens. The document describes a classroom with 20 students, a teacher named Miss Jessica who writes on a whiteboard, and a beautiful picture near the whiteboard. The author states they love their classroom and friends.
This document discusses accuracy, precision, and uncertainty in measurement. It defines systematic and random uncertainties and explains how to determine uncertainty through estimation or by taking multiple measurements and calculating the average deviation. The key ways to express uncertainty are through stating a range of possible values (e.g. 5'6" ± 1/2") or through calculating percentage uncertainty. The document provides examples of calculating uncertainty in addition, subtraction, and multiplication operations. Expressing uncertainty is important for determining if two measured values are the same or different.
This lecture discusses scientific measurements and units. It covers the metric system and SI units, dimensional analysis, unit conversions, and significant figures. Key points include:
1. The metric system uses meters, grams, and seconds as fundamental units. There are seven base SI units including the meter for length and gram for mass.
2. Dimensional analysis uses conversion factors to change between units while maintaining the correct dimensions. It is useful for solving chemistry problems.
3. Significant figures reflect the precision of a measurement and determine how many digits are reported in calculations. Rules for significant figures depend on the operation being used.
This document provides an overview of key concepts from the first chapter of a physics textbook. It introduces why physics is studied, important terminology in physics, use of mathematics in physics, scientific notation and significant figures, units and dimensional analysis, problem-solving techniques, and graphing. Examples are provided for many topics to illustrate physics concepts and calculations involving units, proportions, percentages, and graphing patient temperature data.
The document provides an outline of topics covered in Chapter 6 of The Pharmacy Technician 4E including basic pharmaceutical measurements, calculations, and conversions. Key areas discussed include numbers, fractions, decimals, ratios, proportions, percents, and metric and household conversions. Examples are provided for calculating common denominators, multiplying fractions, setting up and solving proportions, and converting between ratios, percents, and fractions.
This document provides an overview of scientific measurement and units. It discusses qualitative vs quantitative measurements, scientific notation, accuracy and precision, significant figures, and the International System of Units (SI). Some key points covered include:
- Quantitative measurements provide numeric results with defined units, while qualitative measurements use descriptive terms.
- Scientific notation expresses numbers as a coefficient and exponent of 10.
- Accuracy refers to how close a measurement is to the accepted value, while precision describes how consistent repeated measurements are.
- Significant figures determine the precision expressed in a measurement based on the precision of the measuring tool.
- The SI system standardizes units of length, mass, volume, temperature and more based on powers of 10.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
2nd Law of Motion and Free Body DiagramsJan Parker
The document provides instructions for a lab on Newton's Second Law of Motion. Students are told to get their lab books and materials ready. They will review a previous quiz, read about an investigation, make predictions, take notes, and conduct the investigation. The investigation involves measuring force, mass and acceleration of carts. Students will use an equation relating these variables and take quantitative measurements.
1) The document discusses various measures used to analyze grouped data, including mean, median, mode, range, and standard deviation.
2) It provides examples of calculating these measures from frequency distribution tables showing heights of students and exam marks of math students.
3) The document also discusses histograms and how they can visually display the distribution of data in grouped intervals.
The summary discusses criteria for evaluating qualitative research, including credibility/trustworthiness and consistency/dependability. It explains how these criteria relate to philosophical assumptions about knowledge (epistemology) and the nature of reality (ontology). It also identifies potential ethical issues that could influence research design, such as protecting participants' privacy, and discusses how some topics are well-suited to qualitative study by allowing exploration of meanings and experiences.
This document discusses the central limit theorem through simulations in R. It shows how drawing multiple samples from a normal distribution with mean 100 and standard deviation 10 results in the distribution of sample means being normal, even for small sample sizes (n=5). The distribution of sample means becomes narrower as the sample size increases (n=10). Key ideas are that the distribution of sample means will be normal and have the same mean as the original population, and increasing the sample size narrows the spread of this distribution. Homework exercises are suggested to further experiment with these concepts.
The document discusses ratios, proportions, and percentages. It provides examples of calculating ratios, proportions, and percentages from word problems. It also explains the key differences between ratios and proportions. Ratios compare quantities using units while proportions express the relationship between two ratios as an equation. Percentages express a number or amount as a fraction of 100.
The document outlines objectives and concepts related to scientific measurement in chemistry, including defining units in the International System of Units (SI) such as meters, kilograms, and kelvins; distinguishing between accuracy and precision in measurements; and explaining the proper handling of significant figures in measurements and calculations.
The document is a set of teaching slides that introduces key concepts related to taking measurements, including variation, range, mean, accuracy, and precision. It uses examples like timing a falling parachute and weighing objects to explain these terms and illustrate the difference between accuracy and precision. The goal is for students to understand these measurement concepts as explained in a physics textbook.
This document discusses scientific measurement and units. It covers accuracy, precision, and error in measurements. It introduces the International System of Units (SI) including the base units for length, volume, mass, temperature, and energy. It discusses significant figures and proper handling of calculations and conversions between units using dimensional analysis and conversion factors.
This document discusses units of measurement, scientific notation, significant figures, and percent error in chemistry. It provides examples for converting between standard and scientific notation. The prefixes used in units are explained. Rules are given for determining the number of significant figures in measurements and calculations. An example problem demonstrates calculating percent error by comparing a measured value to the accepted value.
This document discusses measurement and units in physics. It begins by explaining that all measurements relate to fundamental quantities like length, mass and time. The International System of Units (SI units) was developed to standardize measurement across countries. There are two types of physical quantities - base quantities that cannot be defined in terms of other quantities, and derived quantities that can be expressed in terms of base quantities using mathematical equations. The document then discusses various base and derived quantities like length, area, volume, time and mass, and defines their respective SI units.
Rencana Pelaksanaan Pembelajaran (RPP) ini membahas pembelajaran bahasa Inggris untuk siswa kelas VII tentang mengenalkan dan menyebutkan nama benda di lingkungan rumah. RPP ini mencakup tujuan pembelajaran, materi, metode, dan langkah-langkah kegiatan selama 4 pertemuan untuk mengidentifikasi, menyebutkan, dan mempraktikkan ungkapan tentang berbagai benda di rumah.
I Love things arounds me : Things in the Classroomdeedewi
There are six chairs in the classroom but no cupboards or erasers. There is one map and three pens. The document describes a classroom with 20 students, a teacher named Miss Jessica who writes on a whiteboard, and a beautiful picture near the whiteboard. The author states they love their classroom and friends.
RPP BAHASA INGGRIS KURTILAS INI DISUSUN UNTUK REKAN-REKAN GURU YANG MENGAJAR DI KELAS 7. ISI DARI RPP INI YAITU TENTANG RPP SEMESTER 1 DAN 2 LENGKAP DENGAN INSTRUMEN PENILAIANNYA BERDASARKAN PANDUAN PENILAIAN KURIKULUM 2013. SEMOGA BERMANFAAT DAN BISA DIGUNAKAN SETIAP KALI MENGAJAR DI KELAS.
Emi is a game center that is a nice place for friends to hang out. It has many games like dancing, shooting, racing, and catching games. Players must buy coins to play the games, and can earn tickets to exchange for prizes. Both kids and adults enjoy playing games at Emi.
This document defines and describes descriptive text. Descriptive text shows how something is done or what something looks like through careful observation. The main goals are to inform or express feelings. Descriptive text is common in journalism, manuals, and other areas. It typically includes describing the subject, its qualities, and parts. Description can be subjective, expressing the author's perspective, or objective, in an impersonal way. Linguistic techniques include using spatial indicators, adjectives, short sentences, and varied descriptive terms. Description varies based on perspective, order, and purpose. Examples are given from literature and brochures.
Powerpoint Presentation for Teaching Descriptive TextBob Septian
Lenka Kripac is a famous Australian singer and songwriter. She was born in 1978 in Australia and raised in New South Wales until age 7, when her family moved to Sydney. She is known for her song "The Show". Lenka has a beautiful face with long hair and is very feminine, with noticeable dimples when she smiles. She is a creative and multi-talented artist who can sing, write songs, play instruments, dance, and act.
Descriptive text provides information by describing a person, place, or thing. It identifies the subject and then describes parts, qualities, and characteristics using adjectives in the present tense. Common features described for people include name, age, nationality, appearance like hair color, eye color, height, and personality traits. Examples are given describing actors Daniel Radcliffe, Emma Watson, and Rupert Grint from Harry Potter including their physical appearance and characters.
This document discusses various elements of promotions and communication processes, including the sender, message, media, decoding, and feedback. It also covers promotions mix, the AIDA model of grabbing attention, exciting interest, creating desire and prompting action. Execution styles and media choice are examined, as well as promotional objectives like supporting sales, encouraging trial, and creating awareness.
The document provides guidance on selecting carbon accounting software, outlining key considerations such as establishing long-term goals, defining success metrics, ensuring the software provides a single source of truth for data, allows for integration with existing systems, and provides insights and reporting capabilities. It emphasizes choosing software that streamlines carbon accounting workflows, is intuitive to use, and helps turn insights into meaningful actions.
The document summarizes the key requirements and benefits of obtaining United States citizenship. It outlines that citizenship can be obtained either by birth or through the naturalization process. The main requirements for naturalization include being 18 years of age or older, having a green card for 5 years, continuous residency in the US, good moral character, basic English skills, and knowledge of US history and government. Benefits of citizenship include the right to vote, unlimited travel in and out of the US, eligibility for a US passport and federal jobs. Sources of additional information on citizenship requirements are also provided.
The document analyzes the characters, iconography, setting, and technical codes used in a Manchester newspaper. On the front cover, a man convicted of animal abuse represents the villain, while citizens who restrained him are heroes. Inside, a fraud convict and two arrested hotel guests also portray villains. Iconic Manchester images like footballers and the Hilton Hotel anchor the local audience. The setting of Manchester is conveyed through images of the city and football team logos. Headlines use techniques like creating narrative enigmas to engage readers.
This document discusses concepts related to sampling and sampling distributions. It begins with definitions of key terms like population, sample, parameter, and statistic. It then explains different sampling methods, focusing on simple random sampling. Different measures of central tendency and variability are outlined like mean, median, mode, range, variance, and standard deviation. The central limit theorem is introduced, which states that the sampling distribution of the mean will approximate a normal distribution for large sample sizes regardless of the population distribution. Examples are provided to illustrate these concepts.
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
This document provides an overview of descriptive statistics. It defines key terms like population, sample, measures of central tendency, and types of data. It discusses how to calculate and interpret the mean, median, and mode for both raw and grouped data. Examples are provided to demonstrate calculating the mean, median, and mode from raw data sets. It also discusses how to determine the mode from a grouped data set presented in a frequency distribution table, including using graphs to identify the modal class. The document covers important concepts in descriptive statistics for summarizing and describing numerical data.
SAMPLING MEAN DEFINITION The term sampling mean is.docxagnesdcarey33086
The document provides definitions and explanations of statistical concepts including:
- Sampling mean, which is an estimate of the population mean based on a sample.
- Sample variance, which measures the spread or variation of values in a sample from the sample mean.
- Standard deviation, which is the square root of the sample variance and measures how dispersed the values are from the mean.
- Hypothesis testing, which determines the validity of claims about a population by distinguishing rare events that occur by chance from those unlikely to occur by chance.
- Decision trees, which use a tree structure to systematically layout and analyze decisions and their potential consequences.
1. The document discusses key concepts in statistics including population, sampling, random sampling, standard error, and standard error of the mean.
2. A population is the total set of observations, while a sample is a subset selected from the population. Random sampling selects subjects entirely by chance so each member has an equal chance of being selected.
3. The standard error is the standard deviation of a statistic's sampling distribution and indicates how much a statistic may vary between samples. It decreases with larger sample sizes. The standard error of the mean specifically measures how much the sample mean may differ from the population mean.
This document discusses sampling distributions and their importance in inferential statistics. It provides examples of constructing sampling distributions of sample means both with and without replacement from a population. Key steps include determining possible sample sizes, listing samples and computing their means, constructing the sampling distribution as a frequency distribution of sample means, and displaying it as a histogram. Sampling distributions are important as they allow making inferences about large populations based on analyzing sample data.
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxanhlodge
The document defines key statistical terms and concepts including:
- Sampling mean is an estimate of the population mean based on a sample. It is calculated by adding all values and dividing by the sample size.
- Sample variance measures the variation or spread of values in a sample. It is calculated by finding the mean of squared differences from the sample mean.
- Standard deviation is the square root of the variance, providing a measure of dispersion from the mean.
- Hypothesis testing uses sample data to determine the validity of claims about a population. The null hypothesis is tested against an alternative using statistical significance.
- Decision trees visually represent decision problems by showing possible choices, outcomes, and probabilities to
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxagnesdcarey33086
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the result
This is t.
1. Rentang
2. Rentang antarkuartil
3. Rentang semikuartil
4. Rentang a-b persentil
5. Simpangan baku
6. Variansi
7. Ukuran penyebaaran relatif
8. Bilangan baku
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How to describe things
1. How to describe things: Descriptive statistics.
Excel is convenient for calculating many descriptive statistics, and for doing some
analyses.
The Excel file “Statistics In 1 Hour” at walkerbioscience.com shows how to load the
Excel data analysis toolpak and do many common analyses.
The Excel file “Descriptive Statistics Examples” at the website illustrates some of the
topics we’ll cover today.
Random variables
• birth weight of next baby born
• outcome of next coin flip - heads or tails
• number of otters you observe in Monterey Bay in 1 day.
If we observe baby births for a year, we will have a collection of birth weights. That
collection will have a distribution with characteristics such as the mean, median, range,
and standard deviation.
1. A typical value: the mean
Suppose that you are in the maternity ward of your local hospital, following the birth of
your first child. You happen to look in the nursery at the newborn babies.
Like many anxious parents, you wonder how the weight of your baby compares to the
weight of the other newborns. Is your baby in the normal range?
You ask the other parents the birth weights of their babies, and collect the data in Table
<birth weights>.
Table <birth weights>.
Baby’s crib number Baby’s birth weight (kilograms)
1
3.3
2
3.4
3
3.7
4
3.9
5
4.1
We’d like to describe both what a typical value of birth weight is, and how much the
babies vary around that typical value. To do that, we’ll use the mean and standard
deviation.
2. The mean of a group of numbers gives us an idea of a typical value.
If you have N numbers, add up all the N numbers and divide by N. For the five birth
weights in Table <birth weights>, N is 5.
The sum of all 5 numbers is 18.4, so the mean birth weight is 18.4/5 = 3.68 kg:
Mean birth weight = X
= (3.3+ 3.4+ 3.7+ 3.9+ 4.1) / 5
= 18.4/5
= 3.68 kg.
Notice that we use an X with a bar over the top, X , as the symbol for the mean.
You might be interested in comparing the birth weight of your baby to the birth weights
of the other babies, to see if your baby is near the typical weight, or is much above or
below typical weights for newborn babies.
We could describe the variability of the birth weights by giving the highest and the
lowest values (the range of values). But the range is not a very good descriptor of
variability, because it can be greatly affected by a single unusual point. For example a
pre-mature baby might have very low birth weight, which would greatly increase the
range and the apparent variability. The most widely used descriptors of variability are the
variance and the standard deviation.
2 Adding things up: Sigma (Σ) notation
Before we look at variance and the standard deviation, it will be useful to have some
shorthand notation for adding up a set of numbers without having to write them all out.
The notation we’ll use is the Greek symbol Sigma (Σ) When we see Σ it means to take
the sum.
Let’s look again at calculating the mean of the baby’s weights, but now we’ll use sigma
notation. There were 5 babies, and we could assign each of them a label:
Baby’s crib number
X1
X2
X3
X4
X5
Baby’s birth weight (kilograms)
3.3
3.4
3.7
3.9
4.1
The letter X represents the variable, in this case birth weight, and the subscripts 1 through
5 indicate which baby we are considering. We use the annotation Xi (X sub i) to indicate
3. any individual baby without specifying which one. So, if i=2, then we are considering
baby X2, whose birth weight is 3.4 kg.
To indicate that we are adding up the 5 birth weights, we could write as follows.
Sum of 5 birthweights = 3.3+ 3.4+ 3.7+ 3.9+ 4.1.
Or we could write:
Sum of 5 birthweights = X1+ X2+ X3+ X4+ X5.
It would get tedious to write out this formula, so instead we use the notation:
Sum of 5 birthweights
5
=∑Xi
i =1
= sum of Xi for i from1 to 5
= X1+ X2+ X3+ X4+ X5
= 3.3+ 3.4+ 3.7+ 3.9+ 4.1
= 18.4
Sometimes we won’t write out the subscript “i=1” or the superscript “5” if the meaning is
clear. In that case, we might just write ΣXi .
Finally, to calculate the mean of the 5 birthweights using sigma notation, we write the
following.
Mean of 5 birthweights = X
5
=∑Xi
i =1
5
= 3.68
Notice again that the symbol for the mean is X-bar, X .
3. Descriptors of variability: variance and standard deviation
We can describe variability of a group, such as the five babies, using the variance, which
we define as follows. The symbol for variance is σ2, sigma squared.
Population variance = σ2
4. N
(
∑ X i− X
i =1
)2
N
=
= [(3.3 – 3.68)2 + (3.4– 3.68) 2 + (3.7– 3.68) 2 + (3.9– 3.68) 2 + (4.1– 3.68) 2] /5
= 0.448 kg2/5
= 0.0896 kg2
Notice that the variance has units of kg2, kilograms squared. We’d like to have a measure
of variability in kilograms, the same units as the original measurements. A measure of
variability in the same units as the original measurements is the standard deviation, σ,
sigma. The standard deviation, σ, is the square root of the variance, σ2.
Population standard deviation = Square root (population variance)
= square root (σ2)
=σ
= square root (0. 0896 kg2)
= 0.299 kg.
Notice that we’ve used the terms population variance and population standard deviation.
If we are only interested in these 5 babies, and not in any other babies, then these 5 are
our entire population.
Alternatively, we may be interested in information about all of the babies that are in the
hospital in a given year. In that case, these 5 babies are just a sample of the babies that
are in the hospital in a given year.
Take a random sample from a population
n = number of observations in the sample.
Sample variance and the Sample standard deviation much as we do for the population,
with a small change.
For the population variance, we divide by N, while for the sample variance we divide by
N-1. Thus, the sample variance is slightly larger than the population variance.
Sample variance = S2
N
2
∑ X i− X
i =1
=
(
)
N −1
5. = [(3.3 – 3.68)2 + (3.4– 3.68)2 + (3.7– 3.68)2 + (3.9– 3.68)2 + (4.1– 3.68)2/(5-1)
= (0.448 kg2)/4
= 0.112 kg2
Notice that the sample variance has its own symbol, S2. The sample standard deviation, S,
is the square root of the sample variance, S2.
Sample standard deviation = S
= Square root (sample variance)
= Square root (S2)
= Square root (0.112)
= 0.335 kg.
Most software programs, including Excel, give you the sample variance and sample
standard deviation by default.
4. How well can we estimate the mean? Standard Error of the Mean (SEM)
Suppose we want to evaluate a drug to treat blood pressure.
•
Give to one patient. BP is 2 units lower. Effective?
•
Give to two patients. Mean BP is 3 units lower. Effective?
How can we be confident that the drug is better than placebo?
Let’s do a thought experiment. The 5 babies we looked at the day that we were in the
hospital were only a small fraction of all the babies that might be in the maternity ward in
a year. Their mean birth weight is 3.68 kg.
If we took a different sample of 5 babies from the same hospital on another day, would
their mean birth weight also be exactly 3.68 kg?
Most likely, it would be a little higher or a little lower than 3.68 kg.
The mean birth weight for any given sample, which contains only part of the whole
population, is an estimate of the population mean, and will likely be a little different from
the true population mean.
The difference between the population mean and the sample mean is the error in
estimating the population mean.
6. If we take many samples from the population, we will get many different estimates of the
population mean.
The sample mean is a statistic; the value of the sample mean depends on which
observations are included in the random sample.
So the sample mean is itself a random variable. It has its own mean and standard
deviation.
The average of the set of sample means is equal to the population mean (Law of large
numbers)
The standard deviation of the set of sample means is equal to the standard deviation of
the population divided by the square root of n, where n is the number of observations in
the sample (Central Limit Theorem). Provided n is sufficiently large, the Central Limit
Theorem tells us that the sampling distribution of the mean is asymptotically normal.
The standard deviation of the sample mean has a special name: the standard error of the
mean (SEM).
We can estimate how close the mean for a given sample is to the population mean using
the Standard Error of the Mean (SEM). The symbol for SEM is σ X . We calculate SEM
as follows.
Standard Error of the Mean = SEM = σ X
= (Population standard deviation)/(Square root of N)
However, we usually don’t know the population standard deviation, σ, so instead we use
the sample standard deviation, s. Because they differ only in the denominator being N
versus N-1, it makes little difference which we use when N is sufficiently large.
So, for a single sample from a population, we estimate SEM as follows using the sample
standard deviation.
Standard Error of the Mean = SEM
= (Sample standard deviation)/(Square root of N)
s
=
N
For our baby example, we calculate SEM as follows.
Sample standard deviation = s = .335
N=5
7. Standard Error of the Mean = SEM
0.335
5
=
= 0.1497
The SEM depends on both the sample standard deviation, S and of the number of
observations in our sample, N.
Not surprisingly, the more observations N we have in our sample, the better our estimate
of the population mean.
If we only have N = 1 or N = 2, we’re not very confident about the population mean.
On the other hand, if we have N = 100 or N = 1000, we start to be a lot more confident
that the mean of the sample is close to the population mean.
If the population has very small variability, giving us a small sample standard deviation,
then most samples will be pretty tightly clustered around the population mean, and a
small SEM.
If the population has high variability, giving us a large standard deviation, then samples
may be scattered widely, giving us a large SEM.
We’ll use SEM in statistical tests such as t-tests and analysis of variance to compare
groups.
The concept of the standard error of a statistic (such as the standard error of the sample
mean, or the standard error of coefficients in a regression model) is critical to
determining the significance of the statistic.
8. Extra topic 1. Robust descriptors, median, rank and non-parametric tests
The mean of a group can be greatly affected by a single extreme value. Suppose we
calculate the average income of all the people in Redmond, Washington, the headquarters
of Microsoft. The mean is going to be greatly affected by the income of Bill Gates, and
may not give us a very representative idea about the income of a typical person working
in Redmond. An alternative way to describe the typical income is the median, which is
the middle observation in a set of observations (if there are an odd number of
observations) or the average of the two middle observations (if there are an even number
of observations). For the birth weight example, we had 5 observations, so the middle
observation is the 3rd observation, so the median, is the value of the 3rd observation,
which is 3.7 kg.
Table <birth weights with a single extreme value> shows the same birth weights, but now
the 5th baby has a weight of 6.0 kg. This single baby changes the mean for the sample
from 3.68 kg to 4.06 kg, which is greater than the weight of all the other babies, and thus
is not really very representative. By contrast, the median is unchanged at 3.7 kg.
Table <birth weights with a single extreme value>.
Baby’s crib number Baby’s birth weight (kilograms)
1
3.3
2
3.4
3
3.7
4
3.9
5
6.0
The median is an example of a robust statistic, which means it is affected relatively little
by extreme values. The median depends on the relative rank (order) of the observations.
Many standard statistical tests, such as the t-test we'll see shortly, use the mean, so they
may be affected by extreme values. For most of these tests, there are alternative statistical
tests based on ranks, and these alternative tests are often called non-parametric tests.
Extra topic 2. Variability versus typical value: Coefficient of Variation (CV)
We often are concerned with the magnitude of variability versus the magnitude of a
typical value (the mean). We describe this ratio of variability to typical value using the
coefficient of variation (CV):
Coefficient of variation = CV = (Sample standard deviation)/Mean.
In most laboratory and manufacturing situations, we’d like the variability to be small
compared to the mean value, so a small CV is desirable.
9. Extra topic 3. Representing values on a standardized scale: the z-score
It is sometimes useful to describe an observation in terms of the number of standard
deviations it is from the mean. This measure of distance from the mean is called the zscore and is defined as follows.
z-score = (Xi – X )/S
We can calculate the z-score of each observation in the birth weight data.
Table <z-scores of birth weights>.
Baby’s crib
Baby’s birth weight
number
(kilograms)
1
3.3
2
3.4
3
3.7
4
3.9
5
4.1
z-score
-1.13
-0.83
0.06
0.65
1.25
Extra topic 4. Are error bars on graphs SEM's or Standard deviations?
Graphs often show a mean value for a variable (such as birth weight) along with error
bars. Unfortunately, the graph often fails to tell you what the error bars mean. Does an
error bar represent one standard deviation? Two standard deviations? One SEM? Two
SEM’s? Without this information, it is easy to be mislead into thinking that two groups
are almost the same (if the error bars represent two standard deviations) or completely
different (if the error bars represent one SEM). If someone shows you a graph with error
bars, ask what they mean.
10. Extra topic 3. Representing values on a standardized scale: the z-score
It is sometimes useful to describe an observation in terms of the number of standard
deviations it is from the mean. This measure of distance from the mean is called the zscore and is defined as follows.
z-score = (Xi – X )/S
We can calculate the z-score of each observation in the birth weight data.
Table <z-scores of birth weights>.
Baby’s crib
Baby’s birth weight
number
(kilograms)
1
3.3
2
3.4
3
3.7
4
3.9
5
4.1
z-score
-1.13
-0.83
0.06
0.65
1.25
Extra topic 4. Are error bars on graphs SEM's or Standard deviations?
Graphs often show a mean value for a variable (such as birth weight) along with error
bars. Unfortunately, the graph often fails to tell you what the error bars mean. Does an
error bar represent one standard deviation? Two standard deviations? One SEM? Two
SEM’s? Without this information, it is easy to be mislead into thinking that two groups
are almost the same (if the error bars represent two standard deviations) or completely
different (if the error bars represent one SEM). If someone shows you a graph with error
bars, ask what they mean.