Measures of Central Tendency and their dispersion and applications. Acknowledgement: Dr Muslima Ejaz


 Gordon Smith
 1 years ago
 Views:
Transcription
1 Measures of Central Tendency and their dispersion and applications Acknowledgement: Dr Muslima Ejaz
2 LEARNING OBJECTIVES: Compute and distinguish between the uses of measures of central tendency: mean, median and mode. Compute and list some uses for measures of variation of dispersion: range, variance and standard deviation. Understand the distinction between the population mean and the sample mean. Learn the empirical rule and its application. REFERENCES: Basic Statistics for the Health Sciences, Jan W. Kuzma and Stephen E. Bohnenblust, by Mayfield Publishing Company, An introduction to Statistical Methods and Data Analysis, Lyman Ott PWSKent Publishing Company, /24/2013 2
3 Average speed of a car crossing midtown Manhattan during the day is 5.3 miles /hr Average minutes an American father of 4 yearold spend alone with his child each day is 42 Average American man is 5 feet 9 inches and average women is 5 feet 3.6 inches tall The average American man is sick in bed seven days a year missing 5 days of work 9/24/2013 3
4 Measures of Central Tendency (center of the distribution) Find a single score that is most typical or most representative of the entire group Helpful in comparing groups No single measure representative in every situation  three ways of determining central tendency Mean Median Mode 9/24/2013 4
5 Mean Also called arithmetic mean or average The sum of all scores divided by the number of scores X n i= = 1 n Xi 9/24/2013 5
6 Sample Mean Add up all the observations given in the data, then divide by sample size (n) The sample size n is the number of observations 9/24/2013 6
7 Example; Mean n = 5 Systolic blood pressures ( mmhg) X1 = 120 X2 = 80 X3 = 90 X4 = 110 X5 = 95 9/24/2013 7
8 Example: Mean X n i = 1 = n Xi Mean Systolic Blood Pressure: X = 495 = /24/2013 8
9 Pros and Cons of the Mean Pros Mathematical center of a distribution. Just as far from scores above it as it is from scores below it. Does not ignore any information Cons Influenced by extreme scores and skewed distributions One data point could make a great change in sample mean 9/24/2013 9
10 Example n= 5 Systolic blood pressures ( mmhg) X1 = 120 X2 = 180 X3 = 90 X4 = 110 X5 = 95 Mean Systolic Blood Pressure: X = 595 = /24/
11 Population Versus Sample Mean Population The entire group you want information about For example: The blood pressure of all 18 yearold male Medical college students at AKU 9/24/
12 Cont Sample A part of the population from which we actually collect information and draw conclusions about the whole population For example: Sample of blood pressures N=five 18yearold male college students in AKU 9/24/
13 Mean Population mu Sample X bar µ X N i= = 1 sigma, the sum of X, add up all scores N n i= = 1 Xi N, the total number of scores in a population 9/24/ n sigma, the sum of X, add up all scores Xi n, the total number of scores in a sample
14 The Median The score that divides the distribution exactly in half when observations are ordered The 50 th percentile (50%) Goal: determine the exact midpoint Half of the rank order of observations n+1 / 2 Scores arranged from highest to lowest middle score 9/24/
15 Example: Median 110, 90, 80, 95, , 90, 95, 110, 120 The median is the middle value when observations are ordered. To find the middle, count in (N+1)/2 scores when observations are ordered lowest to highest. Median Systolic BP: (5+1)/2 = 3 9/24/
16 Finding the median with an even number of scores. With an even number of scores, the median is the average of the middle two observations when observations are ordered. 80, 90, 95, 110, 120, 125 ( )/2 = /24/
17 Example; Median 80, 90, 95, 110, 220 Median 9/24/
18 Pros and Cons of Median Pros Not influenced by extreme scores or skewed distributions Easier to compute than the mean. Cons Doesn t take actual values into account. As its value is determined solely by its rank, provides no information about any of the other values within the distribution 9/24/
19 The Mode The highest frequency/most frequently occurring score Applicable to qualitative and quantitative data Could be bimodal or multimodal 9/24/
20 Central Tendency Example: Mode 75, 76, 90, 90, 95, 99, 100, 120, 120, 135,135, 155, 170, 186, 196, 205, 220 Mode: most frequent observation Mode(s) for Blood Pressure: 90, 120, 135 9/24/
21 Pros and Cons of the Mode Pros Easiest to compute and understand. Cons Ignores most of the information in a distribution The score comes from the data set. Small samples may not have a mode 9/24/
22 Using different measures of central tendency Two factors are important in making the decision of which measure of central tendency should be used: Scale of measurement (ordinal or numerical) Shape of the distribution of observations. A distribution can be symmetric or skewed to the right, positively skewed or to the left, negatively skewed. 9/24/
23 Using different measures of central tendency f(x) In a normal distribution, the mean, median, and mode are the same. µ Mean Median Mode x 9/24/
24 The effect of skew on average. In a skewed distribution, the mean is pulled toward the tail. 9/24/
25 Using different measures of central tendency The following guidelines help the researcher decide which measure is best with a given set of data: The mean is used for numerical data and for symmetric distribution. y Frequency Values 9/24/
26 Using different measures of central tendency The following guidelines help the researcher decide which measure is best with a given set of data: The median is used for ordinal data or for numerical data whose distribution is skewed. 9/24/
27 Using different measures of central tendency The following guidelines help the researcher decide which measure is best with a given set of data: The mode is used primarily for nominal or ordinal data or for numerical data with bimodal distribution Frequency Stress Rating 9/24/
28 Measures of Variation Or Measures of dispersion 9/24/
29 Measures of Variability A single summary figure that describes the spread of observations within a distribution. Centrally located at the Same value on the horizontal axis, but have substantially different amount of variability 9/24/
30 Measures of Variability Consider the following two data sets on the ages of all patients suffering from bladder cancer and prostatic cancer. BC PC The mean age of both the groups is 40 years. If we do not know the ages of individual patients and are told only that the mean age of the patients in the two groups is the same, we may assume that the patients in the two groups have a similar age distribution. 9/24/ Variation in the patient s ages in each of these two groups is very different. The ages of the prostatic cancer patients have a much larger variation than the ages of the bladder cancer patients.
31 Measures of Variability Measure the spread in the data Some important measures Range Mean deviation Variance Standard Deviation Coefficient of variation 9/24/
32 Variability The purpose of the majority of medical, behavioural and social science research is to explain or account for variance or differences among individuals or groups. Examples 1. What factors account for the variance (or difference) in IQ among individuals? 2. What factors account for the variance in treatment compliance among different groups of patients? 9/24/
33 Range The range tells us the span over which the data are distributed, and is only a very rough measure of variability Range: The difference between the maximum and minimum scores 80, 90, 95, 110, 120 Range = = 40 9/24/
34 Range Range is the simplest measure of dispersion It depends entirely on the extreme scores and doesn t take into consideration the bulk of the observations 9/24/
35 X Variation X X X = 25 n = 5 X = 5 This is an example of data with no i.e. zero variability 9/24/
36 Variation X X X X = 25 n = 5 X = 5 This is an example of data with low variability 9/24/
37 Variation X X X X = 25 n = 5 = 5 X This is an example of data with high variability 9/24/
38 Mean deviation The best measures of dispersion should: take into account all the scores in the distribution and should describe the average deviation of all observations from the mean. Normally, to find the average we would want to sum all deviations from the mean and then divide by n, i.e., X n x 9/24/
39 Mean Deviation X X x n = 6; ΣX = = 2.50 X = Σ X/n = 0.50 X = 33/ = 3.50 X = = = = 0.50 = 13 Mean Deviation = 13/ 6 = /24/
40 Variance & Standard Deviation However, if we square each of the deviations from the mean, we obtain a sum that is not equal to zero This is the basis for the measures of variance and standard deviation, the two most common measures of variability (or dispersion) of data 9/24/
41 Variance & Standard Deviation (cont) X X X ( X X ) X X X ( ) ( X X ) = 25 = 0.00 = ( X X ) 2 Note: The is called the Sum of Squares 9/24/
42 Steps to calculate Variance Compute the mean. Subtract the mean from each observation. Square each of the deviations. Find the sum of the squares. Divide the sum by N to get the variance Take the square root of the variance to get the standard deviation. 9/24/
43 Few Facts The square root of the variance gives the standard deviation (SD) and vice versa Variance is actually the average of the square of the distance that the each value is from the mean Why the squared distances and not the actual ones! Sum of the distances will always be zero, when each value is squared the negative sign is eliminated Why to take the square root? Since distances were squared, the units of the resultant numbers are the squares of the units of the original raw data. Finding the square root of the variance puts the SD in the same units as the raw data. i.e. standard deviation expresses variability in the same units as the data. 9/24/
44 Sample Variance The sum of squared deviations from the mean divided by the n  1 (an estimate of the population variance) s 2 = ( ) X x n 1 2 9/24/
45 Variance of a Population The sum of squared deviations from the mean divided by the number of scores (sigma squared): ( X ) µ σ 2 = N 2 9/24/
46 Standard Deviation Formulas Population Standard Deviation Sample Standard Deviation σ s = = ( X µ ) N 2 ( ) X x 2 X x n 1 Sample standard deviation usually underestimates population standard deviation. Using n1 in the denominator corrects for this and gives us a better estimate of the population standard deviation. 9/24/
47 Sometimes it is of interest to compare the degree of variability in the distribution of a factor from two different populations or of two different variables from the same populations eg; SBP (factor) among children and adults (two different populations) or among adults the distribution of SBP has more spread than that of DBP 9/24/
48 Coefficient of variation: expresses the SD as proportion of the mean It is a dimensionless measure of the relative variation. Constructed by dividing the standard deviation by the mean and multiplying by 100. CV = (SD/mean) * (100) It depicts the size of standard deviation relative to its mean Used to compare the variability in one data set with that in another when a direct comparison of standard deviation is not appropriate. 9/24/
49 Coefficient of variation The formula is: CV = (s/x) (100) Suppose two samples of human males yield the following results: Mean age Mean wt SD Adults 25 yrs 145lbs 10lbs Childr en 11 yrs 80lbs 10lbs CV 6.9% 12.5% 9/24/
50 Using different measures of dispersion The following guidelines help investigators decide which measure of dispersion is most appropriate for a given set of data: The standard deviation is used when the mean is used i.e., with symmetric distributions of numerical data The range is used with numerical data when the purpose is to emphasize extreme values. The coefficient of variation is used when the intent is to compare two numerical distributions measured on different scales. 9/24/
51 Empirical Rule Specifies the proportion of the spread in terms of the standard deviation It applies to the normal symmetric or bell shaped distribution Approx 68% of the data values will fall within 1 SD of the mean Approx 95% of the data values will fall within 2 SD of the mean Approx 99.7% of the data values will fall within 3 SD of the mean 9/24/
52 Empirical Rule Approximate percentage of area within given standard deviations 99.7% 95% 68% 9/24/ Assume the distribution of underlying variable is symmetric and bell shaped (Normal)
53 Example Scores on a National Achievement Exam have a mean of 480 and a SD of 90. And if these scores are normally distributed, then approximately 68% will fall between 390 & 570 approximately 95% will fall between 300 & 660 approximately 99.7% will fall between 210 & 750 9/24/
54 Application of the Empirical Rule Women participating in a threeday experimental diet regime have been demonstrated to have normally distributed weight loss with mean 600 g and a standard deviation 200 g. a) What percentage of these women will have a weight loss between 400 and 800 g? b) What percentage of women will lose weight too quickly on the diet (where too much weight is defined as >1000g)? 9/24/
55 a) X : (600,200) ~ 68% /24/
56 b) X : (600,200) 2.3% /24/
Unit 2. Describing Data: Numerical
Unit 2 Describing Data: Numerical Describing Data Numerically Describing Data Numerically Central Tendency Arithmetic Mean Median Mode Variation Range Interquartile Range Variance Standard Deviation Coefficient
More informationDescriptive StatisticsI. Dr Mahmoud Alhussami
Descriptive StatisticsI Dr Mahmoud Alhussami Biostatistics What is the biostatistics? A branch of applied math. that deals with collecting, organizing and interpreting data using welldefined procedures.
More informationChapter 3 Data Description
Chapter 3 Data Description Section 3.1: Measures of Central Tendency Section 3.2: Measures of Variation Section 3.3: Measures of Position Section 3.1: Measures of Central Tendency Definition of Average
More informationStatistics I Chapter 2: Univariate data analysis
Statistics I Chapter 2: Univariate data analysis Chapter 2: Univariate data analysis Contents Graphical displays for categorical data (barchart, piechart) Graphical displays for numerical data data (histogram,
More informationStatistics I Chapter 2: Univariate data analysis
Statistics I Chapter 2: Univariate data analysis Chapter 2: Univariate data analysis Contents Graphical displays for categorical data (barchart, piechart) Graphical displays for numerical data data (histogram,
More informationCHAPTER 4 VARIABILITY ANALYSES. Chapter 3 introduced the mode, median, and mean as tools for summarizing the
CHAPTER 4 VARIABILITY ANALYSES Chapter 3 introduced the mode, median, and mean as tools for summarizing the information provided in an distribution of data. Measures of central tendency are often useful
More informationDescriptive Statistics C H A P T E R 5 P P
Descriptive Statistics C H A P T E R 5 P P 1 1 0130 Graphing data Frequency distributions Bar graphs Qualitative variable (categories) Bars don t touch Histograms Frequency polygons Quantitative variable
More informationAlgebra 2. Outliers. Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)
Algebra 2 Outliers Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves) Algebra 2 Notes #1 Chp 12 Outliers In a set of numbers, sometimes there will be
More informationSection 3.2 Measures of Central Tendency
Section 3.2 Measures of Central Tendency 1 of 149 Section 3.2 Objectives Determine the mean, median, and mode of a population and of a sample Determine the weighted mean of a data set and the mean of a
More informationSTT 315 This lecture is based on Chapter 2 of the textbook.
STT 315 This lecture is based on Chapter 2 of the textbook. Acknowledgement: Author is thankful to Dr. Ashok Sinha, Dr. Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit some of their
More informationChapter 3 Statistics for Describing, Exploring, and Comparing Data. Section 31: Overview. 32 Measures of Center. Definition. Key Concept.
Chapter 3 Statistics for Describing, Exploring, and Comparing Data 31 Overview 3 Measures of Center 33 Measures of Variation Section 31: Overview Descriptive Statistics summarize or describe the important
More informationChapter. Numerically Summarizing Data. Copyright 2013, 2010 and 2007 Pearson Education, Inc.
Chapter 3 Numerically Summarizing Data Section 3.1 Measures of Central Tendency Objectives 1. Determine the arithmetic mean of a variable from raw data 2. Determine the median of a variable from raw data
More informationA is one of the categories into which qualitative data can be classified.
Chapter 2 Methods for Describing Sets of Data 2.1 Describing qualitative data Recall qualitative data: nonnumerical or categorical data Basic definitions: A is one of the categories into which qualitative
More informationContinuous Probability Distributions
Continuous Probability Distributions Called a Probability density function. The probability is interpreted as "area under the curve." 1) The random variable takes on an infinite # of values within a given
More informationMEASURES OF LOCATION AND SPREAD
MEASURES OF LOCATION AND SPREAD Frequency distributions and other methods of data summarization and presentation explained in the previous lectures provide a fairly detailed description of the data and
More informationChapter 2. Mean and Standard Deviation
Chapter 2. Mean and Standard Deviation The median is known as a measure of location; that is, it tells us where the data are. As stated in, we do not need to know all the exact values to calculate the
More information8/28/2017. PSY 5101: Advanced Statistics for Psychological and Behavioral Research 1
PSY 5101: Advanced Statistics for Psychological and Behavioral Research 1 Aspects or characteristics of data that we can describe are Central Tendency (or Middle) Dispersion (or Spread) ness Kurtosis Statistics
More informationStat 20 Midterm 1 Review
Stat 20 Midterm Review February 7, 2007 This handout is intended to be a comprehensive study guide for the first Stat 20 midterm exam. I have tried to cover all the course material in a way that targets
More informationGRACEY/STATISTICS CH. 3. CHAPTER PROBLEM Do women really talk more than men? Science, Vol. 317, No. 5834). The study
CHAPTER PROBLEM Do women really talk more than men? A common belief is that women talk more than men. Is that belief founded in fact, or is it a myth? Do men actually talk more than women? Or do men and
More informationStatistics in medicine
Statistics in medicine Lecture 1 part 1: Describing variation, and graphical presentation Outline Sources of variation Types of variables Fatma Shebl, MD, MS, MPH, PhD Assistant Professor Chronic Disease
More informationDescribing Data: Numerical Measures
Describing Data: Numerical Measures Chapter 03 McGrawHill/Irwin Copyright 2013 by The McGrawHill Companies, Inc. All rights reserved. LEARNING OBJECTIVES LO 31 Explain the concept of central tendency.
More informationAfter completing this chapter, you should be able to:
Chapter 2 Descriptive Statistics Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean, median, and mode for a set of data Find the range, variance, standard
More informationLooking at data: distributions  Density curves and Normal distributions. Copyright Brigitte Baldi 2005 Modified by R. Gordon 2009.
Looking at data: distributions  Density curves and Normal distributions Copyright Brigitte Baldi 2005 Modified by R. Gordon 2009. Objectives Density curves and Normal distributions!! Density curves!!
More informationAnswers Part A. P(x = 67) = 0, because x is a continuous random variable. 2. Find the following probabilities:
Answers Part A 1. Woman s heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the probability that a single randomly selected woman will be 67 inches
More informationCHAPTER 5: EXPLORING DATA DISTRIBUTIONS. Individuals are the objects described by a set of data. These individuals may be people, animals or things.
(c) Epstein 2013 Chapter 5: Exploring Data Distributions Page 1 CHAPTER 5: EXPLORING DATA DISTRIBUTIONS 5.1 Creating Histograms Individuals are the objects described by a set of data. These individuals
More informationCOMPLEMENTARY EXERCISES WITH DESCRIPTIVE STATISTICS
COMPLEMENTARY EXERCISES WITH DESCRIPTIVE STATISTICS EX 1 Given the following series of data on Gender and Height for 8 patients, fill in two frequency tables one for each Variable, according to the model
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More informationChapter 3: The Normal Distributions
Chapter 3: The Normal Distributions http://www.yorku.ca/nuri/econ2500/econ2500onlinecoursematerials.pdf graphsnormal.doc / histogramdensity.txt / normal dist table / ch3image Ch3 exercises: 3.2,
More informationLecture 8: Chapter 4, Section 4 Quantitative Variables (Normal)
Lecture 8: Chapter 4, Section 4 Quantitative Variables (Normal) 689599.7 Rule Normal Curve zscores Cengage Learning Elementary Statistics: Looking at the Big Picture 1 Looking Back: Review 4 Stages
More informationClass 11 Maths Chapter 15. Statistics
1 P a g e Class 11 Maths Chapter 15. Statistics Statistics is the Science of collection, organization, presentation, analysis and interpretation of the numerical data. Useful Terms 1. Limit of the Class
More informationECLT 5810 Data Preprocessing. Prof. Wai Lam
ECLT 5810 Data Preprocessing Prof. Wai Lam Why Data Preprocessing? Data in the real world is imperfect incomplete: lacking attribute values, lacking certain attributes of interest, or containing only aggregate
More informationTOPIC: Descriptive Statistics Single Variable
TOPIC: Descriptive Statistics Single Variable I. Numerical data summary measurements A. Measures of Location. Measures of central tendency Mean; Median; Mode. Quantiles  measures of noncentral tendency
More informationKCP elearning. test user  ability basic maths revision. During your training, we will need to cover some ground using statistics.
During your training, we will need to cover some ground using statistics. The very mention of this word can sometimes alarm delegates who may not have done any maths or statistics since leaving school.
More informationA C E. Answers Investigation 4. Applications
Answers Applications 1. 1 student 2. You can use the histogram with 5minute intervals to determine the number of students that spend at least 15 minutes traveling to school. To find the number of students,
More informationMidrange: mean of highest and lowest scores. easy to compute, rough estimate, rarely used
Measures of Central Tendency Mode: most frequent score. best average for nominal data sometimes none or multiple modes in a sample bimodal or multimodal distributions indicate several groups included in
More informationDescriptive statistics
Patrick Breheny February 6 Patrick Breheny to Biostatistics (171:161) 1/25 Tables and figures Human beings are not good at sifting through large streams of data; we understand data much better when it
More informationMODULE 9 NORMAL DISTRIBUTION
MODULE 9 NORMAL DISTRIBUTION Contents 9.1 Characteristics of a Normal Distribution........................... 62 9.2 Simple Areas Under the Curve................................. 63 9.3 Forward Calculations......................................
More informationContinuous random variables
Continuous random variables A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The total area under a density
More informationSUMMARIZING MEASURED DATA. Gaia Maselli
SUMMARIZING MEASURED DATA Gaia Maselli maselli@di.uniroma1.it Computer Network Performance 2 Overview Basic concepts Summarizing measured data Summarizing data by a single number Summarizing variability
More informationST Presenting & Summarising Data Descriptive Statistics. Frequency Distribution, Histogram & Bar Chart
ST2001 2. Presenting & Summarising Data Descriptive Statistics Frequency Distribution, Histogram & Bar Chart Summary of Previous Lecture u A study often involves taking a sample from a population that
More information1.0 Continuous Distributions. 5.0 Shapes of Distributions. 6.0 The Normal Curve. 7.0 Discrete Distributions. 8.0 Tolerances. 11.
Chapter 4 Statistics 45 CHAPTER 4 BASIC QUALITY CONCEPTS 1.0 Continuous Distributions.0 Measures of Central Tendency 3.0 Measures of Spread or Dispersion 4.0 Histograms and Frequency Distributions 5.0
More informationStatistics. Industry Business Education Physics Chemistry Economics Biology Agriculture Psychology Astronomy, etc. GFP  Sohar University
Statistics اإلحصاء تعاريف 31 Definitions Statistics is a branch of Mathematics that deals collecting, analyzing, summarizing, and presenting data to help in the decisionmaking process. Statistics is
More informationChapter 5: Normal Probability Distributions
Probability and Statistics Mrs. Leahy Chapter 5: Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution What is a Normal Distribution and a Normal
More informationadditionalmathematicsstatisticsadditi onalmathematicsstatisticsadditionalm athematicsstatisticsadditionalmathem aticsstatisticsadditionalmathematicsst
additionalmathematicsstatisticsadditi onalmathematicsstatisticsadditionalm athematicsstatisticsadditionalmathem aticsstatisticsadditionalmathematicsst STATISTICS atisticsadditionalmathematicsstatistic
More informationStatistical Methods. by Robert W. Lindeman WPI, Dept. of Computer Science
Statistical Methods by Robert W. Lindeman WPI, Dept. of Computer Science gogo@wpi.edu Descriptive Methods Frequency distributions How many people were similar in the sense that according to the dependent
More informationBusiness Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 4 Student Lecture Notes 41 Business Statistics: A DecisionMaking Approach 6 th Edition Chapter 4 Using Probability and Probability Distributions Fundamentals of Business Statistics Murali Shanker
More informationThe empirical ( ) rule
The empirical (689599.7) rule With a bell shaped distribution, about 68% of the data fall within a distance of 1 standard deviation from the mean. 95% fall within 2 standard deviations of the mean. 99.7%
More informationDescription of Samples and Populations
Description of Samples and Populations Random Variables Data are generated by some underlying random process or phenomenon. Any datum (data point) represents the outcome of a random variable. We represent
More informationBackground to Statistics
FACT SHEET Background to Statistics Introduction Statistics include a broad range of methods for manipulating, presenting and interpreting data. Professional scientists of all kinds need to be proficient
More informationPerformance of fourthgrade students on an agility test
Starter Ch. 5 2005 #1a CW Ch. 4: Regression L1 L2 87 88 84 86 83 73 81 67 78 83 65 80 50 78 78? 93? 86? Create a scatterplot Find the equation of the regression line Predict the scores Chapter 5: Understanding
More information(i) The mean and mode both equal the median; that is, the average value and the most likely value are both in the middle of the distribution.
MATH 183 Normal Distributions Dr. Neal, WKU Measurements that are normally distributed can be described in terms of their mean µ and standard deviation!. These measurements should have the following properties:
More informationDesign of Experiments
Design of Experiments D R. S H A S H A N K S H E K H A R M S E, I I T K A N P U R F E B 19 TH 2 0 1 6 T E Q I P ( I I T K A N P U R ) Data Analysis 2 Draw Conclusions Ask a Question Analyze data What to
More information11. The Normal distributions
11. The Normal distributions The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 11) The Normal distributions Normal distributions The
More informationUnit 1 Summarizing Data
PubHtlth 540 Fall 2014 1. Summarizing Page 1 of 54 Unit 1 Summarizing It is difficult to understand why statisticians commonly limit their enquiries to averages, and do not revel in more comprehensive
More informationRemember your SOCS! S: O: C: S:
Remember your SOCS! S: O: C: S: 1.1: Displaying Distributions with Graphs Dotplot: Age of your fathers Low scale: 45 High scale: 75 Doesn t have to start at zero, just cover the range of the data Label
More informationCHAPTER 3. YAKUP ARI,Ph.D.(C)
CHAPTER 3 YAKUP ARI,Ph.D.(C) math.stat.yeditepe@gmail.com REMEMBER!!! The purpose of descriptive statistics is to summarize and organize a set of scores. One of methods of descriptive statistics is to
More informationPercentile: Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included:
AP Statistics Chapter 2 Notes 2.1 Describing Location in a Distribution Percentile: The pth percentile of a distribution is the value with p percent of the observations (If your test score places you in
More informationLecture 2. Quantitative variables. There are three main graphical methods for describing, summarizing, and detecting patterns in quantitative data:
Lecture 2 Quantitative variables There are three main graphical methods for describing, summarizing, and detecting patterns in quantitative data: Stemplot (stemandleaf plot) Histogram Dot plot Stemplots
More informationHypothesis testing: Steps
Review for Exam 2 Hypothesis testing: Steps Exam 2 Review 1. Determine appropriate test and hypotheses 2. Use distribution table to find critical statistic value(s) representing rejection region 3. Compute
More informationTypes of Information. Topic 2  Descriptive Statistics. Examples. Sample and Sample Size. Background Reading. Variables classified as STAT 511
Topic 2  Descriptive Statistics STAT 511 Professor Bruce Craig Types of Information Variables classified as Categorical (qualitative)  variable classifies individual into one of several groups or categories
More informationDensity Curves and the Normal Distributions. Histogram: 10 groups
Density Curves and the Normal Distributions MATH 2300 Chapter 6 Histogram: 10 groups 1 Histogram: 20 groups Histogram: 40 groups 2 Histogram: 80 groups Histogram: 160 groups 3 Density Curve Density Curves
More informationThe standard deviation as a descriptive statistic
The standard deviation as a descriptive statistic by Von Bing Yap* Department of Statistics and Applied Probability, National University of Singapore Introduction  ':f The bulk of statistks essentially
More informationThe Central Limit Theorem
Introductory Statistics Lectures The Central Limit Theorem Sampling distributions Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission
More informationIntroduction to Statistics
Why Statistics? Introduction to Statistics To develop an appreciation for variability and how it effects products and processes. Study methods that can be used to help solve problems, build knowledge and
More information1. Use Scenario 31. In this study, the response variable is
Chapter 8 Bell Work Scenario 31 The height (in feet) and volume (in cubic feet) of usable lumber of 32 cherry trees are measured by a researcher. The goal is to determine if volume of usable lumber can
More informationFrequency Distribution CrossTabulation
Frequency Distribution CrossTabulation 1) Overview 2) Frequency Distribution 3) Statistics Associated with Frequency Distribution i. Measures of Location ii. Measures of Variability iii. Measures of Shape
More informationHistograms, Central Tendency, and Variability
The Economist, September 6, 214 1 Histograms, Central Tendency, and Variability Lecture 2 Reading: Sections 5 5.6 Includes ALL margin notes and boxes: For Example, Guided Example, Notation Alert, Just
More informationESP 178 Applied Research Methods. 2/23: Quantitative Analysis
ESP 178 Applied Research Methods 2/23: Quantitative Analysis Data Preparation Data coding create codebook that defines each variable, its response scale, how it was coded Data entry for mail surveys and
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Professor Silvia Fernández Chapter 2 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Visualizing Distributions Recall the definition: The
More informationChapter # classifications of unlikely, likely, or very likely to describe possible buying of a product?
A. Attribute data B. Numerical data C. Quantitative data D. Sample data E. Qualitative data F. Statistic G. Parameter Chapter #1 Match the following descriptions with the best term or classification given
More informationVariety I Variety II
Lecture.5 Measures of dispersion  Range, Variance Standard deviation coefficient of variation  computation of the above statistics for raw and grouped data Measures of Dispersion The averages are representatives
More informationEssential Question: What are the standard intervals for a normal distribution? How are these intervals used to solve problems?
Acquisition Lesson Planning Form Plan for the Concept, Topic, or Skill Normal Distributions Key Standards addressed in this Lesson: MM3D2 Time allotted for this Lesson: Standard: MM3D2 Students will solve
More informationMgtOp 215 Chapter 3 Dr. Ahn
MgtOp 215 Chapter 3 Dr. Ahn Measures of central tendency (center, location): measures the middle point of a distribution or data; these include mean and median. Measures of dispersion (variability, spread):
More informationThis is a review packet for the entire fall semester of Algebra I at Harrison.
HARRISON HIGH SCHOOL ALGEBRA I Fall Semester Review Packet This is a review packet for the entire fall semester of Algebra I at Harrison. You are receiving it now so that: you will have plenty of time
More informationChapter 5. Understanding and Comparing. Distributions
STAT 141 Introduction to Statistics Chapter 5 Understanding and Comparing Distributions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 27 Boxplots How to create a boxplot? Assume
More informationMeasures of Central Tendency. Mean, Median, and Mode
Measures of Central Tendency Mean, Median, and Mode Population study The population under study is the 20 students in a class Imagine we ask the individuals in our population how many languages they speak.
More informationPractice Questions for Exam 1
Practice Questions for Exam 1 1. A used car lot evaluates their cars on a number of features as they arrive in the lot in order to determine their worth. Among the features looked at are miles per gallon
More informationa table or a graph or an equation.
Topic (8) POPULATION DISTRIBUTIONS 81 So far: Topic (8) POPULATION DISTRIBUTIONS We ve seen some ways to summarize a set of data, including numerical summaries. We ve heard a little about how to sample
More informationMeasures of Dispersion
Measures of Dispersion MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Introduction Recall that a measure of central tendency is a number which is typical of all
More information6/25/14. The Distribution Normality. Bell Curve. Normal Distribution. Data can be "distributed" (spread out) in different ways.
The Distribution Normality Unit 6 Sampling and Inference 6/25/14 Algebra 1 Ins2tute 1 6/25/14 Algebra 1 Ins2tute 2 MAFS.912.SID.1: Summarize, represent, and interpret data on a single count or measurement
More informationChapter 3: Examining Relationships Review Sheet
Review Sheet 1. A study is conducted to determine if one can predict the yield of a crop based on the amount of yearly rainfall. The response variable in this study is A) the yield of the crop. D) either
More informationIntroduction to Basic Statistics Version 2
Introduction to Basic Statistics Version 2 Pat Hammett, Ph.D. University of Michigan 2014 Instructor Comments: This document contains a brief overview of basic statistics and core terminology/concepts
More informationMEASURING THE SPREAD OF DATA: 6F
CONTINUING WITH DESCRIPTIVE STATS 6E,6F,6G,6H,6I MEASURING THE SPREAD OF DATA: 6F othink about this example: Suppose you are at a high school football game and you sample 40 people from the student section
More informationMeasures of center. The mean The mean of a distribution is the arithmetic average of the observations:
Measures of center The mean The mean of a distribution is the arithmetic average of the observations: x = x 1 + + x n n n = 1 x i n i=1 The median The median is the midpoint of a distribution: the number
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationChapter 2 Descriptive Statistics
Chapter 2 Descriptive Statistics The Mean "When she told me I was average, she was just being mean". The mean is probably the most often used parameter or statistic used to describe the central tendency
More informationLecture 4B: Chapter 4, Section 4 Quantitative Variables (Normal)
Lecture 4B: Chapter 4, Section 4 Quantitative Variables (Normal) Quantitative Sample vs. Population 689599.7 Rule for Normal Curve Standardizing to zscores Unstandardizing Cengage Learning Elementary
More informationChapter 15 Sampling Distribution Models
Chapter 15 Sampling Distribution Models 1 15.1 Sampling Distribution of a Proportion 2 Sampling About Evolution According to a Gallup poll, 43% believe in evolution. Assume this is true of all Americans.
More informationNORMAL CURVE STANDARD SCORES AND THE NORMAL CURVE AREA UNDER THE NORMAL CURVE AREA UNDER THE NORMAL CURVE 9/11/2013
NORMAL CURVE AND THE NORMAL CURVE Prepared by: Jess Roel Q. Pesole Theoretical distribution of population scores represented by a bellshaped curve obtained by a mathematical equation Used for: (1) Describing
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Review Our objective: to make confident statements about a parameter (aspect) in
More informationdownload instant at
Chapter 2 Test B Multiple Choice Section 2.1 (Visualizing Variation in Numerical Data) 1. [Objective: Interpret visual displays of numerical data] For twenty days a record store owner counts the number
More informationIntroductory Statistics
Introductory Statistics OpenStax Rice University 6100 Main Street MS375 Houston, Texas 77005 To learn more about OpenStax, visit http://openstaxcollege.org. Individual print copies and bulk orders can
More informationIB Questionbank Mathematical Studies 3rd edition. Grouped discrete. 184 min 183 marks
IB Questionbank Mathematical Studies 3rd edition Grouped discrete 184 min 183 marks 1. The weights in kg, of 80 adult males, were collected and are summarized in the box and whisker plot shown below. Write
More information9/2/2010. Wildlife Management is a very quantitative field of study. throughout this course and throughout your career.
Introduction to Data and Analysis Wildlife Management is a very quantitative field of study Results from studies will be used throughout this course and throughout your career. Sampling design influences
More informationInference for Distributions Inference for the Mean of a Population
Inference for Distributions Inference for the Mean of a Population PBS Chapter 7.1 009 W.H Freeman and Company Objectives (PBS Chapter 7.1) Inference for the mean of a population The t distributions The
More informationNonparametric methods
Eastern Mediterranean University Faculty of Medicine Biostatistics course Nonparametric methods March 4&7, 2016 Instructor: Dr. Nimet İlke Akçay (ilke.cetin@emu.edu.tr) Learning Objectives 1. Distinguish
More informationLecture (chapter 13): Association between variables measured at the intervalratio level
Lecture (chapter 13): Association between variables measured at the intervalratio level Ernesto F. L. Amaral April 9 11, 2018 Advanced Methods of Social Research (SOCI 420) Source: Healey, Joseph F. 2015.
More informationLecture 26: Chapter 10, Section 2 Inference for Quantitative Variable Confidence Interval with t
Lecture 26: Chapter 10, Section 2 Inference for Quantitative Variable Confidence Interval with t t Confidence Interval for Population Mean Comparing z and t Confidence Intervals When neither z nor t Applies
More information(quantitative or categorical variables) Numerical descriptions of center, variability, position (quantitative variables)
3. Descriptive Statistics Describing data with tables and graphs (quantitative or categorical variables) Numerical descriptions of center, variability, position (quantitative variables) Bivariate descriptions
More informationNotation Measures of Location Measures of Dispersion Standardization Proportions for Categorical Variables Measures of Association Outliers
Notation Measures of Location Measures of Dispersion Standardization Proportions for Categorical Variables Measures of Association Outliers Population  all items of interest for a particular decision
More information