© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D 2/10/2014 Spring 2014 Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 3: Measures of Central Tendency 1

Review of class intervals 2 VariableFrequency Suppose we have this table What are the real interval lower and upper limits? VariableFrequency VariableFrequency Answer: It depends! You must really know the data Book VariableFrequency

Review of class intervals 3 VariableFrequency Conclusion: It was unfair to ask you to know the real limits, i.e. to become an expert in the problem at hand (I guess that authors of the book are) Book Suppose that we are talking about Age, do you think that this are the correct limits? My guess is that the limits are more like: Supposed that we are talking about number persons that live in household, do we really have to worry about where 19.3 will fall? VariableFrequency VariableFrequency

Review of class intervals 4 Jus be consistent: VariableFrequencyMidpoint ( )/2= VariableFrequencyMidpoint ( )/2= VariableFrequencyMidpoint (15+19)/2= In the exam I’ll clearly establish the limits

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Calculate the mode, the median, and the mean Calculate deviations Calculate the weighted mean Calculate the mode, the median, and the mean from a simple frequency distribution Understand what influences a researcher’s decision to use a specific measure of central tendency CHAPTER OBJECTIVES

Introduction 3.1 Measures of Central Tendency

3.1 7 The most frequently occurring value in a distribution Example: 20, 21, 30, 20, 22, 20, 21, 20 –Mode = 20 Sometimes there is more than one mode –Example: 96, 91, 96, 90, 93, 90, 96, 90 –Mode = 90 and 96 This is a bimodal distribution The mode is the only measure of central tendency appropriate for nominal-level variables The Mode

3.1 There is also multimodal, also the modes don’t have to be the same size

3.1 9 The middlemost case in a distribution Appropriate for ordinal or interval level data How to find the median: –Cases must be ordered –If there are an odd number of cases, there will be a single middlemost case –If there are an even number of cases, there will be two middlemost cases – The halfway point between these two cases should be used as the median The Median

The Median: Example What is the median of the following distribution: 1, 5, 2, 9, 13, 11, 4

The Median: Example What is the median of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4

The “center of gravity” of a distribution Appropriate for interval/level data The Mean

3.1 Figure 3.2

The Mean: Example 3.1 What is the mean of the following distribution: 4, 8, 11, 2

The distance and direction of any raw score from the mean The sum of the deviations that fall above the mean is equal in absolute value to the sum of the deviations that fall below the mean. Deviations

The “mean of the means” The overall mean for a number of groups The Weighted Mean

Obtaining the Mode, Median, and Mean from a Simple Frequency Distribution XfcffX Mo Mdn Position of the Mdn When you are given a frequency table instead of the raw data

Comparing the Mode, Median, and Mean 3.5 Three factors in choosing a measure of central tendency

Level of Measurement ModeMedianMean Nominal YesNo Ordinal Yes No Interval Yes

Symmetrical Distributions The mode, median, and mean have identical values Skewed Distributions The mode is the peak of the curve The mean is closer to the tail The median falls between the two Bimodal Distributions Both modes should be used to describe the data Shape of the Distribution

3.5 Figure 3.3

3.5 Figure 3.4

Fast and Simple Research  Mode Skewed Distribution  Median Advanced Statistics Analysis  Mean Research Objective

24 In class exercises

25 Now in MS Excel help/calculate-the-median-of-a-group-of- numbers-HP aspx

26 Homework: (problem 31 Chapter 2) + Chapter 3: Problem #25 and #30

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved There are three measures of central tendency: the mode, the median, and the mean How individual scores compare to the mean of the distribution can be examined by calculating deviations The mean of means, or the weighted mean, can be calculated for multiple groups The mode, the median, and the mean can also be calculated when data are presented in a simple frequency distribution Choosing which measure of central tendency to report is influenced by the level of measurement of the data, the shape of the distribution, as well as the research objective CHAPTER SUMMARY