1. Measures of Central Tendency Levin and Fox Elementary Statistics In Social Research Chapter 3 1

2. Measures of central tendency: Measures of central tendency are numbers that describe what is average or typical in a distribution We will focus on three measures of central tendency: – The Mode – The Median – The Mean (average) Our choice of an appropriate measure of central tendency depends on three factors: (a) the level of measurement, (b) the shape of the distribution, (c) the purpose of the research. 2

3. The Mode The Mode: The mode is the most frequent, most typical or most common value or category in a distribution. Example: There are more protestants in the US than people of any other religion. The mode is always a category or score, not a frequency. The mode is the only measureof available to nominal-level variables. The mode is not necessarily the category with the majority (that is, 50% or more) of cases. It is simply the category in which the largest number (or proportion) of cases falls. 3

4. Look at the figure below and identity the mode. 4% Let’s Practice! 4

5. The pie chart shows answers of 1998 GSS respondents to the question, “Would you say your own health, in general, is excellent, good, fair, or poor?” Note that the highest percentage (49%) of respondents is associated with the answer “good.” The answer “good” is the mode. Remember: The mode is used to describe nominal variables! A Review of Mode 5

6. Another Mode Example : Our question is the following: “What is the most common foreign language spoken in the United States today, as determined by the mode?” To answer this question, let’s look at a list of the ten most commonly spoken foreign languages in the United States and the number of people who speak each foreign language: 6

7. LanguageNumber of Speakers Spanish17,339,000 French1,702,000 German1,547,000 Italian1,309,000 Chinese1,249,000 Tagalog843,000 Polish723,000 Korean626,000 Vietnamese507,000 Portuguese430,000 Ten Most Common Foreign Languages Spoken in the United States, 1990. Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 2000, Table 51. 7

8. Is the mode 17,339,000? NO! Recall: The mode is the category or score, not the frequency!! Thus, the mode is Spanish. A Review of Mode 8

9. The Mode Some additional points to consider about modes: Some distributions have two modes where two response categories have the highest frequencies. Such distributions are said to be bimodal. NOTE : When two scores or categories have the highest frequencies that are quite close, but not identical, in frequency, the distribution is still “essentially” bimodal. In these instances report both the “true” mode and the highest frequency categories. 9

10. Example of a Bimodal Frequency Distribution 10

11. The Median The Median: The median is the score that divides the distribution into two equal parts so that half of the cases are above it and half are below it. The median can be calculated for both ordinal and interval levels of measurement, but not for nominal data. It must be emphasized that the median is the exact middle of a distribution. So, now let’s look at ways we can find the median in sorted data: 11

12. In some cases, we can find the median by simple inspection. Let’s look at the responses (A) to the question: “Think about the economy, how would you rate economic conditions in the country today?” First, we arrange the responses (B) in order from lowest to highest (or highest to lowest). Since we have an odd number of cases, let’s find the middle case. PoorJim GoodSue Only FairBob PoorJorge ExcellentKaren Total (N)5 PoorJim PoorJorge Only FairBob GoodSue ExcellentKaren Total (N)5 A B 12

13. Calculating the median: JimPoor JorgePoor BobOnly Fair SueGood KarenExcellent We can find the median through visual inspection and through calculation. We can also find the middle case when N is odd by adding 1 to N and dividing by 2: (N + 1) ÷2. Since N is 5, you calculate (5 + 1) ÷ 2 = 3. The middle case is, thus, the third case (Bob), the median response is “Only Fair.” 13

14. Calculating the median: StateNumber California1831 Florida93 Virginia105 New Jersey694 New York853 Ohio265 Pennsylvania168 Texas333 North Carolina42 TOTALN = 9 Another example: The following is a list of the number of hate crimes reported in the nine largest U.S. states for 1997. 14

15. Calculating the median: Finding the Median State for Hate Crimes 1.Order the cases from lowest to highest. 2.In this situation, we need the 5 th case: (9 + 1) ÷ 2 = 5 Which is Ohio Remember: (N + 1) ÷2. StateNumber North Carolina42 Florida93 Virginia105 Pennsylvania168 Ohio265 Texas333 New Jersey694 New York853 California1831 N = 9 15

16. Finding the Median State for Hate Crimes out of Eight States 1.Order the cases from lowest to highest. 2.The median is always that point above which 50% of cases fall and below which 50% of cases fall. 3.For an even number of cases, there will be two middle cases. 4.In this instance, the median falls halfway between both cases. 5.However, the circumstances being explained should determine if you use the two middle cases or the point halfway between both cases for your explanation. StateNumber North Carolina42 Florida93 Virginia105 Pennsylvania168 Ohio265 Texas333 New Jersey694 New York853 16

17. The median in frequency distributions: So now, let’s find the median in frequency distributions: Often the data are arranged in frequency distributions. The procedure is a bit more involved: – We have to find the category associated with the observation located in the middle of the distribution. – To do this, we construct a cumulative percentage distribution. So, let’s take a look at a frequency distribution… 17

18. Table: Political Views of GSS Respondents, 1988 Political Views Frequency (f) CfPercentageC% Extremely Liberal 32 2.4 Liberal17520712.915.3 Slightly Liberal 18939613.929.2 Moderate50289837.066.2 Slightly Conservative 211110915.681.8 Conservative203131215.096.8 Extremely Conservative 4413563.2100.00 Total1356100.00 18

19. Cumulative Percentage Distribution: We construct a cumulative percentage distribution to help locate the middle of the distribution. The observation located in the middle of the distribution is the one that has the cumulative percentage value equal to 50%.  Notice that 29.2% of the observations are accumulated below the category of “moderate” and that 66.2% are accumulated up to and including the category “moderate.” The median is the value of the category associated with this observation. This middle observation falls within the category “moderate,” so the median for this distribution is “moderate.” Cumulative Percentage Distribution: 19

20. Table: Political Views of GSS Respondents, 1988 Political Views Frequency (f) CfPercentageC% Extremely Liberal 32 2.4 Liberal17520712.915.3 Slightly Liberal18939613.9 29.2 Moderate50289837.066.2 29.2-66.2 Slightly Conservative 211110915.681.8 Conservative203131215.096.8 Extremely Conservative 4413563.2100.00 Total1356100.00 20

21. The Mean The Mean: The mean is what most people call the average. It find the mean of any distribution simply add up all the scores and divide by the total number of scores. Here is formula for calculating the mean 21

22. Finding the Mean Communicable Diseases -> Tuberculosis (as of 22 March 2007) -> Case detection rate (MDG indicator 24) -> DOTS all new case detection rate (%) -> Total (Periodicity: Year, Applied Time Period: from 2005 to 2005) 2005 Bangladesh37 Bhutan44 Democratic People's Republic of Korea103 India58 Indonesia47 Maldives76 Myanmar119 Nepal64 Sri Lanka71 Thailand61 Timor-Leste71 © World Health Organization, 2008. All rights reserved 22

23. Finding the Mean: To identify the number of new tuberculosis cases found in 2006 by the WHO in this region, – Add up the cases for all of the countries in the region and – Divide the sum by the total number of cases. Thus, the mean rate is (751 ÷ 11) = 68.273. Finding the Mean 23

24. Using a formula to calculate the mean: The Usefulness of Formulas: The mean introduces the usefulness of a formula, which may be defined as a is a shorthand way to explain what operations we need to follow to obtain a certain result. Again, the formula that defines the mean is: 24

25. Deviation: The deviation indicates the distance and direction of any raw score from the mean. To find the deviation of a particular score, we simply subtract the mean from the score: Where X = any raw score in the distribution 25

26. The Weighted Mean When groups differ in size, you can’t just sum their means and divide by the number of groups. Instead, you must weight each group mean by its size, 26

27. Time to practice! Reasons Why Homeowners get a Home Equity Line of Credit. Consolidate debts: 26 Invest in other real estate: 3 Home improvements/repairs: 45 Other purposes: 9 Purchase auto: 9 Pay for education or medical: 4 27

28. So what do you do? And then? We want to know the mo, mdn, and First, let’s arrange the scores from highest to lowest. Home improvements/ repairs 45 Consolidate debts26 Other purposes9 Purchase auto9 Pay for education or medical 4 Invest in other real estate 3 Total96 28

29. What’s the most frequent case (Mo)? - Home improvements/repairs 45. What is the middlemost score (Mdn)? – 9, because (N + 1) ÷2 or (6+1)÷2= 3.5 What is the mean ( )? – 16, because the sum of the scores is 96 and we divide this by 6 to get 16. Home improvements/ repairs 45 Consolidate debts 26 Other purposes 9 Purchase auto9 Pay for education or medical 4 Invest in other real estate 3 Total (N = 6)96 29

30. So what does this tell us? The mode is the peak of the curve. The mean is found closest to the tail, where the relatively few extreme cases will be found. The median is found between the mode and mean or is aligned with them in a normal distribution. 30

31. Did you know? The shape or form of a distribution can influence the researcher’s choice of a measure of tendency. Why is that? Well, let’s see… 31

32. Chapter Three: Review 32

33. Review: The Mode The Mode: The mode is the category with the largest frequency (or percentage) in the distribution. The mode is always a category or score, not a frequency. The mode is not necessarily the category with the majority (that is, 50% or more) of cases. It is simply the category in which the largest number (or proportion) of cases falls. 33

34. Review: The Median The Median: The median is the score that divides the distribution into two equal parts so that half of the cases are above it and half are below it. The median can be calculated for both ordinal and interval levels of measurement, but not for nominal data. It must be emphasized that the median is the exact middle of a distribution. 34

35. Review: The median: JimPoor JorgePoor BobOnly Fair SueGood KarenExcellent Calculating the median: We can find the median through visual inspection and through calculation. We can also find the middle case when N is odd by adding 1 to N and dividing by 2: (N + 1) ÷2. Since N is 5, you calculate (5 + 1) ÷ 2 = 3. The middle case is, thus, the third case (Bob), the median response is “Only Fair.” 35

36. Review: The Mean The Mean: The mean is what most people call the average. It find the mean of any distribution simply add up all the scores and divide by the total number of scores. Here is formula for calculating the mean 36

37. Review: Measures of Central Tendency Reasons Why Homeowners get a Home Equity Line of Credit. Consolidate debts: 26 Invest in other real estate: 3 Home improvements/repairs: 45 Other purposes: 9 Purchase auto: 9 Pay for education or medical: 4 37

38. Review: Measures of Central Tendency We want to know the mo, mdn, and First, let’s arrange the scores from highest to lowest. Home improvements/ repairs 45 Consolidate debts26 Other purposes9 Purchase auto9 Pay for education or medical 4 Invest in other real estate 3 Total96 38

39. What’s the most frequent case (Mo)? – Other purposes and Purchase auto because they both have the score of 9. What is the middlemost score (Mdn)? – 9, because 9 + 9= 18 and if we divide 18 by 2, we get 9. What is the mean ( )? – 16, because the sum of the scores is 96 and we divide this by 6 to get 16. Home improvements/ repairs 45 Consolidate debts 26 Other purposes 9 Purchase auto9 Pay for education or medical 4 Invest in other real estate 3 Total (N = 6)96 39

40. Review: Shape of the Distribution Choosing a Measure of Central Tendency The shape or form of a distribution can influence the researcher’s choice of a measure of tendency. 40

41. Review: Shape of the Distribution The mode is the peak of the curve. The mean is found closest to the tail, where the relatively few extreme cases will be found. The median is found between the mode and mean or is aligned with them in a normal symmetrical/unimodal distribution. 41