1. Measures of Central Tendency: Just an Average Topic in Statistics

2. What’s “Average?” How many times have you been curious about the “average” of different things? What’s the most common car on the road? When is it typical for babies to start walking? What’s the usual amount of sleep for adults? Answers to those types of questions involve averages.

3. The term “average” is actually a general word used to refer to a “measure of central tendency.” Measure of central tendency - a value which indicates the level of performance of a group. We will examine three common measures of central tendency: 1. mode 2. median 3. mean

4. Mode (Mo) The mode is the most frequently occurring value. Consider the data below. Notice a score of 17 occurs more often than any other score. Therefore, Mo = 17. Score f 28 1 19 2 27 0 18 5 26 1 17 7 25 2 16 2 24 2 15 5 23 2 14 0 22 1 13 2 21 2 12 1 20 5 n = 40

5. We can also use the mode when considering qualitative variables. The frequency distribution below reveals “Chemistry” occurs more often than any other college major. Therefore, Mo = Chemistry. Major f Biology 129 Chemistry 221 Geology 106 Physics 58 n = 514

6. Advantages of the Mode The mode: - is easy to calculate - is the only measure of central tendency that can be used for qualitative variables - is easily understood by general audiences

7. Disadvantages of the Mode The mode: - does not always have a unique value (e.g., the distribution may be “bimodal” and, therefore, will have two modes) - is affected by choice of class interval width - is considerably influenced by the effects of random sampling variability - is a “terminal” statistic (i.e., there is little else that can be done with additional analyses)

8. The score of 18 is the middle score of Set 1. Therefore, Mdn = 18. What is the median score for Set 2? Median (Mdn) The median is the value below which 50% of the observations fall… it is the “middle” score. Consider the following sets of scores: Set 1: 13 15 16 18 20 22 25 Set 2: 34 39 40 42 54 57 Even though the score does not actually exist, 41 is the median for Set 2. 41

9. 14 13 12 11 10 9 8 7 6 5 4 3 2 1 33.5 30.5 ? The score at this point is the median. The cum f column shows there are 15 scores below 30.5, so we need an additional 5 scores to reach the 20th score. 15 scores below this point. 20 scores below this point. We start by determining how many scores would be 50% (n *.5 = 20) and by examining the cum f column, identify the class interval that contains the 20th score (30.5-33.5). 14 13 12 11 10 9 8 7 6 5 4 3 2 1 33.5 30.5 We assume the scores within that class interval are spread evenly throughout the interval. All we need to do is add those points to the lower exact limit and that will give us the median (30.5 + 1.07 = 31.57). The median can also be estimated from grouped data through the use of “linear interpolation.” 1.07 points Score f cum f 39.5 - 42.5 1 40 36.5 - 39.5 5 39 33.5 - 36.5 5 34 30.5 - 33.5 14 29 27.5 - 30.5 12 15 24.5 - 27.5 3 3 n = 40 31.57 We have used 5 of the 14 scores (i.e., 35.7%) in this class interval and, therefore, we have used 35.7% of the interval width (i.e.,.357 * 3 = 1.07). 1.07 points

10. Advantages of the Median The median: - is a good choice for highly skewed distributions since is it less affected by extreme scores - is the only relatively stable measure for open-ended distributions

11. Disadvantages of the Median The median: - is somewhat influenced by the effects of random sampling variability - is a “terminal” statistic (i.e., there is little else that can be done with additional analyses)

12. Mean ( ) The most widely used and most well-known measure of central tendency is the mean. What’s the average salary in baseball? What’s the average temperature in Ohio? When people ask about the “average” of something, they are usually asking about the mean. For example: X

13. The mean is the “balance point” of a distribution and is calculated in the following manner: 27 20 21 20 17 19 18 26 20 25 15 12 15 15 13 18 21 22 20 13 15 18 20 23 17 24 17 25 17 24 19 18 16 23 18 16 17 20 15 20 Given the following data, what would be the mean? (read as “X bar”)  X n X = 759 40 = = 18.975

14. Advantages of the Mean The mean: - is the most stable measure of central tendency (i.e., least sensitive to sampling variability) - can be used as an indicator of skewness when used in conjunction with the median - is mathematically tractable (i.e.,can be used to conduct further analyses - is familiar to and understood by most audiences

15. Disadvantages of the Mean The mean: - is responsive to the exact position of each score in a distribution - is sensitive to extreme scores and, therefore, should not be used with highly skewed distributions - cannot be used with open-ended distributions

16. If scores are grouped, the mean is approximated by taking the midpoint of each class interval as the value of each score in that class interval:  (C j f j ) n k j=1 Where: C j = midpoint of class interval j f j = frequency of class j k = number of intervals Estimated Mean From Grouped Data

17.  (C j f j ) n k j=1 Score f 27 - 29 1 24 - 26 5 21 - 23 5 18 - 20 14 15 - 17 12 12 - 14 3 n = 40 Consider the following grouped data: midpoint 28 25 22 19 16 13 xxxxxxxxxxxx C j f j = 28 = 125 = 110 = 266 = 192 = 39  = 760 760 40 = = 19.00

18. Weighted Means There are times when you have means from several groups and wish to find out “what is the mean of the means?” To accomplish that, you will need to calculate the weighted mean. n 1 1 + … + n j j n 1 + … + n j XwXw = XX

19. Consider the following set of means: Group n Mean 1 20 80 2 43 87 3 38 93 (20)(80) + (43)(87) + (38)(93) = 20 + 43 + 38 8875 101 = = 87.87 n 1 j + … + n j j n 1 + … + n j XwXw = XX

20. Symmetry, the mean, median, and mode When a distribution is perfectly symmetrical, the mean, median, and mode are equal. mean median mode median mean mode median mean When a distribution is skewed, however, the mean is always pulled in the direction of the skew and the median falls between the mean and the mode.