The mean is
1. the balance point of a distribution, the point for which ; 2. the preferred measure for relatively symmetrical distributions and quantitative
variables;
3. the measure with the best sampling stability; 4. widely used in advanced statistical procedures; 5. mathematically tractable;
6. the only measure whose value is dependent on the value of every core in the distribution;
7. more sensitive to extreme scores than the median and the mode and, hence, is not recommended for markedly skewed distributions;
8. not appropriate for qualitative data; and 9. not appropriate for open-ended distributions. The median is
1. the point that divides the ordered scores into two samples of equal size; 2. second to the mean in usefulness;
3. widely used for markedly skewed distributions because it is sensitive only to the number rather than to the values of scores above and below it;
4. the most stable measure that can be used with open-ended distributions; 5. more subject to sampling fluctuation than the mean;
6. less mathematically tractable than the mean; and 7. less often used in advanced statistical procedures. The mode is
1. the score that occurs most often and, therefore, the most typical value; 2. the only measure appropriate for unordered qualitative variables;
3. more appropriate than the mean or the median for quantitative variables that are inherently discrete;
4. the easiest measure to compute;
5. much more subject to sampling fluctuation than the mean and the median; 6. less mathematically tractable than the mean and the median;
7. not necessarily existent, as when a distribution has two or more scores with the same maximum frequency; and
8. rarely used in advanced statistical procedures.
gn
3.6 Location of the Mean, Median, and Mode in a Distribution
77
CHECK YOUR UNDERSTANDING OF SECTION 3.5
15. For the following sets of data, what measures of central tendency would you compute? Justify your choices.
a. 9, 6, 5, 7, 1, 6, 7, 8, 10, 6, 5, 4, 3, 6, 9, 7, 4, 5, 6, 8, 3, 2 b. 6, 5, 9, 6, 7, 5, 6, 8, 3, 4, 5, 7, 5, 4, 8, 5
c. 3, 5, 8, 5, 7, 9, 4, 2, 5, 6, 6, 23
16. Rank the three measures of central tendency with respect to the following char- acteristics; let 1 most and 3 least.
a. Sampling stability
b. Appropriateness for qualitative variables 17. Terms to remember:
a. Sampling stability b. Mathematically tractable c. Open-ended distribution
3.6 LOCATION OF THE MEAN, MEDIAN,
AND MODE IN A DISTRIBUTION
If a distribution is unimodal and symmetrical, the mean, median, and mode have the same value. If the distribution is unimodal but skewed, usually the three measures will be arranged in a predictable order. This order is illustrated in Figure 3.6-1. In both examples, the mean is on the side of the distribution that has the longest tail, and the median falls about one-third of the distance from the mean to the mode. To remember the order—mean, median, mode—note that it is alphabetical, starting from the longer tail. This order occurs because the mean is affected by the value of extreme scores. The median is affected by the presence of extreme scores but not by their value. The mode, however, is not affected by extreme scores unless they
10 9 8 7 6 5 4 3 2 1 a. Negatively skewed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 f Mo Mdn X 10 9 8 7 6 5 4 3 2 1 b. Positively skewed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 f Mo X Mdn
happen to have the greatest frequency of occurrence. This ordering of the mean, median, and mode holds for most unimodal distributions.
The relative location of the mean and median can be used to determine whether a distribution is positively or negatively skewed. For negatively skewed distributions, it is virtually always true that Mdn ; for positively skewed distributions, Mdn. If, for example, you know that the median is 25 and the mean is 20, you would strongly suspect that the distribution is negatively skewed. The greater the discrep- ancy between the two values, the greater the departure from symmetry.5
A knowledge of the relative location of the mean, median, and mode in asymmet- rical distributions can be used to intentionally distort the interpretation of data and mislead consumers of statistics. If you were to graph the wages of workers in one of the construction industries, you would probably obtain a positively skewed distribu- tion. If you were negotiating a new contract for the workers, you would want to report the modal salary, a lower figure than the median or mean, in defending your request for a wage increase. However, if you were on the other side of the negotiat- ing table, you would cite the mean, a higher figure, in arguing against the need for an increase. Even though both the mean and the mode are correct as measures of central tendency, they are misleading when the distribution is markedly skewed. The more appropriate measure for such a distribution is the median. This example illus- trates one of the classic ways in which statistics can be used to mislead the unwary.
CHECK YOUR UNDERSTANDING OF SECTION 3.6
18. Determine the shape—for example, symmetrical, positively skewed, and so on—of each distribution from the following measures of central tendency. a. 16,Mdn 10 b. Mdn
c. 34,Mdn34,Mo128, d. 46,Mdn46,Mo46
Mo240
e. Mo19,Mdn12 f. 23,Mdn23,Mo120,
Mo223,Mo327
3.7 MEAN OF TWO OR MORE MEANS
Suppose that two introductory sociology classes obtained the following mean scores on a departmental examination: 80 and 90. What is the mean of the two means? If each class had the same number of students, you could compute the mean of the means If, as is more likely, the classes con- tain different numbers of students, you must weight the means proportional to their respective sample sizes. Assume that and n120 and that 90 and
n240. The weighted mean, , is given by
X5n1X11n2X21 . . .1nnXn n11n21. . .1nn 5 20s80d140s90d 20140 586.7 XW X2 X1580 X5sX11X2d>25s80190d>2585. X X X X X X X
3.8 More About the Summation Operator
79
The weighted mean is closer to 90 than to 80; this reflects the larger n2associated with 90.
CHECK YOUR UNDERSTANDING OF SECTION 3.7
19. For the following data, compute weighted means. a. 30,n110; 50,n220
b. 20,n110; 25,n210; 30,n320 20. Term to remember:
a. Weighted mean
3.8 MORE ABOUT THE SUMMATION OPERATOR
Section 3.3 introduced the summation operator, . You learned that the symbol tells you to perform an operation, namely, add the terms corresponding to i
equals 1 through n. Many proofs6in statistics involve rules for using the summation
operator with variables and constants. This section describes four of these rules and illustrates their use in proving that the sum of the deviation of the mean from each score is equal to zero. Other proofs involving the summation operator are used in Exercise 23 of “Check Your Understanding of Section 3.8” and in Exercise 21 of the Review Exercises for Chapter 3.