"Average:" Measures of Central Tendency

The most familiar descriptive statistic is the (arithmetic) mean, commonly known as the "average." It is only one of several measures of central tendency. All measures of central tendency describe a set of scores by finding a number that shows the "center" of the scores. Another way to say this is that measures of central tendency tell you the "typical" score. The different measures of central tendency do not necessarily give the same number, because they reflect somewhat different aspects of a set of scores. When they do differ, this tells something about how the individual scores are arranged in relation to the "average."

To calculate the mean score for a group of scores, you add up all the scores, and then divide the sum by the number of scores you added. This can be represented by the equation

m = (sum of X)/N

where m is the mean, X stands for the individual scores, and N is the number of scores. For example, your GPA is the sum of all your grades (converted to numbers, often with A=4.0; A- = 3.7, etc.) divided by the number of grades that went into that sum.

The median and the mode are two other measures of central tendency. The median is the "middle score" from a set of scores (recall that the center [middle] strip of a divided highway is called the median). It is the score that has an equal number of scores above and below it. To calculate it, put the scores in rank order, and then count half-way up from the lowest score (or down from the highest). The mode is the score in a set of scores that occurs the most. To calculate it, count how often each score occurs (on a bar graph, it's the tallest bar). The score that occurs the most is the mode.

Table 3 shows descriptive statistics summarizing the 25 scores for Groups 1 and 2 in Table 1.

```
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Table 3.  Descriptive Statistics for Scores from Table 1.

N =         25                      N =         25
Sum =      217                      Sum =      190
Mean =    8.68                      Mean =    7.60
SD =  1.64                      SD =      3.08
Median =     9                      Median =     9
Mode =       9                      Mode =       9
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[N = total number of scores;  Mean = Sum/N;  Median = 13th rank (12 above and 12 below);  SD =
standard deviation];  Mode = most frequent score
```

Match the measures of central tendency with the correct number for this set of scores:
2, 3, 3, 3, 4, 5, 5, 7, 13.
Q4A. Mean
A. 3         B. 4         C. 5         D. 6         E. cannot tell from information provided
Q4B. Median
A. 3         B. 4         C. 5         D. 6         E. cannot tell from information provided
Q4C. Mode
A. 3         B. 4         C. 5         D. 6         E. cannot tell from information provided

All scores are used to calculate the mean, so a few extremely high or a few extremely low scores will have a big effect on the mean. A few very high scores will increase the mean quite a bit; a few very low scores will decrease it quite a bit. Therefore, a few extreme high scores or extremely low scores makes the distribution asymmetrical (not symmetrical). The mean of Group 1 in Table 3 is 8.7, which is quite close to the median. In contrast, the mean for Group 2 is 7.6. Group 2 has some very low scores, which pull the mean down, but do not affect the median. The median of both groups in Table3 is 9. The median depends only on the order of scores from lowest to highest, so extreme scores do not affect it.

Understanding this difference between mean and median is important, because people can use the mean or the median as the "average" and pick whichever one better supports the their position. For example, during the baseball strike a few years ago, the team owners pointed out that the "average" salary of a major league player was about \$1.5 million/year. The players pointed out that half the players made less than \$0.5 million/year (which is not exactly peanuts). If you do not understand why these numbers are so different, re-read the preceding paragraph, or click HERE for the specific explanation.

Answer to Q4 A = 5; B = 4; C = 3

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(asgn1nc)