THE MEAN, MEDIAN, AND MODE

The most frequently used average is the mean, which is the balance point in a distribution. Its computation is simple-just sum (add up) the scores and divide by the number of scores. The most common symbol for the mean in academic journals is M (for the mean of a population) or m (for the mean of a sample). The symbol preferred by statisticians is:

X which is pronounced "X-bar."

Because the mean is very frequently used as the average, let's consider its formal definition, which is the value around which the deviations sum to zero. You can see what this means by considering the scores in Table 1. When we subtract the mean of the scores (which is 4.0) from each of the other scores, we get the deviations (whose symbol is x). If we sum the deviations, we get zero, as shown in Table 1.

Table 1 Scores and deviation scores

X minus M equals x

1 - 4.0 = -3.0

1 - 4.0 = -3.0

1 - 4.0 = -3.0

2 - 4.0 = -2.0

2 - 4.0 = -2.0

4 - 4.0 = 0.0

6 - 4.0 = 2.0

7 - 4.0 = 3.0

8 - 4.0 = 4.0

8 - 4.0 = 4.0

Sum of the deviations (x)=0.0

Note that if you take any set of scores, compute their mean, and follow the steps in Table 1, the sum of the deviations will always equal zero. ^I

Considering the formal definition, you can see why we also informally define the mean as the balance point in a distribution. The positive and negative deviations balance each other out.

A major drawback of the mean is that it is drawn in the direction of extreme scores. Consider the following two sets of scores and their means.

Scores for Group A: 1, 1, 1, 2,3,6, 7, 8, 8M = 4.11

Scores for Group B: 1,2,2,3,4, 7, 9,25,32M= 9.44

Notice that in both sets there are nine scores ang the two distributions are very similar except for the scores of 25 and 32 in Group B, which are much higher than the others and, thus, make skewed distribution. (To review skewed distributions, see Topic 43.) Notice that the two very high scores have greatly pulled up the mean for Group B; in fact, the mean for Group B is more than twice as high as the mean for Group A because of the two high scores.

When a distribution is highly skewed, we use a different average, the median, which is defined as the middle score. To get an approximate median, put the scores in order from low to high as they are for Groups A and B above, and then count to the middle. Since there are nine scores in Group A, the median (middle score) is 3 (five scores up from the bottom). For Group B, the median (middle score) is 4 (five scores up from the bottom), which is more representative of the center of this skewed distribution than the mean, which we noted was 9.44. Thus, one use of the median is to describe the average of skewed distributions. Another use is to describe the average of ordinal data, which we'll explore in Topic 46.

A third average, the mode, is simply the mostfrequently occurring score. For Group B, there are more scores of 2 than any other score; thus, 2 is the mode. The mode is sometimes used in informal reporting but is very seldom used in formal reports of research.

Because there is more than one type of average, it is vague to make a statement such as, "The average is 4.11." Rather, we should indicate the specific type of average being reported with statements such as, "The mean is 4.11."

^l It might be slightly off from zero if you use a rounded mean such as using 20.33 as the mean when its precise value is 20.3333333333.

EXERCISE ON TOPIC

1. Which average is defined as the most frequently occurring score?

2. Which average is defined as the balance point in a distribution?

3. Which average is defined as the middle score?

4. What is the formal definition of the mean?

5. How is the mean calculated?

6. Should the mean be used for highly skewed distributions?

7. Should the median be used for highly skewed distributions?

8. What is a synonym for the term averages?

9. Suppose a fellow student gave a report in class and said, "The average was 25.88." For what additional information should you ask? Why?