**Mean**

The mean for a discrete variable in a distribution is the mathematical average of all the values. If the variable is considered over the entire population, the average is called the *population mean*. If the variable is considered over a particular sample of a population, the average is called the *sample mean*. There can be only one population mean for a population, but there can be many different sample means. The mean is often denoted by the lowercase Greek letter mu, in italics (*μ*). Sometimes it is also denoted by an italicized lowercase English letter, usually *x*, with a bar (vinculum) over it.

Table 2-7 shows the results of a 10-question test, given to a class of 100 students. As you can see, every possible score is accounted for. There are some people who answered all 10 questions correctly; there are some who did not get a single answer right. In order to determine the mean score for the whole class on this test – that is, the population mean, called μ_{p} – we must add up the scores of each and every student, and then divide by 100. First, let's sum the products of the numbers in the first and second columns. This will give us 100 times the population mean:

- (10 × 5) + (9 × 6) + (8 × 19) + (7 × 17) + (6 ×18) + (5 × 11) + (4 × 6) + (3 × 4) + (2 × 4) + (1 × 7) + (0 × 3)

- = 50 + 54 + 152 + 119 + 108 + 55 + 24 + 12 + 8 + 7 + 0

- = 589

Dividing this by 100, the total number of test scores (one for each student who turns in a paper), we obtain μ_{p} = 589/100 = 5.89.

The teacher in this class has assigned letter grades to each score. Students who scored 9 or 10 correct received grades of A; students who got scores of 7 or 8 received grades of B; those who got scores of 5 or 6 got grades of C; those who got scores of 3 or 4 got grades of D; those who got less than 3 correct answers received grades of F. The assignment of grades, informally known as the ''curve,'' is a matter of teacher temperament and doubtless would seem arbitrary to the students who took this test. (Some people might consider the ''curve'' in this case to be overly lenient, while a few might think it is too severe.)

**Median**

If the number of elements in a distribution is even, then the *median* is the value such that half the elements have values greater than or equal to it, and half the elements have values less than or equal to it. If the number of elements is odd, then the median is the value such that the number of elements having values greater than or equal to it is the same as the number of elements having values less than or equal to it. The word ''median'' is synonymous with ''middle.''

Table 2-8 shows the results of the 10-question test described above, but instead of showing letter grades in the third column, the cumulative absolute frequency is shown instead. The tally is begun with the top-scoring papers and proceeds in order downward. (It could just as well be done the other way, starting with the lowest-scoring papers and proceeding upward.) When the scores of all 100 individual papers are tallied this way, so they are in order, the scores of the 50th and 51st papers – the two in the middle – are found to be 6 correct. Thus, the median score is 6, because half the students scored 6 or above, and the other half scored 6 or below.

It's possible that in another group of 100 students taking this same test, the 50th paper would have a score of 6 while the 51st paper would have a score of 5. When two values ''compete,'' the median is equal to their average. In this case it would be midway between 5 and 6, or 5.5.

**Mode**

The *mode* for a discrete variable is the value that occurs the most often. In the test whose results are shown in Table 2-7, the most ''popular'' or often occurring score is 8 correct answers. There were 19 papers with this score. No other score had that many results. Therefore, the mode in this case is 8.

Suppose that another group of students took this test, and there were two scores that occurred equally often. For example, suppose 16 students got 8 answers right, and 16 students also got 6 answers right. In this case there are two modes: 6 and 8. This sort of distribution is called a *bimodal distribution*.

Now imagine there are only 99 students in a class, and there are exactly 9 students who get each of the 11 possible scores (from 0 to 10 correct answers). In this distribution, there is no mode. Or, we might say, the mode is not defined.

The mean, median, and mode are sometimes called *measures of central tendency*. This is because they indicate a sort of ''center of gravity'' for the values in a data set.

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