Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 3.2 | The Mean 73
the distribution in Figure 3.3, for example, what would happen to the mean (balance point)
if a new score were added at X = 10?
■ The Weighted Mean
Often it is necessary to combine two sets of scores and then find the overall mean for the
combined group. Suppose, for example, that we begin with two separate samples. The first
sample has n = 12 scores and a mean of M = 6. The second sample has n = 8 and M = 7.
If the two samples are combined, what is the mean for the total group?
To calculate the overall mean, we need two values:
1. the overall sum of the scores for the combined group (SX), and
2. the total number of scores in the combined group (n).
The total number of scores in the combined group can be found easily by adding the
number of scores in the first sample (n 1
) and the number in the second sample (n 2
). In this
case, there are 12 scores in the first sample and 8 in the second, for a total of 12 1 8 = 20
scores in the combined group. Similarly, the overall sum for the combined group can be
found by adding the sum for the first sample (SX 1
) and the sum for the second sample
(SX 2
). With these two values, we can compute the mean using the basic equation
SX soverall sum for the combined groupd
overall mean 5 M 5
n stotal number in the combined groupd
5 SX 1 1SX 2
n 1
1 n 2
To find the sum of the scores for each sample, remember that the mean can be defined as
the amount each person receives when the total (SX) is distributed equally. The first sample
has n = 12 and M = 6. (Expressed in dollars instead of scores, this sample has n = 12 people
and each person gets $6 when the total is divided equally.) For each of 12 people to get
M = 6, the total must be SX = 12 × 6 = 72. In the same way, the second sample has n = 8
and M = 7 so the total must be SX = 8 × 7 = 56. Using these values, we obtain an overall
mean of
overall mean 5 M 5 SX 1SX 1 2 72 1 56
5
n 1
1 n 2
12 1 8 5 128
20 5 6.4
The following table summarizes the calculations.
First Sample Second Sample Combined Sample
n 5 12 n 5 8 n 5 20 (12 + 8)
SX 5 72 SX 5 56 SX 5 128 (72 + 56)
M 5 6 M 5 7 M 5 6.4
Note that the overall mean is not halfway between the original two sample means.
Because the samples are not the same size, one makes a larger contribution to the total
group and therefore carries more weight in determining the overall mean. For this reason,
the overall mean we have calculated is called the weighted mean. In this example, the
overall mean of M = 6.4 is closer to the value of M = 6 (the larger sample) than it is to
M = 7 (the smaller sample). An alternative method for finding the weighted mean is
presented in Box 3.1.