30, 24, 34, 28, 32, 35, 29, 26, 36, 30, 33
Find the median alignment pin length.
Solution:
1 Step 1. Write the data in an ascending order and rank them from 1 to 11, since .Observations in ascending order2426282930303233343536Ranks1234567891011
2 Step 2. Rank of the median .
3 Step 3. Find the value corresponding to rank 6, which in this case is equal to 30. Thus, the median alignment pin length is mm. This means that at most 50% alignment pins in the sample are of length less than or equal to 30 and at the most 50% are of length greater than or equal to 30 mm.
Example 2.5.4 (Sales data) For the case of
10 8 15 12 17 7 20 19 22 25 16 15 18 250 300 12
Find the median sale of these individuals.
Solution:
1 Step 1. Write the data in an ascending order and rank them from 1 to 16, since .Observations in ascending order78101212151516171819202225250300Ranks12345678910111213141516
2 Step 2. Rank of the median .
3 Step 3. Following our previous discussion, the median in this case is the average of the values corresponding to their ranks of 8 and 9. Thus, the median of this data is . In other words, the median sales of the given individuals is $16,500. We remark that eight observations fall below 16.5, and eight observations fall above 16.5.
It is important to note that the median may or may not be one of the values of the data set as in this case. Whenever the sample size is odd, the median is the center value, and whenever it is even, the median is always the average of the two middle values when the data are arranged in the ascending order.
Finally, note that the data in this example contain the two values $250,000 and $300,000. These large values seem to be the sales of top‐performing sales personnel and may be considered as outliers. In this case, the mean of these data is
Note that the mean of 47.875 is much larger than the median of 16.5. It is obvious that the mean of these data has been adversely affected by the outliers. Hence, in this case, the mean does not adequately represent the measure of centrality of the data set, so that the median would more accurately identify the location of the center of the data.
Furthermore, if we replace the extreme values of 250 and 300, for example, by 25 and 30, respectively, then the median will not change, whereas the mean becomes 16.937, namely $16,937. Thus, the new data obtained by replacing the values 250 and 300 with 25 and 30, respectively, do not contain any outliers. The new mean value is more consistent with the true average sales.
Weighted Mean
Sometimes, we are interested in finding the sample average of a data set where each observation is given a relative importance expressed numerically by a set of values called weights. We illustrate the concept of weighted mean with the following example.
Example 2.5.5 (GPA data) Elizabeth took five courses in a given semester with 5, 4, 3, 3, and 2 credit hours. The grade points she earned in these courses at the end of the semester were 3.7, 4.0, 3.3, 3.7, and 4.0, respectively. Find her GPA for that semester.
Solution: Note that in this example, the data points 3.7, 4.0, 3.3, 3.7, and 4.0 have different weights attached to them; that is, the weights are the credit hours for each course. Thus, to find Elizabeth's GPA, we cannot simply find the arithmetic mean. Rather, in this case, we need to find the mean called the weighted mean, which is defined as
(2.5.4)
where
Mode
The mode of a data set is the value that occurs most frequently. The mode is the least used measure of centrality. When items are produced via mass production, for example, clothes of certain sizes or rods of certain lengths, the modal value is of great interest. Note that in any data set, there may be no mode, or conversely, there may be multiple modes. We denote the mode of a data set by
Example 2.5.6 (Finding a mode) Find the mode for the following data set:
3, 8, 5, 6, 10, 17, 19, 20, 3, 2, 11
Solution: In the data set of this example, each value occurs once except 3, which occurs twice. Thus, the mode for this set is
Example 2.5.7 (Data set with no mode) Find the mode for the following data set:
1, 7, 19, 23, 11, 12, 1, 12, 19, 7, 11, 23
Solution: Note that in this data set, each value occurs twice. Thus, this data set does not have any mode.
Example 2.5.8 (Tri‐modal data set) Find the mode for the following data set:
5, 7, 12, 13, 14, 21, 7, 21, 23, 26, 5
Solution: In this data set, values 5, 7, and 21 occur twice, and the rest of the values occur only once. Thus, in this example, there are three modes, that is,
These examples show that there is no mathematical relationship among the mean, mode, and median in the sense that if we know any one or two of these measures (i.e., mean, median, or mode), then we cannot find the missing measure(s) without using the data values. However, the values of mean, mode, and median do provide important information about the type or shape of the frequency distribution of the data. Although the shape of the frequency distribution of a data set could be of any type, in practice, the most frequently encountered distributions are the three types shown in Figure 2.5.1. The location of the measures
