1 The IQR gives an estimate of the range of the middle 50% of the population.
2 The IQR is potentially a more meaningful measure of dispersion than the range as it is not affected by the extreme values that may be present in the data. By trimming 25% of the data from the bottom and 25% from the top, we eliminate the extreme values that may be present in the data set. Thus, the IQR is often used as a measure of comparison between two or more data sets on similar studies.
2.7.4 Coefficient of Variation
The coefficient of variation is usually denoted by
(2.7.4)
where
Example 2.7.3 A company uses two measuring instruments, one to measure the diameters of ball bearings and the other to measure the length of rods it manufactures. The quality control department of the company wants to find which instrument measures with more precision. To achieve this goal, a quality control engineer takes several measurements of a ball bearing by using one instrument and finds the sample average
Solution: Using formula (2.7.4), we have
The measurements of the lengths of rod are relatively less variable than of the diameters of the ball bearings. Therefore, we can say the data show that instrument 2 is more precise than instrument 1.
Example 2.7.4 (Bus riders) The following data gives the number of persons who take a bus during the off‐peak time schedule (3–4 pm.) from Grand Central to Lower Manhattan in New York City. Using technology, find the numerical measures for these data:
| 17 | 12 | 12 | 14 | 15 | 16 | 16 | 16 | 16 | 17 | 17 | 18 | 18 | 18 | 19 | 19 | 20 | 20 | 20 | 20 |
| 20 | 20 | 20 | 20 | 21 | 21 | 21 | 22 | 22 | 23 | 23 | 23 | 24 | 24 | 25 | 26 | 26 | 28 | 28 | 28 |
MINITAB
1 Enter the data in column C1.
2 From the Menu bar, select Stat Basic Statistics Display Descriptive Statistics:
3 In the dialog box that appears, enter C1 under variables and select the option Statistics.
4 A new dialog box Descriptive Statistics: Statistics appears. In this dialog box, select the desired statistics and click OK in the two dialog boxes. The values of all the desired statistics as shown below will appear in the Session window.
USING R
We can use a few built in functions in R to get basic summary statistics. Functions ‘mean()’, ‘sd()’, and ‘var()’ are used to calculate the sample mean, standard deviation, and variance, respectively. The coefficient of variation can be calculated manually using the mean and variance results. The ‘quantile()’ function is used to obtain three quantiles and the minimum and maximum. The function ‘range()’ as shown below can be used to calculate the range of the data. The task can be completed by running the following R code in the R Console window.
x = c(17,12,12,14,15,16,16,16,16,17,17,18,18,18,19,19,20,20,20,20,20, 20,20,20,21,21,21,22,22,23,23,23, 24,24,25,26,26,28,28,28) #To concatenate resulting mean, standard deviation, variance, and coefficient of variation c(mean(x), sd(x), var(x), 100*sd(x)/mean(x))
| 0% | 25% | 50% | 75% | 100% |
| 12 | 17 | 20 | 23 | 28 |
#To obtain the range we find Max‐Min range(x)[2]‐range(x)[1]
2.8 Box‐Whisker Plot
In the above discussion, several times we made a mention of extreme values. At some point we may wish to know what values in a data set are extreme values, also known as
