**Example**

Find the interquartile range of the population of orchestra members.

**Solution**

The median, 66.85 inches, divided the population into two halves, those members who are less than 66.85 inches tall and those who are greater than 66.85 inches tall. There are 31 members in each half. Because an odd number of values exists in the half, the median of the lower half is the = 16th value in the ordered set of the 31 lower values. Thus, the first quartile is *Q*_{1} = 62.5 inches. Similarly, the third quartile is the median of the 31 upper half values. The 16th value in the ordered set of upper values is *Q*_{3} = 69.5 inches. The interquartile range is now found to be

*Q*_{3} – *Q*_{1} = 69.5 – 62.5 = 7 inches

which is a value much smaller than the range. This is the spread in the middle half of the population of heights.

**Mean Absolution Deviation**

Before describing the next measure of dispersion, we need to define what is meant by a deviation. The quantity, (*X*_{i} – *μ*) , is the deviation of the ith population value from the population. By taking the absolute value, the deviation is a measure of how far the value is from the mean. For the orchestra members, the shortest person has a deviation of 53.5 – 66.4 = –12.9 inches. The negative sign indicates that this person's height is below the mean; that is, the shortest member's height is 12.9 inches below the mean height of all orchestra members. The member who is 74 inches tall has a deviation of 74 – 66.4 = 7.6 inches. He is 7.6 taller than the average height of all orchestra members. If we add all of the deviations together, we get zero.

*Mean absolute deviation* is the mean (or average) distance of the population values from the population mean; that is,

Now, (*X*_{i} – *μ*) is the deviation of the *i*th population value from the mean. The absolute deviation of the *i*th population value from the population mean is the distance of that value from the population mean, |*X*_{i} *μ* |. For the populations of *X*, *Y*, and *Z*, we have *μ* = 15. For a population value of 13, the distance of that value from the mean is |13 –15| = | –2 | =2. The mean absolute deviation is the population mean of these distances, E(|*X* – *μ*|).

The mean absolute deviation of the population distribution of *Y* is 0.75, and that of *Z* is 1.14.Notice that the greater dispersion in the distribution of Z as compared to *Y* is captured in the mean absolute deviation.

**Example**

Find the mean absolute deviation of the heights of the 62 orchestra members. Interpret the mean absolute deviation in the context of the problem.

**Solution**

We begin by finding the deviation and the absolute deviation from the mean for each band member. The mean height was 66.4, so the deviation for a particular orchestra member is that member's height minus 66.4. The absolute deviation is the absolute value of the deviation, or how far the member's height is from the mean. These are given in Table 6.1.

The mean absolute deviation is the average of the absolute deviations. For the orchestra members, the mean absolute deviation is 4.32 inches. This means that, on average, an orchestra member's height is 4.32 inches from the population mean height of 66.4 inches.

**Variance and Standard Deviation**

Two more measures of dispersion are the variance and the standard deviation. The *variance* is the mean squared distance of the population values from the mean; that is,

Notice that we have expressed the variance as an expected value.Here, it represents the average squared deviation of a population value from the mean. For the population distribution of *Y*, displayed in Figure 6.1, the variance is

.

For the population distribution of *Z*, displayed in Figure 6.2, the variance is 2.2. The units associated with the variance are the square of the measurement units of the population values.As an illustration, if the population values are recorded in inches, as they are with the orchestra members' heights, the variance is in inches^{2}. To obtain a measure of dispersion in the same units as the population values, the *standard deviation* is found to be the square root of the variance; that is, σ = √σ^{2}.

Although not technically correct, the standard deviation is often described as being the average distance of a population value from the mean. For the population distribution of *Y*, the standard deviation is √1 = 1. For *Z*, displayed in Figure 6.2, the population standard deviation is √2.2 = 1.5. The mean absolute deviation and the standard deviation are the same for *Y*. This is very unusual.More commonly, the mean absolute deviation is close, but not equal to the standard deviation. For *Z*, the mean absolute deviation is 2, and the standard deviation is 1.5. Although these parameters are different, we will often interpret the standard deviation as we would the mean absolute deviation.We simply need to realize that this interpretation of the standard deviation is only an approximate one.

Which measure of dispersion should we use? It is always good to compute more than one measure of dispersion as each gives you slightly different information. The range is often reported. The mean absolute deviation is an intuitive measure of dispersion, but it is not used much in practice, primarily because it is difficult to answer more advanced statistical questions using mean absolute deviation. Most of the statistical methods are based on the standard deviation. However, the range, mean absolute deviation, and standard deviation are all inflated by unusually large or unusually small values in the population. In this case, the IQR is often the best measure of dispersion.

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