Measuring Variability Two distributions can have the same shape and center, but look quite different. Both distributions are symmetric and single-peaked, with centers around 150. But the variability of these two distributions is quite different.

Measuring Variability There are several ways to measure the variability of a distribution. The three most common are the range, interquartile range, and standard deviation. Range The range of a distribution is the distance between the minimum value and the maximum value. That is, Range = Maximum - Minimum The range is not a resistant measure of variability. It depends only on the maximum and minimum values, which may be outliers.

Measuring Variability Quartiles The quartiles of a distribution divide the ordered data set into four groups having roughly the same number of values. To find the quartiles, arrange the data values from smallest to largest and find the median. The first quartile Q1 is the median of the data values that are to the left of the median in the ordered list. The third quartile Q3 is the median of the data values that are to the right of the median in the ordered list.

Measuring Variability We can avoid the impact of extreme values on our measure of variability by focusing on the middle of the distribution. Order the data values from smallest to largest. Find the quartiles, the values that divide the distribution into four groups of roughly equal size. The first quartile Q1 lies one-quarter of the way up the list. The second quartile is the median, which is halfway up the list. The third quartile Q3 lies three-quarters of the way up the list.

Measuring Variability The interquartile range (IQR) measures the variability in the middle half of the distribution. Interquartile Range (IQR) The interquartile range (IQR) is the distance between the first and third quartiles of a distribution. In symbols, IQR = Q3 – Q1 The quartiles and the interquartile range are resistant because they are not affected by a few extreme values.

IQR Example

Measuring Variability If we summarize the center of a distribution with the mean, we should use the standard deviation to describe the variation of data values around the mean. Standard Deviation The standard deviation measures the typical distance of the values in a distribution from the mean. To find the standard deviation sx of a quantitative data set with n values: Find the mean of the distribution. Calculate the deviation of each value from the mean: deviation = value – mean Square each of the deviations. Add all the squared deviations, divide by n – 1, and take the square root.

Standard Deviation Example

Measuring Variability Properties of the standard deviation as a measure of variability: sx is always greater than or equal to 0. sx = 0 only when there is no variability, that is, when all values in a distribution are the same. Larger values of sx indicate greater variation from the mean of a distribution. sx is not resistant. The use of squared deviations makes sx even more sensitive than x to extreme values in a distribution. sx measures variation about the mean. It should be used only when the mean is chosen as the measure of center.

Measuring Variability Choosing Measures of Center and Variability The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use the mean and sx only for roughly symmetric distributions that don’t have outliers.

LESSON APP 1.7 Have we found the beef? Here are data on the amount of fat (in grams) in 12 different McDonald’s beef sandwiches, along with a dotplot. The mean fat content for these sandwiches is x-bar = 22.833 grams. Find the range of the distribution. Find the interquartile range. Interpret this value in context. Calculate the standard deviation. Interpret this value in context. The dotplot suggests that the Bacon Clubhouse Burger, with its 40g of fat, is a possible outlier. Recalculate the range, interquartile range, and standard deviation for the other 11 sandwiches. Compare these values with the ones you obtained in Questions 1 through 3. Explain why each result makes sense.