So what measure of variability can we use when working with sets of data that contain outliers? One solution is to use the interquartile range (IQR). The IQR, or the middle fifty, is the range for the middle fifty percent of the data. The IQR only considers middle values, so it is not affected by the outliers.
The common measures of variability are the range, IQR, variance, and standard deviation. Data sets with similar values are said to have little variability while data sets that have values that are spread out have high variability. When working to find variability, you'll also need to find the mean and median.
Midterm grades compared to final grades look like this: midterm variance is 224.83, the range is 48 and 14.99 is the standard deviation. For the final grades, the variance is 103.51, 31 is the range and 10.17 is the standard deviation.
Since the range is equal to the highest midterm grade minus the lowest midterm grade, we can easily find the range for this data set. Plugging in 100 for our highest midterm grade and 52 for our lowest midterm grade, we find that the range is equal to 100 minus 52, or 48.
To find the IQR, we first have to find the median and locate Q1 and Q3. For the data set shown, 88 is the median. Then separate the quartiles with brackets. After determining Q1 and Q3, find the medians of those quartiles which will be our values for Q1 and Q3. In each quartile look for the two middle numbers since each quartile has an even data set. We should find that Q1 = 79 and Q3 = 98. Now we can plug in Q1 and Q3 into the formula. If we subtract the median for Q3 from Q1 we'll get 98 - 79, or 19 for the IQR.
The range is the simplest measure of variability. You take the smallest number and subtract it from the largest number to calculate the range. This shows the spread of our data. The range is sensitive to outliers, or values that are significantly higher or lower than the rest of the data set, and should not be used when outliers are present.
Like the variance, the standard deviation measures how close the scores in the data set are to the mean. However, the standard deviation is measured in the exact same unit as the data set. Let's find the standard deviation of the midterm exam grades.