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The interquartile range, or IQR (also called the midspread or middle fifty), is the difference between the third and the first quartile in a data set. The IQR is a measure of how spread out your data is around the mean. The interquartile range formula is:
IQR = Q3 – x1
Where Q3 is the upper quartile and x1 is the lower quartile.
You can calculate the interquartile range using our online interquartile range calculator, or you can read this article which describes how to calculate the interquartile range by hand.
What is the Interquartile Range Formula Used For?
The IQR formula is a measure of spread, it is primarily used to build box plots. It can also be used as a test for normal distribution and to find outliers in a data set.
IQR as a test for normal distribution
The interquartile range formula can also be used in conjunction with the mean and standard deviation to test whether or not a population has a normal distribution. The formula to determine whether or not a population is normally distributed are:
Q1 – (σ z1) + X
Q3 – (σ z3) + X
Where Q1 is the first quartile, Q3 is the third quartile, σ is the standard deviation, z is the standard score (“z score”) and X is the mean. In order to tell whether a population is normally distributed, solve both equations and then compare the results. If there is a significant difference between the results and the first or third quartiles, then the population is not normally distributed.
The IQR as a way to determine outliers
The interquartile range formula can be used to find outliers. Outliers are high or low values that fall below Q1-1.5(IQR) or above Q3+1.5(IQR). These outliers will be beyond the whiskers of a boxplot.