Statistics Index > Basic Statistics> > What is The Interquartile Range Formula?

## What is The Interquartile Range Formula?

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 IQR formula is:

IQR = Q_{3} – Q_{1}

Where Q_{3} is the upper quartile and Q_{1} is the lower quartile.

You can calculate the IQR using our online interquartile range calculator. The calculator also works as a quartile calculator.

Need to calculate the interquartile range formula by hand? Check out this video:

### 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:

Q_{1} – (σ z_{1}) + X

Q_{3} – (σ z_{3}) + X

Where Q_{1} is the first quartile, Q_{3} 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.

Where did that 1.5 came from? is it following a trend? wont it destroy my data?

Christian,

It’s just for figuring out if you have outliers.

there’s no need to use it in the interquartile range formula itself.

Stephanie