Probability & Statistics

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Quartiles and Interquartile Range

The interquartile range is a measure of the middle 50% of data, and therefore represents data in a way that ignores outliers.

Quartiles divide data into quarters, and allow us to find the interquartile range. The interquartile range only takes account of the middle 50% of data, allowing us to ignore extreme results and gain a better understanding of data.

The range can be affected by extreme values. Extreme values are either much higher than the majority of values in the set, or much lower than the majority of values.

The interquartile range only takes account of the middle 50% of data. Therefore, it ignores the lowest 25% and highest 25% of data, and removes any extreme values from the calculation.

In which quartiles do extreme values sit?

Complete this sentence: the interquartile range accounts for the _________ of data

Here is a list of numbers. Let's find the interquartile range! $4, 7, 9, 13, 15, 18, 19, 20, 23$

Find the lower quartile

The lower quartile (Q1) could be understood as the median of the lower half of the set of data. We can find this through the equation $\dfrac{n + 1}{4}$ where n is the number of values in the set.

In this example, there are 9 values

So the position of the lower quartile is $\dfrac{9 + 1}{4} = 2.5th$ value.

The 2.5th value is between 7 and 9

To find the midpoint between these values, add them together and divide by 2. $\dfrac{7+9}{2}=8$

Find the upper quartile

The upper quartile (Q3) could be understood as the median of the upper half of the set of data, and the position of the upper quartile is $3 \times (n + \dfrac{1}{4})$

What is the upper quartile?

The upper quartile is $19.5$

We can use the upper and lower quartiles we have found to calculate the interquartile range.

Find the interquartile range

Interquartile range $(IQR) = 19.5 - 8 = 11.5$

Larger IQR indicates more spread out data

Conversely, a smaller IQR indicates data that is more consistent.

If there are 15 values in a list of data, what is the position of the lower quartile (Q1)?