What if a software package gives a different quartile value?

Different software packages (like Excel or R) use different interpolation methods for computing quartiles. The method shown here is the standard exclusive method, often called Tukey's hinges.

How do quartiles identify outliers?

Values lower than $Q_1 - 1.5 imes ext{IQR}$ or higher than $Q_3 + 1.5 imes ext{IQR}$ are typically defined as outliers.

Yes. If the middle 50% of the dataset consists of identical values, $Q_1$ will equal $Q_3$, making the IQR zero.

How to find quartiles and the interquartile range

Quartiles divide a rank-ordered dataset into four equal parts. The Interquartile Range (IQR) measures the spread of the middle 50% of the data. This method applies to any univariate numerical dataset. Use it to identify outliers and assess data dispersion.

The setup

Order the dataset from smallest to largest. Count the total number of observations, let this be $n$ .

The steps

Find the median (Second Quartile, $Q_2$ ). If $n$ is odd, $Q_2$ is the exact middle value. If $n$ is even, $Q_2$ is the average of the two middle values. 2. Find the First Quartile ( $Q_1$ ). This is the median of the lower half of the data. Exclude $Q_2$ from this half if $n$ is odd. 3. Find the Third Quartile ( $Q_3$ ). This is the median of the upper half of the data. Exclude $Q_2$ from this half if $n$ is odd. 4. Calculate the Interquartile Range (IQR). Subtract $Q_1$ from $Q_3$ : $ext{IQR} = Q_3 - Q_1$ .

Checking the result

Verify that $Q_1 \le Q_2 \le Q_3$ . Approximately 25% of the data points should fall below $Q_1$ , 50% below $Q_2$ , and 75% below $Q_3$ .

Common errors

Forgetting to sort the dataset before finding the quartiles is a standard mistake. Another error is including the median in the upper and lower halves when computing $Q_1$ and $Q_3$ for an odd-numbered dataset.

Worked example

Find the quartiles and IQR for the dataset: 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25.

Order the data: 5, 7, 12, 14, 15, 22, 25, 30, 36, 42, 53. Count $n = 11$ . Find $Q_2$ : The middle value is in the 6th position. $Q_2 = 22$ . Find $Q_1$ : The lower half is 5, 7, 12, 14, 15. The median of this half is 12. $Q_1 = 12$ . Find $Q_3$ : The upper half is 25, 30, 36, 42, 53. The median of this half is 36. $Q_3 = 36$ . Calculate IQR: $ext{IQR} = Q_3 - Q_1 = 36 - 12 = 24$ .

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References: OpenStax Introductory Statistics, Chapter 2 · Khan Academy: Summarizing quantitative data