IQR and box plots

The middle 50% is the half of the ordered data between the lower and upper quartiles.

1. Put the values in order. Quartile positions are meaningless until the data are ordered.
2. Find the median. This is $Q_2$, the centre of the whole list.
3. Find the centres of the two halves. If the list has an odd number of values, leave the overall median out before finding $Q_1$ and $Q_3$.
4. Subtract. Calculate $Q_3-Q_1$ and include the unit.

With an even number of values, split the ordered list into two equal halves immediately. For eight values, $Q_1$ is the median of the first four and $Q_3$ is the median of the last four.

• Whisker ends show the minimum and maximum; box ends show $Q_1$ and $Q_3$.
• The line inside the box is the median.
• Every position must be plotted against the same numerical scale.

A complete comparison usually needs one statement about location and one about variation. Quote the evidence and keep the context.

• Centre — use the median: “Linnets usually scored more because their median was higher.”
• Middle spread — use the IQR: “Linnets were less variable because their IQR was smaller.”
• Full spread — use the range: “Puffins covered a wider overall range.”

The median position can suggest skewness

A box plot is roughly symmetric when the two halves of the box have similar widths. If one half is noticeably longer, the data have a longer tail on that side. This is evidence of skewness, not a proof that every part of the distribution has that shape.

• If $Q_3-Q_2>Q_2-Q_1$, the upper half of the box is longer, which suggests positive skew.
• If $Q_2-Q_1>Q_3-Q_2$, the lower half is longer, which suggests negative skew.
• If the two widths are similar, the central half is roughly symmetric.