Quartiles, percentiles and interpercentile ranges

Quartiles

• Q1 (lower quartile): 25% of values lie below it.
• Q2 (median): 50%.
• Q3 (upper quartile): 75%.

IQR = Q3 - Q1 contains the middle 50% of the data and ignores extreme values. For raw data use the S1 rule (n/4: round up if not whole; average with the next value if whole). For grouped data, interpolate at positions n/4 and 3n/4 (mark schemes also allow (n + 1)/4 and 3(n + 1)/4; the questions here use n).

Percentiles

The P th percentile has P% of the data below it. For grouped data, interpolate at position P/100 × n. The 10th to 90th interpercentile range = P90 - P10 contains the middle 80% of the data.

For the 85 times: P10 is at position 8.5, in 0 ≤ t < 10: 0 + 8.5/12 × 10 = 7.08. P90 at 76.5 in 30 ≤ t < 50: 30 + (76.5 - 75)/10 × 20 = 33.

Interpreting

A larger IQR or interpercentile range means the data are more spread out. The range (largest - smallest) uses only two values and is badly affected by outliers, so the IQR is usually a better measure of spread when outliers are present.

Worked example

Estimate the lower quartile of the 85 times.

1. Position 85/4 = 21.25.
2. Cumulative frequency is 12 below 10 and 30 below 15, so Q1 is in 10 ≤ t < 15.
3. Q1 = 10 + (21.25 - 12)/18 × 5

Answer: Q1 ≈ 12.57 minutes

Check yourself

1. Estimate the upper quartile of the 85 times, using position 3n/4. Give your answer to 3 significant figures.
   Show answer

   24.4 minutes. Position 63.75 lies in 20 ≤ t < 30: 20 + (63.75 - 55)/20 × 10 = 24.4.

2. Estimate the interquartile range of the 85 times, using positions n/4 and 3n/4. Give your answer to 3 significant figures.
   Show answer

   11.8 minutes. Q3 ≈ 24.375 and Q1 ≈ 12.569, so IQR ≈ 11.8.

3. Estimate the 90th percentile of the 85 times, using position 90n/100.
   Show answer

   33 minutes. Position 76.5 lies in 30 ≤ t < 50: 30 + (76.5 - 75)/10 × 20 = 33.

4. Estimate the 10th to 90th interpercentile range of the 85 times, using positions 10n/100 and 90n/100. Give your answer to 3 significant figures.
   Show answer

   25.9 minutes. P90 = 33 and P10 = 8.5/12 × 10 = 7.08, so the range ≈ 25.9.

5. Find the interquartile range of the 10 values 3, 5, 6, 8, 9, 11, 12, 14, 15, 17.
   Show answer

   8. n/4 = 2.5, so Q1 = 3rd value = 6; 3n/4 = 7.5, so Q3 = 8th value = 14. IQR = 8.

6. Why is the IQR often preferred to the range as a measure of spread?
   • It uses every data value
   • It is not affected by extreme values
   • It is always larger than the range
   • It can be found without ordering the data
   Show answer

   It is not affected by extreme values. It depends only on the middle 50% of the data.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1

Last updated 7 November 2026. Spotted a mistake? Tell us.