Percentiles and Quartiles

A percentile is a position, not a percent correct

A percentile describes location within a distribution. A test score at the 90th percentile is relatively high in a reference group; it does not mean 90% of questions were answered correctly. Always state the reference group, period and variable.

For a population distribution, quantiles can be defined through its cumulative distribution. In a finite sample, software uses different interpolation conventions. Ties and small samples mean that “exactly 25% below the first quartile” is not generally true.

Worked example with an explicit rule

Sort eight hypothetical durations: 2, 3, 4, 7, 8, 10, 12, 20 minutes. Using the median-of-halves rule for this even-sized dataset:

• Median Q₂ = (7 + 8)/2 = 7.5.
• Lower half: 2, 3, 4, 7; Q₁ = (3 + 4)/2 = 3.5.
• Upper half: 8, 10, 12, 20; Q₃ = (10 + 12)/2 = 11.
• Interquartile range IQR = 11 − 3.5 = 7.5 minutes.

The IQR summarizes the middle portion of the distribution and is less affected by extreme values than the range. The range here is 20 − 2 = 18 minutes. Neither measures sampling uncertainty in an estimated mean.

Why a calculator may give another answer

A common linear-interpolation convention places percentile p at zero-based index h = (n − 1)p, then interpolates between adjacent sorted values. For Q₁, h = 7 × 0.25 = 1.75, so the value is 3 + 0.75(4 − 3) = 3.75. For Q₃, h = 5.25 gives 10.5.

These differ from the median-of-halves values without either calculation being an arithmetic error. State the convention and use it consistently. For odd samples, median-of-halves implementations can also differ on whether the overall median is included in the halves.

Box-plot fences are screening rules

Using the first convention, the lower fence is 3.5 − 1.5(7.5) = −7.75 and the upper fence is 11 + 1.5(7.5) = 22.25. The value 20 is below the upper fence, so it is not flagged by that rule.

A flag does not prove an observation is wrong; an unflagged value is not automatically typical. Check units, records, subgroup differences and the process generating the data. Do not delete values simply to improve a result. A box plot also hides detail such as multiple clusters, so pair it with a distribution plot when useful.

Practice

For 1, 2, 4, 6, 8, 10, calculate Q₁ and Q₃ using the median of each three-value half.

Show the explanation

The lower half is 1, 2, 4, so Q₁ = 2. The upper half is 6, 8, 10, so Q₃ = 8. IQR = 6. State this convention instead of assuming all software uses it.

Continue

Read data visualization and spread. OpenStax introduces percentile and quartile interpretation.