Quartiles, percentiles and cumulative frequency

Range and the middle-half spread

Range is the maximum minus the minimum. For 2,3,3,4,18 it is 16. It uses only the two extremes, so different distributions can share that range. The interquartile range instead describes the width of the middle half using Q3 − Q1.

Quartiles locate positions in an ordered set

Use this stated convention for the example: the median splits an ordered set into lower and upper halves; Q1 and Q3 are the medians of those halves. For 1, 2, 3, 4, 6, 8, 9, 11, the median is 5, Q1 is 2.5 and Q3 is 8.5. The interquartile range is 8.5-2.5 = 6.

A percentile locates a proportion of observations at or below a position. Conventions for finite samples can differ, especially at boundaries; follow a task’s specified method. An interpolated percentile need not equal an observed value.

Cumulative means “up to here”

For travel-time classes with frequencies 4, 8, 6 and 2, the cumulative frequencies at 10, 20, 30 and 40 minutes are 4, 12, 18 and 20. Start at (0,0) when no observations are at or below zero.

The graph cannot decrease, because extending a threshold cannot remove observations already counted. At 20 minutes the cumulative count is 12, so 8 observations exceed 20 minutes. Read differences of cumulative counts to recover class frequencies.

Grouped percentile readings are estimates

For 20 observations, read the median near cumulative frequency 10. It lies between 10 and 20 minutes. Linear interpolation within that class gives 10 + (10-4)/8 × 10 = 17.5 minutes. This assumes a uniform spread within the class for the estimate; the exact original median is not known from the grouped table alone.

Likewise Q1 is read near cumulative frequency 5 and Q3 near 15. A larger IQR indicates more spread in the middle half, not necessarily a larger full range.

Check your understanding 1 Cumulative count is not the last class count

A cumulative graph records 18 observations at or below 30 minutes and 12 at or below 20 minutes. How many lie in 20 < t ≤ 30?

Your answer or drawing on paper

Show answer

There are 18 − 12 = 6 observations in that class. Eighteen includes all earlier classes as well; it is not the frequency of the final class alone.

Check the limitation

Can two raw data sets with these same class frequencies have different exact medians? Yes: observations can move within their classes while preserving every grouped count. A detailed-looking curve does not restore information that was never retained.