Construct a cumulative frequency curve and use it to estimate the median, quartiles, interquartile range and percentiles

What this dot point is asking

SEAB wants you to build a cumulative frequency curve from grouped data and read off the median, the lower and upper quartiles, the interquartile range, and percentiles. The curve turns grouped data into estimates of position and spread that a frequency table alone cannot give.

The answer

Cumulative frequency

Cumulative frequency is a running total of the frequencies, the number of values up to and including the upper boundary of each class. You build it by adding each class frequency to the sum of all the earlier ones, so the final cumulative frequency equals the total number of values.

The cumulative frequency curve

Plot the cumulative frequency against the upper class boundary and join the points with a smooth curve. The result is an S-shaped (ogive) curve that rises from zero to the total frequency, and reading across and down from it estimates positions in the data.

Median and quartiles

For $n$ values, read the curve at these cumulative frequency positions:

• median (middle) at $\dfrac{n}{2}$,
• lower quartile at $\dfrac{n}{4}$,
• upper quartile at $\dfrac{3n}{4}$.

Read across from the position on the vertical axis to the curve, then down to the value on the horizontal axis.

Interquartile range and percentiles

The interquartile range measures spread using the middle half of the data:

A percentile is the value below which a given percentage of the data lies; the $90$th percentile is read at the cumulative frequency $0.9n$. The IQR is less affected by outliers than the full range.

Examples in context

Example 1. Comparing two classes. Drawing two cumulative frequency curves on the same axes lets a teacher compare the medians and interquartile ranges of two classes' marks. A curve further to the right with a smaller IQR shows higher and more consistent scores.

Example 2. Setting a pass mark. To pass the top $30\%$ of candidates, an examiner reads the $70$th percentile off the curve, setting the boundary so that $70\%$ fall below it. Percentiles turn a target proportion into a cut-off value.

Try this

Q1. State the cumulative frequency position for the median of $160$ values. [1 mark]

• Cue. $\dfrac{160}{2} = 80$.

Q2. The lower quartile is $22$ and the upper quartile is $35$. Find the interquartile range. [1 mark]

• Cue. $35 - 22 = 13$.

Q3. For $240$ values, state the position used to read the upper quartile. [1 mark]

• Cue. $\dfrac{3 \times 240}{4} = 180$.