Cumulative Frequency Curves

Grade 6

What cumulative frequency means

Cumulative frequency is a running total of frequencies. Each value in the cumulative frequency column tells you how many data values are less than or equal to the upper boundary of that class.

You plot the cumulative frequency against the upper class boundary — not the midpoint. That's the most common error.

Building the table and plotting the curve

EXAMPLE
Exam marks: 0 ≤ x < 20 (freq 5), 20 ≤ x < 40 (freq 12), 40 ≤ x < 60 (freq 18), 60 ≤ x < 80 (freq 10), 80 ≤ x < 100 (freq 5). Build a cumulative frequency table.
Up to 20: 5 Up to 40: 5 + 12 = 17 Up to 60: 17 + 18 = 35 Up to 80: 35 + 10 = 45 Up to 100: 45 + 5 = 50 (total) Plot points at (20, 5), (40, 17), (60, 35), (80, 45), (100, 50). Join with a smooth S-shaped curve. The curve starts at (0, 0).

Reading off the median

The median is the value at the halfway point of the data. For 50 values, go across from cf = 25 on the y-axis to the curve, then down to the x-axis.

For n values, the median is at n ÷ 2 on the cumulative frequency axis. Lower quartile is at n ÷ 4. Upper quartile is at 3n ÷ 4.

Watch out for the curve shape

The curve must be smooth and increasing — it can never go down. If your plotted points don't form a rising S-shape, check your running total: you may have added incorrectly or plotted against the midpoint instead of the upper boundary.
Nova
Try this one

A cumulative frequency graph is drawn for the heights of 60 plants. A student reads the following values from the graph:

  • lower quartile \( = 24 \) cm
  • median \( = 33 \) cm
  • upper quartile \( = 41 \) cm

(a) Work out the interquartile range. [2]

(b) Explain what the interquartile range tells you about the heights of the plants. [1]

Nova's hint:

This tests the meaning and calculation of the interquartile range.

Ask yourself: 'which two quartiles do I subtract, and which order?' The interquartile range describes how spread out the middle half of the data is — a smaller value means more consistent data.

Try to Solve it with Nova?