4.7.1 Cumulative frequency

Cumulative frequency refers to the total number of times that a particular value or a range of values occurs in a dataset.

It is calculated by adding up the frequencies of all the values up to a certain point in the data.
For example, if you have a set of test scores for a class of \( 30 \) students, and the scores range from \( 50 \) to \( 100 \), with intervals of \( 10 \), you could construct a frequency table to show how many students received each score.
The cumulative frequency for a particular score is the total number of students who scored that value or lower.

To draw a cumulative frequency diagram

Create a frequency table that shows the data values and their corresponding frequencies.
Add a new column to the frequency table for the cumulative frequency.
- Calculate the cumulative frequency by adding up the frequencies for each data value, as well as the frequencies for all the values before it.
Plot the cumulative frequency on the \( y \)-axis of a graph and the data values on the \( x \)-axis.
Mark the cumulative frequency for each data value with a point on the graph.
Join the points with a line, starting from the left-hand side of the graph.

Draw a cumulative frequency diagram for the following set of data:

Calculate the cumulative frequency table

Plot the data on a graph

Mark the data values on the \( x \)-axis and the cumulative frequency on the \( y \)-axis.

Mark the cumulative frequency for each data value with a point

Join the points with a smooth line

To calculate median from cumulative frequency diagram

Find the position of the median
- For \( n \) data values this will be \( \frac{n}{2} \)
Draw a horizontal line from \( \frac{n}{2} \) on the cumulative frequency axis until it hits the curve.
Draw a vertical line from the curve to the horizontal axis and take a reading
- This reading is the median.

To calculate upper quartile from cumulative frequency diagram

Find the position of the upper quartile
- For \( n \) data values this will be \( \frac{3n}{2} \)
Draw a horizontal line from \( \frac{3n}{2} \) on the cumulative frequency axis until it hits the curve.
Draw a vertical line from the curve to the horizontal axis and take a reading
- This reading will be the upper quartile.

To calculate lower quartile from cumulative frequency diagram

Find the position of the upper quartile
- For \( n \) data values this will be \( \frac{n}{4} \)
Draw a horizontal line from \( \frac{n}{4} \) on the cumulative frequency axis until it hits the curve.
Draw a vertical line from the curve to the horizontal axis and take a reading
- This reading will be the lower quartile.

To calculate \( p^{th} \) percentile from cumulative frequency diagram

Percentiles split the data into \( 100 \) parts
Find the position of the percentile
- For \( n \) data values this will be \( \frac{np}{100} \)
Draw a horizontal line from \( \frac{np}{100} \) on the cumulative frequency axis until it hits the curve.
Draw a vertical line from the curve to the horizontal axis and take a reading
- This reading will be the \( p^{th} \) percentile.