 10.06 Cumulative frequency tables and graphs

Lesson

We have already seen how the frequencies of data values can be used to create a histogram. The cumulative frequencies can also be plotted to create another type of chart, called a cumulative frequency graph. This graph will be used in the next chapter for finding values such as the median and interquartile range from a set of grouped data.

Cumulative frequency is a 'running total' of the frequencies. To calculate it, we add an additional column to the frequency distribution table:

class interval frequency cumulative frequency
$50\le t<55$50t<55 $5$5 $5$5
$55\le t<60$55t<60 $10$10 $5+10=15$5+10=15
$60\le t<65$60t<65 $25$25 $15+25=40$15+25=40
$65\le t<70$65t<70 $26$26 $40+26=66$40+26=66
$70\le t<75$70t<75 $40$40 $66+40=106$66+40=106
$75\le t<80$75t<80 $49$49 $106+49=155$106+49=155
$80\le t<85$80t<85 $28$28 $155+28=183$155+28=183
Total $183$183

• The first value in the cumulative frequency column will always be the same as the first value in the frequency column.
• To get the second cumulative frequency value we add the second frequency to the first cumulative frequency value, $5+10=15$5+10=15.
• The second cumulative frequency value tells us there are $15$15 values in the interval $50\le p<60$50p<60.
• The third cumulative frequency value tells us there are $40$40 values in the interval $50\le p<65$50p<65, and so on.
• The final cumulative frequency value is always equal to the sum of the frequencies. In this case, there are $183$183 values in the entire data set, represented by $50\le p<85$50p<85.

Worked example

example 1

The frequency distribution table below shows the heights ($h$h), in centimetres, of a group of children aged $5$5 to $11$11.

Child's height in cm frequency cumulative frequency
$9090<h100$5$5$5$5$100100<h110 $22$22 $27$27
$110110<h120$30$30$57$57$120120<h130 $31$31 $88$88
$130130<h140$18$18$106$106$140140<h150 $6$6 $112$112

Use the table to answer the following questions:

1. How many children were in the group?
2. How many children had heights greater than $130$130 cm but less than or equal to $140$140 cm?
3. Which class interval contained the most children?
4. How many children had a height less than or equal to $120$120 cm?
5. How many children had a height greater than $130$130 cm?

Solution

1. The final cumulative frequency value tells us there were $112$112 children in the group. This is equal to the sum of the values in the frequencies column.
2. The frequency column indicates there are $18$18 children with heights in the range $130130<h140. 3. The class interval with the highest frequency is$120120<h130.

Outcomes

MS11-2

represents information in symbolic, graphical and tabular form

MS11-7

develops and carries out simple statistical processes to answer questions posed