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$50≤t<55 | $5$5 | $5$5 |
$55\le t<60$55≤t<60 | $10$10 | $5+10=15$5+10=15 |
$60\le t<65$60≤t<65 | $25$25 | $15+25=40$15+25=40 |
$65\le t<70$65≤t<70 | $26$26 | $40+26=66$40+26=66 |
$70\le t<75$70≤t<75 | $40$40 | $66+40=106$66+40=106 |
$75\le t<80$75≤t<80 | $49$49 | $106+49=155$106+49=155 |
$80\le t<85$80≤t<85 | $28$28 | $155+28=183$155+28=183 |
Total | $183$183 |
The frequency distribution table below shows the heights ($h$h), in centimetres, of a group of children aged $5$5 to $11$11.
Child's height in cm | frequency | cumulative frequency |
---|---|---|
$90 |
$5$5 | $5$5 |
$100 |
$22$22 | $27$27 |
$110 |
$30$30 | $57$57 |
$120 |
$31$31 | $88$88 |
$130 |
$18$18 | $106$106 |
$140 |
$6$6 | $112$112 |
Use the table to answer the following questions:
Solution
Using the values in the cumulative frequency column, we can create a cumulative frequency histogram.
class interval | frequency | cumulative frequency |
---|---|---|
$50\le t<55$50≤t<55 | $5$5 | $5$5 |
$55\le t<60$55≤t<60 | $10$10 | $15$15 |
$60\le t<65$60≤t<65 | $25$25 | $40$40 |
$65\le t<70$65≤t<70 | $26$26 | $66$66 |
$70\le t<75$70≤t<75 | $40$40 | $106$106 |
$75\le t<80$75≤t<80 | $49$49 | $155$155 |
$80\le t<85$80≤t<85 | $28$28 | $183$183 |
Total | $183$183 |
Notice that the columns in a cumulative frequency histogram will always increase in size from left to right. The frequency represented by any particular column will be equal to the difference in height between that column and the one before it.
A cumulative frequency polygon, also known as an ogive, is a line graph connecting cumulative frequencies at the upper endpoint of each class interval. Sometimes the cumulative frequency histogram and polygon are displayed together:
The cumulative frequency polygon can also be displayed on its own.
A principal wants to investigate the performance of students at his school in Performing Arts. To do this, he has the marks of each student studying Performing Arts collected into groups and put into a frequency table. Each group of marks is assigned a grade.
The frequency table for this is shown below.
Complete the cumulative frequency column.
Grade | Score $\left(x\right)$(x) | Frequency $(f)$(f) | Cumulative Frequency $(cf)$(cf) |
---|---|---|---|
$E$E | $0\le x<20$0≤x<20 | $7$7 | $\editable{}$ |
$D$D | $20\le x<40$20≤x<40 | $14$14 | $\editable{}$ |
$C$C | $40\le x<60$40≤x<60 | $32$32 | $\editable{}$ |
$B$B | $60\le x<80$60≤x<80 | $97$97 | $\editable{}$ |
$A$A | $80\le x<100$80≤x<100 | $62$62 | $\editable{}$ |
Calculate the total frequency.
Identify the class size.
Complete the sentence below.
Approximately three quarters of the scores recorded are greater than $\editable{}$.
Consider the table.
Score ($x$x) | Cumulative Frequency ($cf$cf) |
---|---|
$10$10 | $7$7 |
$11$11 | $15$15 |
$12$12 | $18$18 |
$13$13 | $20$20 |
$14$14 | $26$26 |
How many scores were there in total?
How many scores of $14$14 were there?
How many scores of less than $13$13 were there?
Consider the histogram attached.
How many scores were there in total?
How many scores of $46$46 occured?