Ogives (Cumulative Frequency Graphs)

Q: What is an ogive used for?

To estimate the median, quartiles and percentiles of grouped data, and to find how many values are above or below a point.

Q: Why plot at the upper class boundary?

Cumulative frequency counts everything up to the end of each class.

Q: How accurate are ogive readings?

They are estimates. Memos accept answers within a small range of the expected value.

An ogive is a cumulative frequency curve. It lets you estimate the median, quartiles and percentiles of grouped data, and it is one of the most common graphs in Paper 2 statistics.

Cumulative frequency

The cumulative frequency of a class is the total frequency up to the end of that class.

Example. Test marks (%) of 80 learners:

$0 < x \leq 20$ : 4, cumulative 4
$20 < x \leq 40$ : 12, cumulative 16
$40 < x \leq 60$ : 28, cumulative 44
$60 < x \leq 80$ : 26, cumulative 70
$80 < x \leq 100$ : 10, cumulative 80

The last cumulative frequency equals the total.

Drawing ogives

Plot cumulative frequency against the upper boundary of each class, and start at the lower boundary of the first class with 0.

For the data above:

$(0; 0), (20; 4), (40; 16), (60; 44), (80; 70), (100; 80)$

Join the points with a smooth curve. The ogive always rises (or stays level), never falls.

Estimating the median, quartiles and percentiles

With $n$ values, read from the curve at:

$Q_{1}$ : $\frac{n}{4}$ ; median: $\frac{n}{2}$ ; $Q_{3}$ : $\frac{3 n}{4}$
The $k$ th percentile: $\frac{k}{100} \times n$

Example. $n = 80$ :

Median at 40: in $40 < x \leq 60$ , about $40 + \frac{40 - 16}{28} \times 20 \approx 57, 1%$
$Q_{1}$ at 20: about $42, 9%$
$Q_{3}$ at 60: about $72, 3%$
90th percentile at 72: about $84%$

Reading the other way. To find how many learners scored below 40%, go up from 40 on the horizontal axis to the curve: 16 learners.

Key terms

Ogive: A cumulative frequency graph.
Percentile: The value below which a given percentage of the data lies.
Quartiles: The 25th, 50th and 75th percentiles.

Common mistakes

Plotting at midpoints instead of upper boundaries.
Not starting the curve at 0.
Reading the median at $n$ instead of $\frac{n}{2}$ .
Reading off the wrong axis.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

The test marks (%) of 80 learners are:

$0 < x \leq 20$ : 4
$20 < x \leq 40$ : 12
$40 < x \leq 60$ : 28
$60 < x \leq 80$ : 26
$80 < x \leq 100$ : 10

Complete the cumulative frequency column.

Worked solution

Hidden - try the question first

Question 2

5 marks total

routine

Draw the ogive for the data in Question 1. Write down the points you plot.

Worked solution

Hidden - try the question first

Solution

Start point $(0; 0)$
$(20; 4)$ , $(40; 16)$
$(60; 44)$ , $(80; 70)$ , $(100; 80)$
Points plotted at upper boundaries
Smooth increasing curve with labelled axes

Question 3

6 marks total

complex

Use your ogive to estimate:

(a) the median (2)

(b) the lower and upper quartiles (3)

Worked solution

Hidden - try the question first

Solution

(a) Read at cumulative frequency 40
Median $\approx 57, 1%$
(b) $Q_{1}$ at 20: $\approx 42, 9%$
$Q_{3}$ at 60
$\approx 72, 3%$
(c) IQR $\approx 72, 31 - 42, 86 \approx 29, 5$ (round only at the end; graph readings from about 29 to 30 are accepted)

Question 4

8 marks total

problem-solving

Use the same data.

(a) Estimate the 90th percentile and explain what it means. (3)

(b) What percentage of learners scored 40% or less? (2)

Worked solution

Hidden - try the question first

Solution

(a) $(0; 0)$
Reading at 72: about 84%
90% of learners scored 84% or less
(b) 16 learners
$\frac{16}{80} = 20%$
(c) 75% must score at or above the pass mark, so 25% may be below it
This is $Q_{1}$ , at cumulative frequency 20
$Q_{1} \approx 42, 9%$ , so the highest whole-number pass mark is 42%

FAQs

What is an ogive used for?

To estimate the median, quartiles and percentiles of grouped data, and to find how many values are above or below a point.

Why plot at the upper class boundary?

Cumulative frequency counts everything up to the end of each class.

How accurate are ogive readings?

They are estimates. Memos accept answers within a small range of the expected value.

Cumulative frequency

Drawing ogives

Estimating the median, quartiles and percentiles

Key terms

Common mistakes

Practice questions

FAQs

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