Histograms, Frequency Polygons and Ogives

Q: Where do I plot points on an ogive?

At the upper boundary of each class, against the cumulative frequency, starting at zero on the lower boundary of the first class.

Q: How accurate do my ogive readings need to be?

Draw carefully on the grid provided. Memos accept readings within a small range of the expected value.

Q: What is the difference between a histogram and a bar graph?

A histogram shows continuous grouped data, so the bars touch. A bar graph shows separate categories with gaps between bars.

Grouped data is shown with histograms, frequency polygons and ogives. Ogive questions are the most common in Paper 2: you complete a cumulative frequency table, draw the curve and read estimates of the median, quartiles and percentages from it.

Cumulative frequency

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

Example. Times (minutes) taken by 40 learners:

$10 < t \leq 20$ : frequency 4, cumulative 4
$20 < t \leq 30$ : frequency 9, cumulative 13
$30 < t \leq 40$ : frequency 15, cumulative 28
$40 < t \leq 50$ : frequency 8, cumulative 36
$50 < t \leq 60$ : frequency 4, cumulative 40

The last cumulative frequency must equal the total number of data values. That is your check.

Drawing and reading ogives

An ogive (cumulative frequency curve) plots cumulative frequency against the upper boundary of each class.

For the data above, plot:

$(10; 0), (20; 4), (30; 13), (40; 28), (50; 36), (60; 40)$

Start at the lower boundary of the first class with cumulative frequency 0.
Join the points with a smooth curve or straight lines (follow the question).
Label both axes.

Reading off. To find how many values are below a given $x$ , go up from $x$ to the curve and across to the vertical axis. To find a percentage above a value, subtract from the total first.

Histograms and frequency polygons. A histogram has touching bars. When the classes have equal widths, as in IEB questions, the bar heights are the frequencies. A frequency polygon joins the midpoints of the tops of the bars.

Estimating the median and quartiles

With $n$ values, read from the ogive at these cumulative frequencies:

$Q_{1}$ : $\frac{n}{4}$
Median: $\frac{n}{2}$
$Q_{3}$ : $\frac{3 n}{4}$

Example. With $n = 40$ , the median is at cumulative frequency 20. It lies in the $30 < t \leq 40$ class. Reading from the ogive, or estimating within the class:

$median \approx 30 + \frac{20 - 13}{15} \times 10 \approx 34, 7 minutes$

Similarly, $Q_{1} \approx 26, 7$ minutes and $Q_{3} \approx 42, 5$ minutes.

Estimates read from a hand-drawn graph will differ a little. Memos usually accept a small range.

Key terms

Cumulative frequency: The running total of frequencies.
Ogive: A graph of cumulative frequency against upper class boundaries.
Frequency polygon: A line graph joining the midpoints of the tops of histogram bars.
Class boundary: The value that separates two neighbouring classes.

Common mistakes

Plotting cumulative frequency at the class midpoint instead of the upper boundary.
Not starting the ogive at zero on the lower boundary of the first class.
Reading the median at $n$ instead of $\frac{n}{2}$ .
Forgetting to subtract from the total when asked how many are above a value.

Practice questions

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

Question 1

3 marks total

knowledge

The times taken by 40 learners to complete a task are:

$10 < t \leq 20$ : 4
$20 < t \leq 30$ : 9
$30 < t \leq 40$ : 15
$40 < t \leq 50$ : 8
$50 < t \leq 60$ : 4

Complete the cumulative frequency column.

Worked solution

Hidden - try the question first

Question 2

5 marks total

routine

Use the table in Question 1.

(a) Write down the coordinates of the points you would plot to draw the ogive. (3)

(b) Draw the ogive. (2)

Worked solution

Hidden - try the question first

Solution

(a) $(10; 0)$ as the starting point
$(20; 4)$ , $(30; 13)$ , $(40; 28)$
$(50; 36)$ , $(60; 40)$
(b) Points plotted at upper boundaries
Smooth curve, axes labelled

Question 3

6 marks total

complex

Use the ogive from Question 2 to estimate:

(a) the median time (2)

(b) the lower and upper quartiles (3)

Worked solution

Hidden - try the question first

Solution

(a) At cumulative frequency 20
median $\approx 34, 7$ minutes
(b) $Q_{1}$ at cumulative frequency 10: $\approx 26, 7$ minutes
$Q_{3}$ at cumulative frequency 30
$\approx 42, 5$ minutes
(c) IQR $\approx 42, 5 - 26, 7 = 15, 8$ minutes

Question 4

8 marks total

problem-solving

Use the same data.

(a) Estimate how many learners took more than 45 minutes. (2)

(b) What percentage of learners is this? (1)

(d) Is the data roughly symmetric? Compare your mean with the median to decide. (2)

Worked solution

Hidden - try the question first

Solution

(a) At $(10; 0)$ , cumulative frequency $\approx 32$
$40 - 32 = 8$ learners
(b) $\frac{8}{40} = 20%$
(c) Midpoints 15, 25, 35, 45, 55: $\sum f x = 60 + 225 + 525 + 360 + 220 = 1390$
$\overset{x}{ˉ} \approx \frac{1 390}{40} = 34, 75$ minutes
(d) The mean (34,75) and median (about 34,7) are almost equal
so the data is roughly symmetric

FAQs

Where do I plot points on an ogive?

At the upper boundary of each class, against the cumulative frequency, starting at zero on the lower boundary of the first class.

How accurate do my ogive readings need to be?

Draw carefully on the grid provided. Memos accept readings within a small range of the expected value.

What is the difference between a histogram and a bar graph?

A histogram shows continuous grouped data, so the bars touch. A bar graph shows separate categories with gaps between bars.

Cumulative frequency

Drawing and reading ogives

Estimating the median and quartiles

Key terms

Common mistakes

Practice questions

FAQs

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