15C

Grouped data

Grouping continuous and discrete data into classes, class centres, histograms and polygons for grouped data, and estimating the median from a cumulative frequency polygon.

Grouping data

When there are many different values, group them into classes of equal width. The class centre is the average of the class boundaries, and the modal class has the highest frequency. Heights of $30$ students, in cm:

Height (cm)	150–155	155–160	160–165	165–170	170–175
Class centre	152.5	157.5	162.5	167.5	172.5
Frequency	3	7	12	6	2
Cumulative frequency	3	10	22	28	30

Each class includes its lower boundary but not its upper one, so $160$ goes in 160–165.

Estimating the median

For continuous data, plot each cumulative frequency at the upper boundary of its class (starting from $0$ at the first lower boundary) and join the points with straight lines. Read across from $\dfrac{n}{2}$ to estimate the median.

Cumulative frequency polygon

Each segment rises by the class frequency over the width of the class. With k = 15 (half of 30), the reading is the median estimate.

Try this

Set $k = 15$ and read off the median estimate. What does $k = 22$ give, and why is it a class boundary?

Example 1●●●

Estimate the median height.

$\dfrac{n}{2} = 15$, which falls in 160–165 (cumulative frequency $10$ to $22$).

Median $\approx 160 + \dfrac{5}{12} \times 5 \approx 162.1$ cm

Go 5 of the 12 values into the class, along a class of width 5.

Discrete data can be grouped too, for example scores 0–4, 5–9, 10–14. The class centre of 5–9 is $7$.

Flashcards

Class centre

Average of the class boundaries

Modal class

The class with the highest frequency

Median from an ogive

Read across from $\dfrac{n}{2}$

Practice

Find the class centre of 160–165.
The modal class of the height data is:
How many students are shorter than $165$ cm?
How many students were measured?
Estimate the median height, to 1 decimal place.
How many students are shorter than $160$ cm?
Why is a median from grouped data only an estimate?
Find the relative frequency of the class 165–170.
Scores are grouped as 0–4, 5–9, 10–14. Find the class centre of 5–9.