Linear Interpolation (Edexcel GCSE Statistics)

Revision Note

Author

Roger B

Expertise

Maths

Median from Grouped Data

How do I find the median for grouped data?

Grouped data doesn’t contain the individual data values
- So we can’t find the exact median in the usual way
We can estimate the median for grouped data using linear interpolation
- This assumes the data is evenly spread across the class containing the median
STEP 1
Identify the class interval containing the median
- This is the class containing the ‘ $\frac{n}{2}$ ^th’ data value
  - Divide the total number of values, n, by 2
- e.g. if there are 80 data values, $\frac{n}{2} = \frac{80}{2} = 40$
  - Consider cumulative frequencies until you find the class interval containing the 40^th value
STEP 2
Find ‘how far into’ that class interval the $\frac{n}{2}$ ^th data value is
- e.g. if the interval with the median contains the 36^th through 43^rd data values
  - then the 40^th value is ‘5 values in’ (36, 37, 38, 39, 40)
  - and there are 8 values in the interval
  - so the 40^th value is $\frac{5}{8}$ of the way into the interval
STEP 3
Multiply the class width of the class interval containing the median by the fraction found in Step 2
- e.g. if the interval with the median is $50 \leq x \leq 70$
  - the class width is $70 - 50 = 20$
  - $20 \times \frac{5}{8} = 12.5$
STEP 4
Add the result from Step 3 to the lower boundary of the class interval containing the median
- The result is the estimated median
- e.g. lower bound of $50 \leq x \leq 70$ is 50
  - The estimated median is $50 + 12.5 = 62.5$
The estimated median can also be found using the following formula:
- $estimated median = L + \frac{\frac{n}{2} - C}{f} \times w$
  - L is the lower boundary of the class interval containing the median
  - n is the total number of data values
  - C is the cumulative frequency of all the class intervals before the one containing the median
  - f is the frequency of (i.e. the number of values in) the class interval containing the median
  - w is the width of the class interval containing the median (upper boundary minus lower boundary)
- This formula combines all four steps of the process
  - but it is not given to you on the exam
  - So if you want to use it you’ll need to remember it

Exam Tip

The formula can be tricky to remember correctly
- It’s better to understand how the method works
- Then you don’t need to remember the formula
Remember that the median found this way is an estimate
- You can’t find the exact median without knowing all the data values

Worked Example

A student collected data about the length of time (x hours) students in his school spent listening to music in a given week. He collected data from 50 students in total. The following table summarises the data:

Time spent, x (hours)	Number of students
0 ≤ x ≤ 10	3
10 < x ≤ 20	19
20 < x ≤ 30	12
30 < x ≤ 40	10
40 < x ≤ 50	5
50 < x ≤ 60	1

Work out an estimate for the median amount of time spent listening to music by the students.

STEP 1: Identify the class interval containing the median

Divide the total number of values, n, by 2

Here n = 50

$\frac{50}{2} = 25$

So we’re looking for the interval with the 25^th value

Note that this is different from finding the median from a set of data values

In that case we would be looking for the value halfway between the 25^th and 26^th values

With linear interpolation we don’t have to worry about that!

Add a cumulative frequency column to the table and work out the cumulative frequencies

Time spent, x (hours)	Number of students	Cumulative Frequency
0 ≤ x ≤ 10	3	3
10 < x ≤ 20	19	22
20 < x ≤ 30	12	34
30 < x ≤ 40	10	44
40 < x ≤ 50	5	49
50 < x ≤ 60	1	50

The second class interval goes up to the 22^nd data value and the third class interval goes up to the 34^thdata value

So the median is in the third class interval

The median is in the $20 < x \leq 30$ class interval

STEP 2: Find how far into the class interval the $\frac{n}{2}$ ^th data value is

The interval with the median contains the 23^rd through 34^th data values

The 25^th data value is ‘3 values in’ to the interval (23, 24, 25)

And there are 12 data values in the interval

$\frac{3}{12} = \frac{1}{4}$

So the median is 1/4 of the way into the interval

STEP 3: Multiply the class width by the fraction found in Step 2

Subtract the lower boundary from the upper boundary to find the class width of the $20 \leq x \leq 30$ interval

30-20=10

Multiply by the fraction in Step 2

$10 \times \frac{1}{4} = 2.5$

STEP 4: Add the result from Step 3 to the lower boundary of the class interval

20 + 2.5 = 22.5

Don’t forget the units in your answer!

Estimated median = 22.5 hours

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