  # Find Quartiles from a Distribution (Interpolation)

In this worksheet, students find the quartiles and inter quartile range from a small data set. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR,

Curriculum topic:   Statistics

Curriculum subtopic:   Statistics Analysing Data

Difficulty level:   #### Worksheet Overview

When you studied cumulative frequency diagrams, you will have come across finding the interquartile range.

The reason we find the interquartile range is to give a measure of consistency between two sets of data.

If we knock out the top 25 % and the bottom 25% we are only looking at how spread out the central 50% of the data is. The lower this value, the more consistent the data is.

How to find the Quartiles.

Quartile just means quarter. If we are doing this in a small data set, we need to find the numbers that are in the position 1/4 and 3/4 of the way along this.

We find the position of the Lower and upper Quartiles by using the formula

LQ =
 n+1 4
UQ =
 3 (n+1) 4

Where n is the number of pieces of data in the list.

Example: If I had a list of 11 pieces of data.

The LQ would be the (11 + 1 / 4) 3rd number

The UQ would be the 3(11 + 1) / 4) 9th number.

Exampl2 1: Finding the IQR

What is the interquartile range of the data set

21, 23, 20, 27, 25, 24,, 30, 21,22, 28,30

Step 1: Order the data

20, 21, 21, 22, 23, 24, 25, 27, 28, 30, 30

Step 2: Find the position of the quartiles.

I have 11 pieces of data, this means the LQ is the 3rd number and the UQ is the 9th

Step 3: Find the value of the LQ and UQ

LQ = 3rd number = 21

UQ = 9th number = 28

Step 4: Find the IQR

IQR = UQ - LQ

= 28 - 21 = 7

Example 2: Finding the IQR from a frequency table

This frequency table shows the results of a survey into the number of people in a car that passes a point.

 People in a car 1 2 3 4 5 6 Frequency 45 198 121 76 52 13

Find the IQR for this data.

Step 1: Find out the total frequency is for this problem,

We just add up all the frequencies for this. This gives us 505 cars

Step 2: Find the position of the quartiles.

Using the formula above, the LQ is the 126.5th number and the UQ is the 379.5th number

Step 3: Finding the values of the quartiles.

This is a bit complicated.

If I wrote the first column out as a list, I would write 45 numbers out. (have I gone past 126.5th yet? - No)

If I then write out the second column as a list I would have 45 + 198 = 243 numbers (Have I gone past 126.5th yet? Yes!)

My LQ must be in the second column so it must be 2

If you do the same for the UQ you find that UQ = 4

Step 4: Find the IQR

IQR = UQ - LQ = 4 - 2 = 2

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