# 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:

### QUESTION 1 of 10

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

To find the IQR we...

Find the UQ for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

Is the IQR for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

5

-5

Find the LQ for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

Find the IQR for the data set...

1,2,0,2,5,3,1,0,1,2,3,4,5,1,2

The data  about the shoe sizes of 111 people are shown.

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Number of people 5 7 9 14 21 27 16 6 5 1

The data  about the shoe sizes of 111 people are shown.

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Number of people 5 7 9 14 21 27 16 6 5 1

What is the IQR for this data?

The data  about the amount of kids for 187 families

 Kids 0 1 2 3 4 5 Frequency 6 21 72 84 3 1

The data  about the amount of kids for 187 families

 Kids 0 1 2 3 4 5 Frequency 6 21 72 84 3 1

Is it possible to find the exact IQR from a grouped frequency table?

Yes

No

• Question 1

To find the IQR we...

EDDIE SAYS
IQR means Inter-Quartile-Range Range is always a difference, so we need to subtract.
• Question 2

Find the UQ for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

8
EDDIE SAYS
If we order these numbers, we get... 2,3,3,4,5,6,7,7,8,8,9 We have 11 digits here, so the UQ will be the 9th number. Just count along
• Question 3

Is the IQR for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

5
EDDIE SAYS
If we order these numbers, we get... 2,3,3,4,5,6,7,7,8,8,9 We have found that the UQ = 8 and the LQ = 3, so the IQR = 8 - 3 (Remember that a difference can never be negative - it's just how far they are apart)
• Question 4

Find the LQ for 7, 8, 3, 6, 7, 3, 8, 5, 4, 9, 2

3
EDDIE SAYS
If we order these numbers, we get... 2,3,3,4,5,6,7,7,8,8,9 We have 11 digits here, so the LQ will be the 3rd number. Just count along
• Question 5

Find the IQR for the data set...

1,2,0,2,5,3,1,0,1,2,3,4,5,1,2

2
EDDIE SAYS
Our first order of business here is to order these numbers. 0,0,1,1,1,1,2,2,2,2,3,3,4,5,5 We have 15 pieces of data here, so the LQ is the 4th and the UQ is the 12th. This gives a LQ of 1 and a UQ of 3
• Question 6

The data  about the shoe sizes of 111 people are shown.

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Number of people 5 7 9 14 21 27 16 6 5 1

EDDIE SAYS
If we have 111 people - We add 1 and divide by 4 to find the LQ (28th) - We multiply this by 3 to get the UQ (84th)
• Question 7

The data  about the shoe sizes of 111 people are shown.

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Number of people 5 7 9 14 21 27 16 6 5 1

What is the IQR for this data?

3
EDDIE SAYS
The LQ is in position 28, this is in the column for shoe size 6 The UQ is in position 84, this is in the column for shoe size 9 All we have to do now is find the difference
• Question 8

The data  about the amount of kids for 187 families

 Kids 0 1 2 3 4 5 Frequency 6 21 72 84 3 1
EDDIE SAYS
If we have 187 people - We add 1 and divide by 4 to find the LQ (47th) - We multiply this by 3 to get the UQ (141st)
• Question 9

The data  about the amount of kids for 187 families

 Kids 0 1 2 3 4 5 Frequency 6 21 72 84 3 1
1
EDDIE SAYS
The LQ is in position 47, this is in the column for 2 kids The UQ is in position 141, this is in the column for 3 kids All we have to do now is find the difference
• Question 10

Is it possible to find the exact IQR from a grouped frequency table?

No
EDDIE SAYS
This goes back to the fact that we don\'t know where each value lies in a group. If we don\'t know this, we cant do the subtraction
---- OR ----

Sign up for a £1 trial so you can track and measure your child's progress on this activity.

### What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started