 # Explain Median from a Frequency Table

In this worksheet, students practise finding the median from frequency tables and grouped frequency tables. 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

The definition of median is 'the middle number when the numbers are in order,

This is incredibly easy of the numbers are just a list. All you need to do is order them and cross them off from each end.

It does get a bit harder when they are in a frequency table.

To do this we have to follow three steps.

1) Find the total frequency (this is the same as how many numbers would be there if we wrote it out as a list)

2) Find the position of the middle number

3) Work through the frequencies until we find the position we are looking for.

Example 1: I collect information on shoe sizes and summarise it as shown.

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Frequency 5 7 9 14 21 28 15 6 5 3

Step 1: Find the total frequency.

All we need to do is add up all the frequencies. This gives us 113.

Step 2: Find the position of the median.

To find the position of the median, we add 1 to the total ad half it

(113 + 1) ÷ 2 = 57th

This means the median the 57th number in the list

Step 3: Find the value of the median.

To do this, the easiest way is to find the cumulative frequencies then see where the 57th number lies

 Shoe Size 3 4 5 6 7 8 9 10 11 12 Frequency (Cum 5 12 21 35 56 84 99 105 110 113

We can now see that shoe size 7 starts at the 56th number and ends at the 98th number.

The 57th number must therefore be in the shoe size 7 column.

Example 2: I collect information on the pocket money of 45 students and summarise it in a grouped frequency table.

 Pocket Money (p) 0 ≤ p <10 10 ≤ p <20 20 ≤ p <30 30 ≤ p <40 40 ≤ p <50 40 ≤ p <60 Frequency 4 6 11 10 8 6

a) Find the median group and estimate the median pocket money.

To do these two tasks we start the same way as we did before.

Step 1: Find the total frequency.

All we need to do is add up all the frequencies. This gives us 45.

Step 2: Find the position of the median.

To find the position of the median, we add 1 to the total ad half it

(45 + 1) ÷ 2 = 23

This means the median the 23 rd number in the list

Step 3: Find the value of the median.

To do this, the easiest way is to find the cumulative frequencies then see where the 23rd number lies

 Pocket Money (p) 0 ≤ p <10 10 ≤ p <20 20 ≤ p <30 30 ≤ p <40 40 ≤ p <50 40 ≤ p <60 Frequency 4 6 11 10 8 6 Frequency (cum) 4 10 21 31 39 45

We can now see that 30 ≤ p <40 starts at the 21th number and ends at the 30th number.

The 23rd number must therefore be in the column 30 ≤ p <40 - This is the median group

Estimating the median (Highest level only)

To estimate the median, we need to know where in this group the 23rd lies.

We know the group starts at the 21st position and we are looking for the 23rd, so we need to go 2 positions into this group.

The groups starts at 30 and finishes at 40 so we know it is worth 10.

We can therefore go 2/10 or (1/5th into the group).

As the group is 10 wide, we need to find 1/5 of 10 which is 2.

Adding this onto the start of the group gives an extimated median of 30 + 2 = £32

To find the median for a frequency table...

For a grouped frequency table, if we are estimating the median...

I record the number of children in a family who are visiting the dentist.

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

What is the position of the median for this data?

(Just put the number)

94

I record the number of children in a family who are visiting the dentist.

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

What is the median for this data?

(Just put the number)

0

1

2

3

4

5

I survey a group of teachers about how long they spend marking in a week

 Hours marking 1 2 3 4 5 6 Frequency 5 7 14 21 3 1

What position is the median in for this data?

(just put in the number)

I survey a group of teachers about how long they spend marking in a week

 Hours marking 1 2 3 4 5 6 Frequency 5 7 14 21 3 1

What position os the median in for this data?

(just put in the number)

1

2

3

4

5

6

I measure the heights of 100 trees and put the data into a grouped frequency table

 Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160 Frequency 5 9 22 28 25 10

What position is the median in for this data?

(just put in the number)

I measure the heights of 100 trees and put the data into a grouped frequency table

 Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160 Frequency 5 9 22 28 25 10

What group is the median in for this data?

(just put in the number)

40 - 60

60 - 80

80 - 100

100 - 120

120 - 140

140 - 160

I record the amount of hours a bike tyre is ridden before it gets a puncture.

 Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400 Frequency 24 45 18 10 2

Estimate the median for this data. Give your answer to 1 decimal places.

I record the amount of hours a bike tyre is ridden before it gets a puncture.

 Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400 Frequency 24 45 18 10 2

What position is the median in for this data?

(just put in the number)

• Question 1

To find the median for a frequency table...

EDDIE SAYS
Finding the position of the median is the hardest part of this. Remember that the number you get for finding ( n+ 1 )/2 is the position of the median and not the median itself.
• Question 2

For a grouped frequency table, if we are estimating the median...

EDDIE SAYS
You'll only ever be asked to estimate the median in the second half of an exam paper. Remember, find the group just like when you are using a frequency table and then work out what proportion of the group you need to go into.
• Question 3

I record the number of children in a family who are visiting the dentist.

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

What is the position of the median for this data?

(Just put the number)

94
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position
• Question 4

I record the number of children in a family who are visiting the dentist.

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

What is the median for this data?

(Just put the number)

2
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position 3) Work through the frequencies until you pass the position. If I do this for this question, I get to 99 when ive added the 72 of the frequencies. This means the median is 2
• Question 5

I survey a group of teachers about how long they spend marking in a week

 Hours marking 1 2 3 4 5 6 Frequency 5 7 14 21 3 1

What position is the median in for this data?

(just put in the number)

26
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position
• Question 6

I survey a group of teachers about how long they spend marking in a week

 Hours marking 1 2 3 4 5 6 Frequency 5 7 14 21 3 1

What position os the median in for this data?

(just put in the number)

3
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position 3) Work through the frequencies until you pass the position. If I do this for this question, I get to 26 when I've finished adding the frequencies up to 3 hours. This means the median is 3
• Question 7

I measure the heights of 100 trees and put the data into a grouped frequency table

 Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160 Frequency 5 9 22 28 25 10

What position is the median in for this data?

(just put in the number)

50
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position
• Question 8

I measure the heights of 100 trees and put the data into a grouped frequency table

 Height (cm) 40 ≤ h ≤ 60 60 ≤ h ≤ 80 80 ≤ h ≤ 100 100 ≤ h ≤ 120 120 ≤ h ≤ 140 140 ≤ h ≤ 160 Frequency 5 9 22 28 25 10

What group is the median in for this data?

(just put in the number)

100 - 120
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position
• Question 9

I record the amount of hours a bike tyre is ridden before it gets a puncture.

 Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400 Frequency 24 45 18 10 2

Estimate the median for this data. Give your answer to 1 decimal places.

227.8
EDDIE SAYS
We know the median here is in the 50th position. Working through the frequencies tells us that the median is in the second group (200 < t < 250). This group starts at the 25th position and we are looking for the 50th so we need to go 25 positions into this group. The group is worth 45 positions so we need to go 25/45 into the group. The group is worth 50 in total so we are going to go 25/45 x 50 into the group which gives 27.8. The median will therefore be 250 + 27.8 = 277.8 hours
• Question 10

I record the amount of hours a bike tyre is ridden before it gets a puncture.

 Time (hrs) 150 < t < 200 200 < t < 250 250 < t < 300 300 < t < 350 350 < t < 400 Frequency 24 45 18 10 2

What position is the median in for this data?

(just put in the number)

50
EDDIE SAYS
Remember the rule for finding the position of the median. 1) Add up all the frequencies 2) Add one and half it to find the position
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