# Find the Mode and Median from a Frequency Table

In this worksheet, students practise finding the median and mode from 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

When we are finding averages, we should easily be able to find the median and mode from a list but it gets a bit more complicated when we are dealing with frequency tables.

Definitions.

Median: The value that is in the middle position of a list of ordered data.

Mode: The value with the highest frequency.

The good news when we are dealing with the median is that in a frequency table, the data is already ordered.

Example 1: Find the median and modal shoe size for this frequency table.

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

Mode: The mode is defined as the value with the highest frequency. In this table, the highest frequency is 28.

This means the modal shoe size is size 8.

Median: We need to find the number in the middle position, to do this, we first need to find the position of this number.

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: Find the median and modal groups for this 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

Mode: The mode is defined as the value with the highest frequency. In this table, the highest frequency is 11.

This means the modal pocket money is 20 ≤ p <30

Median: We need to find the number in the middle position, to do this, we first need to find the position of this number.

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 20 ≤ p <30 starts at the 21th number and ends at the 30th number.

The 23rd number must therefore be in the column 20 ≤ p <30 - This is the median group

The mode for a frequency table is the column with the...

To find the median for a frequency table by...

I survey a group of students about how many pets they have.

I display the data in a frequency table.

 Pets 0 1 2 3 4 5 Frequency 3 7 9 6 2 1

Which of these is the mode of this data?

2

9

I survey a group of students about how many pets they have.

I display the data in a frequency table.

 Pets 0 1 2 3 4 5 Frequency 3 7 9 6 2 1

Find the position of the median and the value of the median

 Median Position Value

I survey a group of adults about how many children they have.

I display the data in a frequency table.

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

What is the modal number of kids in a family?

I survey a group of adults about how many children they have.

I display the data in a frequency table.

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

What is the median number of kids in a family?

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 27 26 11

What is the modalgroup for the height of the trees?

40 ≤ h ≤ 60

60 ≤ h ≤ 80

80 ≤ h ≤ 100

100 ≤ h ≤ 120

120 ≤ h ≤ 140

140 ≤ h ≤ 160

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 27 26 11

What is the median group for the height of the trees?

40 ≤ h ≤ 60

60 ≤ h ≤ 80

80 ≤ h ≤ 100

100 ≤ h ≤ 120

120 ≤ h ≤ 140

140 ≤ h ≤ 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 3

Find the modal group for the bike tyres.

150 < t < 200

200 < t < 250

250 < t < 300

300 < t < 350

350 < t < 400

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 3

Find the median group for the bike tyres.

150 < t < 200

200 < t < 250

250 < t < 300

300 < t < 350

350 < t < 400

• Question 1

The mode for a frequency table is the column with the...

EDDIE SAYS
Nice and simple, the definition for mode is 'the value with the highest frequency' All we need to do is look for the largest frequency and read off the value in the other column.
• Question 2

To find the median for a frequency table by...

EDDIE SAYS
The biggest mistake people make on this is counting up, adding 1 and halfing but then putting this down as the median. This is only the position of the median.
• Question 3

I survey a group of students about how many pets they have.

I display the data in a frequency table.

 Pets 0 1 2 3 4 5 Frequency 3 7 9 6 2 1

Which of these is the mode of this data?

2
EDDIE SAYS
The highest frequency is 9, but this represents 2 pets 9 times. The mode is therefore the 2 and not the 9.
• Question 4

I survey a group of students about how many pets they have.

I display the data in a frequency table.

 Pets 0 1 2 3 4 5 Frequency 3 7 9 6 2 1

Find the position of the median and the value of the median

 Median Position Value
EDDIE SAYS
If we add up all the frequencies, we get 28. The position of the median is (28 + 1) ÷ 2 = 14.5 Don't worry that this is a decimal. All this means is we are looking for the number between the 14th and 15th numbers. If we count through the frequencies, we find both the 14th and 15th numbers are in the 2 pet column.
• Question 5

I survey a group of adults about how many children they have.

I display the data in a frequency table.

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

What is the modal number of kids in a family?

3
EDDIE SAYS
The modal number is just the number that has the highest frequency. For this table, 84 is the highest frequency, so the mode is...
• Question 6

I survey a group of adults about how many children they have.

I display the data in a frequency table.

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

What is the median number of kids in a family?

3
EDDIE SAYS
The 94th number in this list is the median value. If we count through the frequencies, we get to the 94th in the 2 kids column.
• 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 27 26 11

What is the modalgroup for the height of the trees?

100 ≤ h ≤ 120
EDDIE SAYS
Once again, we\'re just looking for the column with the highest frequency. The highest frequency is 27 which makes the modal group...
• 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 27 26 11

What is the median group for the height of the trees?

80 ≤ h ≤ 100
EDDIE SAYS
We have 100 pieces of information in this list so the median is in the 50.5th position. We count through the frequencies and find the 50.5th number in the group...
• 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 3

Find the modal group for the bike tyres.

200 < t < 250
EDDIE SAYS
The highest frequency here is 45. This gives us the modal group...
• 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 3

Find the median group for the bike tyres.

200 < t < 250
EDDIE SAYS
As we have 100 pieces of data, we are trying to find the group that has the 50.5th number. After the first column, we get to 24. After the second column, we get to 24 + 45 = 69 Because we hadn't gone past it in the first column and we have after the second. The median group must be...
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