How to find frequency distribution?

Here we will learn about the mean from a frequency table, including what it is and how to calculate it. We will also learn how to find an estimate for the mean from a grouped frequency table.

There are also averages from frequency tables worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

Mean from a frequency table is when we find the mean average from a data set which has been organised into a frequency table.

To calculate the mean we find the total of the values and divide the total by the number of values. The number of values is the total frequency. This can be abbreviated to $n$ .

\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{\text{total}}{n}

We can use an extra column to help.

Mean is a measure of central tendency, it is a value that can be used to represent a set of data.

E.g. The frequency table shows the number of people living in $16$ flats.

There are $5$ flats with $1$ person living there, so we work out $1\times5=5$

There are $6$ flats with $2$ people living there, so we work out $2\times6=12$

There are $3$ flats with $3$ people living there, so we work out $3\times3=9$

There are $2$ flats with $4$ people living there, so we work out $2\times4=8$

The number of values $(n)$ s the total frequency, here $n=16$

\text{Mean}=\frac{\text{total}}{n}=\frac{(1\times 5)+(2\times 6)+(3\times 3)+(4\times 2)}{16}=\frac{34}{16}=2.125

The mean is $2.125$ people

When the data has been grouped together and put into a grouped frequency table we cannot find the actual mean because we only have a range of possible values.

Instead we can find an estimate for the mean using the midpoints of each group. We can add more columns to the grouped frequency table to help.

E.g.
The frequency table shows the marks scored in a test by $20$ students.

There are $3$ people who scored between $0-9$ marks. As we don’t know the exact number of marks that each of these people scored we will use the midpoint of $4.5$ marks.

We can find the midpoint by adding the smallest value and the largest value together and dividing by $2$

\text{Mean}=\frac{\text{total}}{n}=\frac{420}{20}=21

The estimated mean is $21$ marks

In order to calculate the mean from a frequency table:

Multiply the number values by the frequencies.
Find the totals.
Divide the total by n.

Get your free mean from a frequency table worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Get your free mean from a frequency table worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

The frequency table shows the number of goals scored in 10 football matches. Find the mean number of goals.

Multiply the number values by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals.

2Find the totals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

3Divide the total by n.

The last step is to divide the total by $n$ .

\text{Mean}=\frac{\text{total}}{n}=\frac{11}{10}=1.1

The mean is $1.1$ goals

The frequency table shows the number of people on $20$ buses. Find the mean number of people.

Multiply the number values by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals. to help us find the subtotals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

The last step is to divide the total by $n$ .

\text{mean}=\frac{\text{total}}{n}=\frac{191}{20}=9.55

The mean is $9.55$ people

The frequency table shows the ages, in years of $25$ children in a club. Find the mean age.

Multiply the number values by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

The last step is to divide the total by $n$ .

\text{Mean}=\frac{\text{total}}{n}=\frac{123}{25}=4.92

The mean is $4.92$ years

In order to estimate the mean from a grouped frequency table:

Find the midpoints of the groups.
Multiply midpoints by the frequencies.
Find the totals.
Divide the total by n.

The grouped frequency table shows the number of people in $10$ train carriages. Estimate the mean number of people.

Find the midpoints of the groups.

We do not know the actual values of each item in the group, so we use the midpoint. We can find the midpoint by adding the smallest value and the largest value together and dividing by $2$ . We can add a column for the midpoints.

Multiply midpoints by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

The last step is to divide the total by $n$ .

\text{mean}=\frac{\text{total}}{n}=\frac{110}{10}=11

But remember that this is an estimate for the mean.

The estimated mean is $11$ people

The grouped frequency table shows the number of marks in a test taken by $25$ students. Estimate the mean number of marks.

Find the midpoints of the groups.

Multiply midpoints by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

The last step is to divide the total by $n$ .

\text{Mean}=\frac{\text{total}}{n}=\frac{542.5}{25}=21.7

But remember that this is an estimate for the mean.

The estimated mean is $21.7$ marks

The grouped frequency table shows the weight in grams of $30$ potatoes.

Estimate the mean weight.

Give your answer to $1$ decimal place.

Find the midpoints of the groups.

Multiply midpoints by the frequencies.

We can add an extra column next to the frequency column to help us find the subtotals.

Find the total of the frequency column, $n$ . Add up the subtotals to find the total.

The last step is to divide the total by $n$ .

\text{Mean}=\frac{\text{total}}{n}=\frac{1300}{30}=43.333… = 43.3 \ \text{(to 1 dp)}

But remember that this is an estimate for the mean.

The estimated mean is $43.3 g (to 1 dp)$

Check which average you are being asked for

Check if you have been asked for the median, mode or mean average.

Divide by the total frequency

Remember to divide by the total frequency, not by the number of rows in the table. Always check to see if your final answer is sensible.

The mean does not have to be a whole number, it can be a decimal or a fraction.

Classes for grouped frequency tables can be written in different ways

The class intervals used in grouped frequency tables can be written in different ways. Be careful when finding the midpoint.

E.g.

$0$ to $5$

0-5

0\le x <5

Estimated mean from grouped frequency tables

We only use the estimated mean when we have a grouped frequency table because we do not have the actual values for the data.

Frequency distribution table

A frequency distribution table is the same as a frequency table.

Practice mean from a frequency table questions

We need to multiply the numbers by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{Mean}=\frac{\text{total}}{n}=\frac{19}{15}=1.2666… = 1.27 \ \text{(to 3 sf)}

We need to multiply the numbers by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{Mean}=\frac{\text{total}}{n}=\frac{141}{21}=6.714… = 6.71 \ \text{(to 3 sf)}

We need to multiply the numbers by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{Mean}=\frac{\text{total}}{n}=\frac{957}{16}=59.81… = 59.8 \ \text{(to 3 sf)}

We need to use midpoints. Then we multiply the midpoints by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{mean}=\frac{\text{total}}{n}=\frac{53}{17}=3.117… = 3.12 \ \text{(to 3 sf)}

We need to use midpoints. Then we multiply the midpoints by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{Mean}=\frac{\text{total}}{n}=\frac{717}{14}=51.214… = 51.2 \ \text{(to 3 sf)}

We need to use midpoints. Then we multiply the midpoints by the frequencies and find the total. The we divide the total by $n$ , the frequency total

\text{Mean}=\frac{\text{total}}{n}=\frac{71.75}{18}=3.4166… = 3.42 \ \text{(to 3 sf)}

Mean from a frequency table GCSE questions

1. Alex works in a restaurant.

At 1 pm one day she records the number of customers sitting at the tables in the restaurant. Here are the results.

(a) Work out the total number of tables in the restaurant. (b) Work out the total number of customers sitting at the tables in the restaurant.

(5 marks)

(a)
$25$
For the correct answer

(1)

(b)
$(0\times 3)+(1\times 4)+(2\times 10)+(3\times 3)+(4\times 5)$
For the sum of the products

(1)

$=53$
For the correct answer

(1)

$=2.12$
For the correct answer

(1)

2. Here is some information about $10$ buses leaving a station.

Work out an estimate for the mean number of minutes late.

(3 marks)

$5, 15$ and $25$
For the midpoints

(1)

$\frac{(5\times 5)+(3\times 15)+(2\times 25)}{10}$
For the sum of the products divided by 10

(1)

$=\frac{120}{10}=12$
For the correct answer

(1)

3. The table shows the weight, $w$ kg of $90$ bags that people took on a coach trip.

Calculate an estimate for the mean weight of the $90$ bags.
Give your answer to $3$ significant figures.

(4 marks)

$2.5, 7.5, 12.5, 17.5$ and $22.5$
For the midpoints

(1)

$(12\times 2.5)+(19\times 7.5)+(23\times 12.5)+(31\times 17.5)+(5\times 22.5)$
For the sum of the products

(1)

$=\frac{1115}{90}=12.388…$
For the dividing the sum of the products by $90$

(1)

$=12.388…=12.4 \ \text{(to 3 sf)}$
For the correct answer

(1)

You have now learned how to:

Find the mean from a frequency table
Find an estimate for the mean from a grouped frequency table

Cumulative frequency
Histogram
Types of sampling methods

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