Averages from a grouped table

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Key points about averages from a grouped table

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  • For , which contains lots of unique values, it is usually more efficient to group the data. This allows for fewer classes and categories of data to be used.

  • are often used to describe the groups in the table. For example, the interval 0 < 𝑡 ≤ 5 means that 𝑡 is larger than 0 but less than or equal to 5.

  • If data is organised into groups, the exact value of each item of data is not known, just which group it belongs to.

    • This means the exact value for the , or cannot be found. However, the modal group and the group that contains the median can still be identified.

  • An for the mean can be found by assuming all data values within a group take the of the group.

To support you in finding averages from a grouped frequency table, make sure you are confident in working with averages from an ungrouped table.

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Check your understanding

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How to find the mode and median from a grouped frequency table

The method for finding the mode and median for a grouped frequency table is no different from the method for an ungrouped table.

To find the mode in a grouped frequency table, identify the row with highest frequency.

The modal class is the group or interval the highest frequency relates to.

To find the median in a frequency table, use the \( \frac{n+1}{2} \) to find the position of the median, where 𝑛 is the number of pieces of data.

Add up the frequencies in the frequency table to identify which group or interval contains the median.

Find out more below

GCSE exam-style questions

  1. The table shows the time Simon takes to get home from school over a term.

Find the modal class.

A two‑column table with purple headers labelled ‘Time (minutes)’ and ‘Frequency’. Six grouped time intervals with their corresponding frequencies are listed: • ‘16 < t ≤ 18’ — frequency 3 • ‘18 < t ≤ 20’ — frequency 8 • ‘20 < t ≤ 22’ — frequency 10 • ‘22 < t ≤ 24’ — frequency 11 • ‘24 < t ≤ 26’ — frequency 2 • ‘26 < t ≤ 28’ — frequency 1

  1. The table shows the time Simon takes to get home from school over a term.

Which group contains the median time?

A two‑column table with purple headers labelled ‘Time (minutes)’ and ‘Frequency’. Six grouped time intervals and their frequencies are listed: • ‘16 < t ≤ 18’ — frequency 3 • ‘18 < t ≤ 20’ — frequency 8 • ‘20 < t ≤ 22’ — frequency 10 • ‘22 < t ≤ 24’ — frequency 11 • ‘24 < t ≤ 26’ — frequency 2 • ‘26 < t ≤ 28’ — frequency 1

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How to estimate the mean for a grouped frequency distribution

Since no individual values are known for each piece of data in a grouped frequency table, the exact mean cannot be calculated. Instead, an estimate of the mean can be found using midpoints.

To find the estimated mean, when data is presented in a grouped frequency table, use the following method:

  1. Add two additional columns to the frequency table called Midpoints (𝑚) and 𝑓 × 𝑚.

  2. Complete the midpoint column by finding the midpoint of each group interval. This can be found by adding the lower and upper bounds of the group and dividing the total by two.

  3. Multiply each frequency, 𝑓, by its midpoint, 𝑚. Write these values in the new column, 𝑓 × 𝑚.

  4. Add up the total of the 𝑓 × 𝑚 column.

  5. Add up the total of the frequency column, 𝑓.

  6. Divide the total from Step 4 by the total from Step 5.

In some exam questions, the additional columns may already have been created.

Find out more below

GCSE exam-style questions

  1. Alice is playing a game where she listens to a piece of music and guesses the artist.

The table shows the time it takes in seconds for each of her 50 correct guesses.

Estimate the mean time for her correct guesses.

A two‑column table with purple headers labelled ‘Time (sec)’ and ‘Frequency’. Four grouped time intervals with their frequencies are listed: • ‘0 < t ≤ 2’ — frequency 8 • ‘2 < t ≤ 4’ — frequency 21 • ‘4 < t ≤ 6’ — frequency 11 • ‘6 < t ≤ 8’ — frequency 10

  1. Ben is measuring the height (cm) of 100 sunflowers.

The grouped frequency table shows the results.

Estimate the mean height.

A two‑column table with purple headers labelled ‘Height (cm)’ and ‘Frequency’. Five grouped height intervals with their frequencies are listed: • ‘150 < s ≤ 200’ — frequency 28 • ‘200 < s ≤ 250’ — frequency 47 • ‘250 < s ≤ 300’ — frequency 19 • ‘300 < s ≤ 350’ — frequency 4 • ‘350 < s ≤ 400’ — frequency 2

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Quiz - Averages from a grouped table

Practise what you've learned about averages from a grouped table with this quiz.

Now you've revised averages from a grouped table, why not look at Higher – Venn diagrams?

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