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Estimating the Mean of Grouped Data

FoundationHigherAQAEdexcel

Learn Estimating the Mean of Grouped Data for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. The mean of grouped data is estimated using the midpoint of each class.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

When data is grouped, the individual values are lost, so the mean can only be estimated using the midpoint of each class.

The midpoint is assumed to represent every value in that class. Multiply each midpoint by its frequency, add these products, and divide by the total frequency. This gives \(\frac{\sum fx}{\sum f}\), where \(x\) is now the midpoint rather than an exact value.

The answer is always an estimate, and the word must be used. The estimate is only exact if the values in each class happen to be evenly spread around the midpoint, which is an assumption rather than a fact. Using the class width or the upper boundary instead of the midpoint is the most common error.

Revision notes

Finding the midpoints

Midpoint \(= \frac{\text{lower boundary} + \text{upper boundary}}{2}\).

For the class \(10 \leq x < 20\), the midpoint is \(\frac{10 + 20}{2} = 15\). Calculate the midpoint for every class before doing anything else.

The calculation

Multiply each midpoint by its frequency to get \(fx\).

Add the \(fx\) column and the frequency column, then divide: estimated mean \(= \frac{\sum fx}{\sum f}\).

Why it is an estimate

The midpoint is assumed to represent every value in the class.

In reality the values may not be evenly spread around it. The answer is therefore an estimate, and questions expect the word to be used explicitly.

Key points

  • Grouped data loses individual values.
  • Use the midpoint of each class.
  • Midpoint = (lower + upper) ÷ 2.
  • Estimated mean = sum of fx ÷ sum of f.
  • The answer is always an estimate.
  • Values are assumed evenly spread in each class.

Worked examples

Example 1

A class \(0 \leq x < 10\) has frequency 6. Work out the fx value for this class. [2 marks]

Working

Midpoint \(= \frac{0 + 10}{2} = 5\)find the class midpoint
\(fx = 5 \times 6 = 30\)multiply the midpoint by the frequency

Example 2

A grouped table gives \(\sum fx = 850\) and \(\sum f = 40\). Work out an estimate of the mean. [2 marks]

Working

\(\frac{850}{40}\)divide the total of fx by the total frequency
\(= 21.25\)work out the estimated mean

Example 3

Explain why the mean of grouped data can only be estimated. [2 marks]

Working

The individual data values are lost when the data is groupedstate what is lost
so the midpoint must be assumed to represent every value in the class, which may not be accurateexplain the assumption

Common mistakes

  • Using the class width instead of the midpoint.

    The midpoint is (lower + upper) ÷ 2.

  • Forgetting to say estimate.

    The word is usually required for the final mark.

  • Dividing by the number of classes.

    Divide by the total frequency.

  • Using the upper boundary as x.

    That would overestimate every class.

Exam tips

  • Calculate all midpoints before starting.
  • Add an fx column to the table.
  • Always write estimate in your answer.
  • Divide by the total frequency, not the number of classes.

Key terms

Midpoint
The middle value of a class interval.
Estimated mean
An approximate mean calculated using midpoints.
Grouped data
Data placed into class intervals.
Assumption
Something taken to be true, here that midpoints represent values.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.