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Median from a Frequency Table

FoundationHigherAQAEdexcelOCR

Median from a Frequency Table is a key statistics topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on finding the median from a frequency table, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Use the running total to locate the middle value.

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

Foundation

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Higher

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Topic overview

Finding the median from a frequency table means locating the middle value once all the data is considered in order.

First find the total frequency, then work out the position of the middle value using \(\frac{n+1}{2}\). With \(n = 25\) values, the median is the \(13\)th.

Then count through the frequencies until you pass that position. The values in a frequency table are already in order, so a running total tells you which value the \(13\)th item falls on without listing everything out.

Revision notes

Finding the position

Add the frequencies to get the total, then use \(\frac{n+1}{2}\) to find the median's position.

For a total of \(31\), the median is the \(16\)th value. For an even total such as \(20\), the position is \(10.5\), meaning the mean of the \(10\)th and \(11\)th values.

Using a running total

Add the frequencies down the table, keeping a cumulative count.

If the running totals are \(4, 11, 22, 28, 31\) and you want the \(16\)th, it falls in the third group, because \(16\) is more than \(11\) but not more than \(22\).

Reading off the answer

The median is the value of the row where the running total first reaches or exceeds the position.

So in the example above, the median is the third value in the table.

Key points

  • Add the frequencies to find the total.
  • The median position is \(\frac{n+1}{2}\).
  • Use a running total down the table.
  • The median is the value where the running total reaches the position.
  • For an even total, average the two middle values.
  • Values in a frequency table are already in order.

Worked examples

Example 1

Frequencies are \(3, 8, 6, 4\). Find the total and the median position.

Working

\[3+8+6+4 = 21\]add the frequencies
\[\frac{21+1}{2} = 11\]use the position formula
\[\text{The 11th value}\]state the position

Example 2

Using running totals \(3, 11, 17, 21\), find which group contains the \(11\)th value.

Working

\[11 \le 11\]the second running total reaches 11
\[\text{The second group}\]the median is the second value

Example 3

A frequency table has total \(20\). Find the median position.

Working

\[\frac{20+1}{2} = 10.5\]use the position formula
\[\text{Average the 10th and 11th values}\]an even total needs two middle values

Common mistakes

  • Finding the median of the frequencies.

    The median is a data value, found using the frequencies, not a median of the frequency column.

  • Using \(\frac{n}{2}\) instead of \(\frac{n+1}{2}\).

    The position formula adds one before halving.

  • Not using a running total.

    Counting through cumulatively is what locates the median without listing all the data.

  • Stopping at the first frequency larger than the position.

    Compare the position against the running total, not the individual frequency.

Exam tips

  • Add a running total column to the table.
  • Work out the position before looking at any values.
  • Compare the position against the running totals, not the frequencies.
  • Check the median is one of the listed values.

Key terms

Median
The middle value when data is ordered.
Running total
A cumulative sum down the table.
Frequency
How often each value occurs.
Position
Where the median sits in the ordered data.

Related topics

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