Averages and charts: revision notes and practice

What you will learn

• Calculate and compare mean, median, mode and range.
• Read discrete values and frequencies from a bar chart.
• Use weighted totals and cumulative frequency positions.
• Choose appropriate rounding and explain what an average describes.

Key rules

Mean

Add all observations and divide by their number. For frequency data, use Σ(value × frequency)/Σfrequency.

Median and mode

Order data for the median. The mode is the most frequent value; ties may give more than one mode.

Range

Subtract the smallest observed value from the largest observed value.

Worked examples

Example 1: Find the mean, median and mode of 2, 4, 4, 6, 9.

1. Total = 25; there are 5 observations, so mean = 5.
2. The ordered middle value is 4.
3. The value 4 appears twice, more often than any other.

Mean 5, median 4, mode 4

Example 2: Values 1, 2 and 3 have frequencies 2, 3 and 1. Find the mean.

1. Weighted total = 1 × 2 + 2 × 3 + 3 × 1 = 11.
2. Total frequency = 2 + 3 + 1 = 6.

11/6, approximately 1.83

A practice question

The recorded values are 10, 7, 14, 10, 16, 12. Find their mean.

1. Distinguish typical value from spread

An average summarises a typical or central value, while a measure of spread describes how varied the observations are. Mean, median and mode are different averages and need not give the same result. Range is a simple measure of spread. State which statistic is requested before calculating; finding the mean when the question asks for the median does not answer the question.

For the data 2, 4, 4, 6, 9, the mean is 5, the median is 4 and the mode is 4. The range is 7. Each number describes a different property of the same five observations. An average does not need to be one of the observed values: nobody in this list has the value 5, yet 5 is still the correct mean.

2. Calculate the ordinary mean

Add all observations, including repetitions, and divide by the number of observations. For 2, 4, 4, 6, 9, the total is 25 and the count is 5, giving mean 25/5 = 5. Count the data carefully before dividing. Repeated values still represent separate observations, so the second 4 contributes to both the total and the count.

Negative data are handled using ordinary arithmetic. The mean of −3, 1 and 5 is (−3 + 1 + 5)/3 = 1. Keep the signs during addition. The mean must lie between the smallest and largest observations when each observation has equal positive weight. A result outside that interval signals a likely addition, counting or division error and gives you a useful quick check.

Mean = sum of observations / number of observations

3. Find the median by ordering data

The median is the middle of the ordered observations. Put the data in ascending order before locating it. With an odd number n of observations, the middle position is (n + 1)/2. For seven observations, that is the fourth value. Sorting does not change the observations or their mean; it makes their positions visible so that the median can be identified correctly.

With an even number of observations, use the mean of the two central values. For 1, 3, 3, 5, 7, 9, the third and fourth values are 3 and 5, giving median 4. Choosing just one of them is incorrect. The median can therefore be a number that does not occur in the list, and it need not equal the mode even when one central value is repeated.

4. Identify mode and calculate range

The mode is the value occurring most often. Count frequencies rather than choosing the greatest numerical value. In 1, 3, 3, 5, 7, 9, the mode is 3 because it occurs twice. If two values tie for the greatest frequency, both can be reported as modes. If every value occurs equally often, the data do not have a single uniquely most frequent value.

The range is the largest observed value minus the smallest observed value. For the same list, range = 9 − 1 = 8. Include negative signs carefully: values from −4 to 7 have range 7 − (−4) = 11. A frequency of zero means a value is not observed, so it should not determine the minimum or maximum when calculating the range of a frequency distribution.

Range = largest observed value − smallest observed value

5. Read a discrete frequency chart

A discrete frequency chart separates the possible observed values from how many times they occurred. The horizontal labels might be 0, 1, 2 and 3 books, while the vertical scale records the number of students with each value. A bar above 2 reaching frequency 4 means four students each reported two books. It does not mean that the observation itself has value 4.

Read the vertical scale and any gaps between tick marks before recording frequencies. Keep the horizontal labels in their correct order and check that categories have not been skipped. The total number of observations is the sum of the frequencies, not the number of bars. These questions use discrete ungrouped values; histogram areas and grouped-class midpoints involve additional ideas not needed for this chart type.

6. Use a weighted total for the mean

For each value, multiply it by its frequency to find its contribution to the total. If values 1, 2 and 3 have frequencies 2, 3 and 1, the weighted total is 1 × 2 + 2 × 3 + 3 × 1 = 11. There are 2 + 3 + 1 = 6 observations, so the mean is 11/6, approximately 1.83.

Dividing by the three rows would give 11/3 and ignore the repeated observations. Taking the mean of the three value labels also ignores their different frequencies. A frequency of zero contributes zero to both the weighted total and the observation count. Write both totals before dividing, and keep the exact fraction or full calculator value until the final rounding instruction has been applied.

Mean = Σ(value × frequency) / Σfrequency

7. Locate the median using cumulative counts

A frequency table need not be expanded into a long list to find the median. First find the total frequency and the middle position or positions. Then accumulate frequencies from the lowest value upward. For values 0, 1, 2 and 3 with frequencies 1, 3, 4 and 2, the cumulative totals are 1, 4, 8 and 10. Positions five and six both contain the value 2.

There are ten observations in that example, so the median is the mean of the fifth and sixth observations, giving 2. If the two middle positions fall in different value rows, average those two values. Do not average their position numbers or frequency heights. The cumulative totals identify where an ordered observation belongs; they are a way of counting, rather than a new set of observed values.

8. Compare summaries and report them accurately

An extreme observation can affect the mean and range strongly. The median depends on central positions and can be less sensitive to one unusually large value. A mode is useful for identifying the most common category or value, but may fail to provide a unique summary. When explaining which average is appropriate, refer to the actual data and purpose rather than treating one measure as always best.

Give the requested accuracy and retain units where relevant. A mean rounded to two decimal places needs two digits after the decimal point, including trailing zeros. State whether a result is exact or rounded, and do not round intermediate weighted totals unnecessarily. Finally, check that the frequencies total the stated observation count and that the median lies among the central ordered values. These checks complement the calculation rather than replacing it.

Check your understanding

How to earn the marks

• Order observations before finding a median.
• For charts, keep values separate from their frequencies.
• Show the weighted total and total frequency before dividing.
• Round the final mean as requested and retain relevant units.

A common mistake

Do not divide a frequency-table total by the number of rows. Each frequency counts observations.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — S2, S4

  Discrete-data averages, range, frequency tables and simple charts are available at both tiers. Histograms and cumulative frequency are outside this bank.

• Edexcel GCSE Mathematics — S2, S4

  Discrete-data averages, range, frequency tables and simple charts are available at both tiers. Histograms and cumulative frequency are outside this bank.

• OCR GCSE Mathematics — 12.02, 12.03

  Discrete-data averages, range, frequency tables and simple charts are available at both tiers. Histograms and cumulative frequency are outside this bank.

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