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Averages and SpreadEdexcel GCSE Maths: Revision notes

Section 1

Mean, Median, Mode and Range

There are three types of average, plus one measure of spread:

MeasureHow to find it
MeanSum of all values ÷ number of values
MedianThe middle value when data is ordered (average of the two middle values if there is an even number of values)
ModeThe value(s) that occur most often
RangeLargest value − smallest value

For data given in a frequency table, the mean is found using: mean = (Σ (value × frequency)) ÷ (total frequency).

Key termsmeanmedianmoderange
Common mistake

Always order the data before finding the median — forgetting to order the values is a very common error.

Section 2

Estimating the Mean from Grouped Data

When data is grouped into class intervals, the exact original values are lost, so the mean can only be estimated, using the midpoint of each class as a representative value.

Method:

  1. Find the midpoint of each class interval
  2. Multiply each midpoint by its frequency
  3. Add these products together and divide by the total frequency

The modal class is simply the class interval with the highest frequency — no midpoint calculation is needed for this.

Key termsmidpointmodal class
Example

For the class 10–20 with frequency 6, the midpoint is (10+20)÷2 = 15, contributing 15 × 6 = 90 to the total.

Section 3

Outliers and Their Effect

An outlier is a value that is unusually far from the rest of the data. Outliers can distort averages and the range significantly:

  • The mean is heavily affected by outliers, since every value contributes to the sum
  • The median is much less affected, since it only depends on the middle position
  • The range is very sensitive to outliers, since it only uses the two extreme values

When a data set contains an outlier, the median (rather than the mean) is often a more representative average.

Key termsoutlier
Exam tip

If a question mentions one unusually high or low value, expect it to test whether you know the mean is more affected than the median.

Section 4

Comparing Data Sets: Averages and Spread

To compare two data sets fully, you must comment on both an average and a measure of spread — one alone is not enough:

  1. Compare an average (mean or median) — which data set is typically higher/lower?
  2. Compare a measure of spread (range or interquartile range) — which data set is more consistent/varied?
  3. Interpret both comparisons in the context of the question

Example: 'Class A has a higher mean score than Class B, but Class A also has a larger range, showing Class A's results are less consistent.'

Common mistake

Comparing only the average and ignoring the spread (or vice versa) will lose marks — always compare both.

Section 5

Interquartile Range and Box Plots

The interquartile range (IQR) = upper quartile (Q3) − lower quartile (Q1), and measures the spread of the middle 50% of the data. It is less affected by outliers than the range.

A box plot displays the minimum, lower quartile, median, upper quartile and maximum. When comparing box plots:

  • A higher median line shows a higher typical value
  • A wider box (bigger IQR) shows greater spread in the middle 50% of the data
  • Skew can be seen from the position of the median within the box and the length of the whiskers on each side
Key termsinterquartile rangebox plot

Must Know

  • Mean = sum ÷ count; median = middle value (order the data first); mode = most frequent value; range = largest − smallest
  • Estimate the mean from grouped data using midpoints: Σ(midpoint × frequency) ÷ total frequency
  • The modal class is the class interval with the highest frequency
  • The mean and range are heavily affected by outliers; the median and IQR are more resistant
  • Always compare BOTH an average and a measure of spread when comparing two data sets
  • IQR = upper quartile − lower quartile; box plots show minimum, Q1, median, Q3, maximum

That's the notes covered.

Carry on to the next subtopic.