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S2.1StatisticsFoundation

Measures of Central Tendency

A measure of central tendency is a single value that represents the “middle” of a data set. The three you need are the mean, median and mode — and knowing which one to use is as important as calculating it.

25 min Measures of Location and Spread
Edexcel AS Level Maths: 2.1 Measures of Central TendencyCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Calculate the mean, median and mode
  • Find these from a frequency table
  • Choose the most appropriate average
  • Estimate the mean of grouped data
1

Mean, median, mode

The mean\textbf{mean} is the total divided by how many values; the median\textbf{median} is the middle value when ordered; the mode\textbf{mode} is the most frequent value.

xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
Mean = sum of values ÷ number of values.
Example 1Find the mean of 4, 7, 7, 10, 12.
1Sum =40= 40, count =5= 5.
Answer88
2

From a frequency table

With frequencies, the mean uses ∑fx÷∑f\sum fx \div \sum f. The median is the (n+12)\left(\tfrac{n+1}{2}\right)th value (use the cumulative frequencies to locate it), and the mode is the value with the highest frequency.

xˉ=∑fx∑f\bar{x} = \dfrac{\sum fx}{\sum f}
Multiply each value by its frequency, then divide by the total frequency.
3

Grouped data and choosing an average

For grouped data, use the midpoint\textbf{midpoint} of each class as xx to estimate the mean. Choose the median\textbf{median} when data has extreme values (it is not distorted by outliers), the mode\textbf{mode} for categorical data, and the mean\textbf{mean} when data is fairly symmetric.

Tip — Outliers present? The median is the safest average — the mean gets dragged towards extremes.

Formula recap

xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
Mean of a list.
xˉ=∑fx∑f\bar{x} = \dfrac{\sum fx}{\sum f}
Mean from a frequency table.
median position=n+12\text{median position} = \dfrac{n+1}{2}
Locating the median.

Common mistakes to avoid

Forgetting to weight by frequency in a frequency table.
Use Σfx/Σf, not just the mean of the distinct values.
Using the mean when there are big outliers.
The median resists outliers; prefer it for skewed data.

Key takeaways

  • Mean = Σx/n (or Σfx/Σf for tables); median = middle value; mode = most frequent.
  • For grouped data, use class midpoints to estimate the mean.
  • Median is best with outliers; mode for categories; mean for symmetric data.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Measures of Central Tendency. Use the feedback to decide what needs another look.

4 past-paper questions · 6 marks

Practise this topic

Related lessons

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