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 Video by Zeeshan Zamurred Measures of Location and Spread

Edexcel AS Level Maths: 2.1 Measures of Central TendencyWatch the full walkthrough 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

Mean, median, mode

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

\overset{x}{ˉ} = \frac{\sum x}{n}

Mean = sum of values ÷ number of values.

1Sum

= 40

, count

= 5

Answer

8

From a frequency table

With frequencies, the mean uses $\sum f x \div \sum f$ . The median is the $(\frac{n + 1}{2})$ th value (use the cumulative frequencies to locate it), and the mode is the value with the highest frequency.

\overset{x}{ˉ} = \frac{\sum f x}{\sum f}

Multiply each value by its frequency, then divide by the total frequency.

Grouped data and choosing an average

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

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

Formula recap

\overset{x}{ˉ} = \frac{\sum x}{n}

Mean of a list.

\overset{x}{ˉ} = \frac{\sum f x}{\sum f}

Mean from a frequency table.

median position = \frac{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.

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