Edexcel GCSE Statistics Summarising data: averages, spread, standard deviation, standardised scores, skewness and outliers

What this topic demands

Summarising data is about reducing a data set to a few meaningful numbers: an average (central tendency) and a measure of spread (dispersion), plus a description of its shape (skewness and outliers). Edexcel tests accurate calculation, the right choice of measure for the context, and clear interpretation. The biggest calculation levers are the grouped mean, the median by interpolation, the IQR, and (at Higher tier) the standard deviation and standardised score.

This guide covers the four dot-point pages on summarising data, then sets out the exam patterns Edexcel repeats.

Measures of central tendency

Measures of central tendency covers the mode, median and mean for discrete and grouped data. The mean of frequency data is $\frac{\sum fx}{\sum f}$; the grouped mean uses class midpoints (an estimate); the grouped median uses linear interpolation. At Higher tier you also meet the weighted mean $\frac{\sum wx}{\sum w}$ and the geometric mean, and the effect of changes and transformations on each average.

Measures of dispersion

Measures of dispersion covers the range, quartiles, the interquartile range $IQR = Q_3 - Q_1$, percentiles, and (at Higher) the interpercentile and interdecile range. The IQR pairs with the median because it ignores extreme values; pairing the mean with the IQR is treated as wrong.

Standard deviation and standardised scores

Standard deviation and standardised scores (Higher tier) covers the standard deviation, the typical distance of values from the mean, using the formulae sheet. The mean pairs with the standard deviation. The standardised score $\frac{x - \mu}{\sigma}$ rescales a value to standard-deviation units so values from different distributions can be compared; it is not on the formulae sheet.

Skewness and outliers

Skewness and outliers covers determining skewness by inspection and (Higher) by the formula $\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}$, interpreting positive and negative skew, and identifying outliers using the $1.5 \times IQR$ rule or the $\mu \pm 3\sigma$ rule, then commenting on them in context.

How this topic is examined

A typical Edexcel profile:

• Averages. Mean of frequency and grouped data, and median by interpolation.
• Spread. Range, IQR, and interpercentile range, with the correct pairing.
• Standard deviation. Calculating it from a list or grouped data, and standardising scores.
• Shape. Skewness from the averages or formula, and outlier tests.

Check your knowledge

Attempt these under timed conditions, then check against the solutions.

1. Find the mean of values $3, 5, 7, 9$. (1 mark)
2. State which average is best for data with extreme outliers. (1 mark)
3. A data set has $Q_1 = 20$, $Q_3 = 32$. Find the IQR. (1 mark)
4. The formula for the standardised score is what? (1 mark)
5. A data set has mean $60$, median $55$, standard deviation $10$. Find the skewness. (2 marks)
6. State the upper outlier boundary for $Q_3 = 40$, $IQR = 12$. (2 marks)
7. Which measure of spread pairs with the median? (1 mark)
8. After taking the square root, what does the standard deviation measure? (1 mark)