Mean, median & mode
The mean is the arithmetic average, the median the middle value once the data are ordered, and the mode the most frequent. They answer different questions: the mean uses every value and so is dragged by outliers, while the median ignores magnitudes and is robust. That is why house prices and salaries are reported as medians and exam marks as means.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Sort the data firstNothing about the median or quartiles works on unsorted data, and sorting also makes the mode visible.
- Mean = total ÷ countΣx/n. Keep it exact as a fraction unless a decimal is requested.
- Median = middle valueFor odd n it is the ((n+1)/2)th value; for even n average the two middle ones.
- Mode = most frequent valueThere may be none, one, or several. Say "bimodal" rather than picking one arbitrarily.
Worked example
For the data 4, 7, 7, 9, 13, 40, find the mean, median and mode, and comment.
- Already sorted. n = 6, sum = 4+7+7+9+13+40 = 80.
- Mean = 80/6 ≈ 13.33.
- Median: average the 3rd and 4th values, (7 + 9)/2 = 8.
- Mode: 7 appears twice, everything else once.
Answer. Mean ≈ 13.33, median 8, mode 7. The mean exceeds every value but one — the outlier 40 has pulled it far above the bulk of the data.
Where marks get dropped
These are the specific errors that cost credit on mean, median & mode questions — QED's rubric penalises each of them separately.
- Computing the median without sorting. The middle of the unsorted list is meaningless.
- Reporting the mean for strongly skewed data without comment. Here it describes no typical observation at all.
- Assuming there is always exactly one mode. Data can be bimodal or have no repeated value at all.
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Mean, median & mode — frequently asked questions
Which average should I report?
Median for skewed data or when outliers are present; mean for roughly symmetric data, since it uses all the information; mode for categorical data where averaging is meaningless.
What happens to each if I add 5 to every value?
Mean, median and mode all increase by 5. If you multiply every value by 3, all three triple — both are consequences of the averages being location measures.
Can the mean be a value that never occurs?
Yes, routinely — the mean of 1 and 2 is 1.5. It is a summary, not an observation.
The rest of Statistics
Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Mean, median & mode
- 2Variance, standard deviation & spread
- 3Shape, skew & outliers
- 4Boxplots, histograms & quartiles
- 5The normal distribution & z-scores
- 6Sampling, bias & the sampling distribution
- 7The central limit theorem
- 8Confidence intervals for a mean
- 9Hypothesis testing & p-values
- 10t-tests & comparing two means
- 11Chi-square tests for independence
- 12Correlation & least-squares regression
- 13Type I / Type II errors & power
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