Read data and averages without losing the denominator
Calculate mean, median and weighted averages, then distinguish what a small dataset does and does not establish.
In this guide
Read the labels before the values
A table or chart is meaningful only with its units, categories and time period. A column headed 'registrations' counts registrations, which may differ from unique people or actual attendance. A total for one month cannot automatically be compared with a total for a whole year. Before calculating, restate the question as an operation on labelled quantities: for example, 'Tuesday attendance divided by total attendance across the listed days.' This prevents a correct division from using the wrong denominator.
Worked example: three ways to describe one list
Five volunteers record these travel times in minutes: 12, 15, 15, 18 and 40. The mean is (12 + 15 + 15 + 18 + 40)/5 = 100/5 = 20 minutes. The median is the middle value after ordering, or 15 minutes. The mode is also 15 because it occurs most often. The range is 40 - 12 = 28 minutes. Each statistic answers a different question; a mean of 20 does not mean most volunteers travelled for exactly 20 minutes.
An extreme value can pull the mean
If the final travel time in that list were 90 instead of 40, the mean would become 150/5 = 30 minutes, while the median would remain 15. The median's stability here makes it useful for describing the middle observation when one value is unusually large. That does not make the mean wrong. For estimating the total travel minutes across these five people, the mean multiplied by five recovers the total exactly. Choose a statistic that matches the purpose rather than memorizing one as universally best.
Combine groups with their sizes
A group of eight students has a mean score of 70, and a second group of 12 has a mean of 85 on the same assessment. Their score totals are 8 × 70 = 560 and 12 × 85 = 1,020. The combined mean is (560 + 1,020)/(8 + 12) = 1,580/20 = 79. Simply averaging 70 and 85 would give 77.5 and incorrectly assign equal weight to groups of different sizes. The group sizes supply the necessary weights.
Stay within the evidence
Suppose attendance rises after a club introduces reminder messages. The table shows a sequence in time; it does not by itself prove the messages caused the rise. A new venue, a different activity or seasonal availability might also matter. Likewise, a survey of volunteers from one club cannot automatically describe all Filipino students. Sample selection and missing data affect how far a conclusion can reach.
- Order values before selecting a median; for an even count, average the two middle values.
- Distinguish an absolute increase from a percentage increase.
- Check whether chart axes begin at zero or use a shortened range before judging visual size.
Try it yourself
Find the mean, median and mode of 9, 11, 14, 18, 18 and 20. Then combine a group of six participants with a mean of 80 and another group of four with a mean of 65. What is the overall mean, and why should the two group means not receive equal weight?
Reveal the worked answer
For the ordered values 9, 11, 14, 18, 18 and 20, the mean is 90/6 = 15, and the median is (14 + 18)/2 = 16. The mode is 18. Next, combine six participants with a mean of 80 and four with a mean of 65. Their total is 480 + 260 = 740, so the overall mean is 740/10 = 74. Explain why averaging 80 and 65 directly would answer a different, equally weighted comparison.
For the ordered values 9, 11, 14, 18, 18 and 20, the mean is 90/6 = 15, and the median is (14 + 18)/2 = 16. The mode is 18. Next, combine six participants with a mean of 80 and four with a mean of 65. Their total is 480 + 260 = 740, so the overall mean is 740/10 = 74. Explain why averaging 80 and 65 directly would answer a different, equally weighted comparison.
Put one idea into practice.
Try a short mathematics session, then use the explanations to choose what to review next.
Start a short sessionSources & further reading
OpenStax Introductory Statistics 2e — Measures of the center of the dataSomething unclear or incorrect? Suggest a correction to this guide.
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