Statistical interpretation: compare groups before combining them
Check group composition, spread and sampling before turning an average or percentage into a broad conclusion.
In this guide
A summary number needs a defined comparison
An average or percentage compresses many observations. To interpret it, identify who or what was counted, the time period, the denominator and the conditions under which the observations arose. Two percentages can use different mixes of easy and difficult cases. Two equal means can conceal very different spreads. A large sample can still overrepresent people likely to respond. These are separate issues: arithmetic accuracy does not establish comparability, representativeness or a causal explanation. Start by defining the claim the data are supposed to support, then ask what evidence that claim requires.
Read within-group and overall results separately
Combining groups gives each group influence according to its size. A team handling many easy tasks can have a high overall completion rate even if another team performs better within each difficulty category. This is not a contradiction in arithmetic. It means the overall rates answer a different question from the within-category rates. A fair comparison may need consistent case definitions and information about the distribution of case types. The right analysis depends on the intended question; blindly discarding either the overall or grouped results loses information.
Worked example 1: an overall ranking reverses
Two fictional teams handle tasks classified in advance as easy or hard. Team A completes 36 of 40 easy tasks and 12 of 20 hard tasks. Team B completes 19 of 20 easy tasks and 26 of 40 hard tasks. Every task belongs to one category. Compare completion rates within the categories and across all tasks.
- For easy tasks, A completes 36/40 = 90%, while B completes 19/20 = 95%. B has the higher rate.
- For hard tasks, A completes 12/20 = 60%, while B completes 26/40 = 65%. B again has the higher rate.
- Overall, A completes 48 of 60 tasks, or 80%. B completes 45 of 60 tasks, or 75%. A now has the higher rate.
- A's workload contains more easy tasks; B's contains more hard tasks. The overall percentages mix performance with workload composition. They do not erase the within-category results.
Worked example 2: equal means, unequal spread
Two small populations have values 48, 50, 52 and 20, 50, 80. Both have mean 50. To compare their spread, use population variance: the mean of the squared deviations from the population mean. The word population matters because the standard sample-variance estimator uses a different denominator. No distributional shape needs to be assumed merely to calculate these descriptive values.
- For the first population, deviations are -2, 0 and 2. Squared deviations total 8, giving population variance 8/3 and standard deviation about 1.63.
- For the second, deviations are -30, 0 and 30. Squared deviations total 1,800, giving variance 600 and standard deviation about 24.49.
- The equal means describe equal centers, not equal consistency. The second population has much larger departures from its mean.
- Standard deviation has the original measurement unit; variance has its square. Neither number identifies a cause of the variation.
Separate sampling from assignment and measurement
Random sampling concerns how observations enter a study and can support generalization to a defined population. Random assignment concerns how participating cases enter comparison conditions and can support causal interpretation. Neither automatically solves nonresponse, missing data or a flawed measurement. For example, randomly inviting users to answer a survey does not make the eventual respondents representative if response depends strongly on their experiences. Describe the actual responding group and investigate the response process before treating its average as the population's average.
Misconception: a larger number settles the explanation
A larger overall success rate does not alone establish a better method, just as a larger sample does not automatically remove selection bias. Ask whether plausible alternative explanations were addressed. Also avoid interpreting a small difference between sample means as definitely meaningful without considering variability and study design. This lesson does not provide a universal threshold for significance; a formal uncertainty analysis requires additional assumptions and information.
Try it yourself
A service invites 5,000 people to complete a survey. Most responses come from recent users, while many long-time users do not reply. Does the invitation count alone establish that the answers represent all users? What additional information would help?
Reveal the worked answer
No. A large invitation count does not remove selection or nonresponse differences. Examine response rates and characteristics across relevant user groups, how people were invited, and whether missing respondents may have systematically different experiences. More responses can improve precision within a sample while leaving a biased selection process unresolved.
No. A large invitation count does not remove selection or nonresponse differences. Examine response rates and characteristics across relevant user groups, how people were invited, and whether missing respondents may have systematically different experiences. More responses can improve precision within a sample while leaving a biased selection process unresolved.
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 — Data and samplingOpenStax Introductory Statistics 2e — Measures of spreadSomething unclear or incorrect? Suggest a correction to this guide.
Original FreeReviewer.ph guidance, informed by the linked sources. Exam requirements and procedures should always be checked with the examining body.