The core idea
The mean shares a total equally, the median locates the middle of ordered data, and the mode identifies the most frequent value. Measures of spread describe how far observations vary. A useful report combines a suitable centre, a spread measure and the context of collection.
1. Start with what each observation means
A data set is a collection of recorded values. Before calculating, identify the variable, unit, population of interest and collection method. If the variable is travel time to a library, record minutes, not a mixture of minutes and hours. Define whether waiting time is included. Eight measurements of one person's journey answer a different question from one measurement for each of eight people.
Missing data are not automatically zero. A blank travel-time entry means the value is unknown; zero means no elapsed travel time. Check repeated entries, impossible negative durations and inconsistent units before summarising, but do not delete an inconvenient value simply because it changes the answer. Keep a record of any correction and the evidence behind it.
Sources: OpenStax: Introductory Statistics 2e: Measures of the Center of the Data ↗ · OpenStax: Introductory Statistics 2e: Data, Sampling, and Variation in Data and Sampling ↗
2. Solved example: the mean shares the total
Consider these illustrative travel times in minutes: 8, 10, 10, 11, 12, 13, 16, 40. Let n denote the number of observations, so n = 8. The arithmetic mean is the sum of values divided by n. Their sum is 120 minutes, giving 120 ÷ 8 = 15 minutes. The mean is the equal share if all eight journey times were pooled.
No traveller needs to have taken exactly 15 minutes. The mean uses every value, including the 40-minute journey. Without that observation the total would be 80 over seven observations, giving about 11.43 minutes. This comparison demonstrates sensitivity; it is not a reason to discard 40. A missed bus could make that long journey a real and relevant part of the data.
Sources: OpenStax: Introductory Statistics 2e: Measures of the Center of the Data ↗
3. Locate the median and mode
Arrange values from least to greatest before finding the median. With eight observations there are two middle positions, fourth and fifth, containing 11 and 12. Their mean is (11 + 12)/2 = 11.5 minutes. With an odd number of observations there is one middle position. The median describes a central location without being pulled as strongly by how extreme the largest observation is.
The mode is 10 minutes because it appears twice while every other value appears once. A data set may have more than one mode, or no uniquely most frequent value. For a category such as favourite activity, a mean usually has no meaning, but a mode can identify the most common response. Avoid saying simply “the average” when your reader needs to know which summary you used.
Sources: NCERT Class VII Mathematics Exemplar: Data Handling ↗ · OpenStax: Introductory Statistics 2e: Measures of the Center of the Data ↗
4. Measure spread, including the middle half
The range is maximum minus minimum: 40 − 8 = 32 minutes for the travel times. It quickly shows the full span but depends only on two observations. Quartiles describe the lower and upper portions. For this lesson, split an ordered even-sized data set into equal halves and take each half's median. This stated convention matters because software can use different quartile rules.
The lower half is 8, 10, 10, 11, giving lower quartile Q₁ = 10. The upper half is 12, 13, 16, 40, giving upper quartile Q₃ = 14.5. The labels 1 and 3 identify the quartiles. Interquartile range, abbreviated IQR, is Q₃ − Q₁ = 4.5 minutes. It describes the span between the quartiles and is less affected by the single 40-minute value than the full range.
The same data, different summaries
| Measure | For 2, 4, 4, 6, 9 |
|---|---|
| Mean | (2 + 4 + 4 + 6 + 9) ÷ 5 = 5 |
| Median | 4 — the middle ordered value |
| Mode | 4 — the most frequent value |
| Range | 9 − 2 = 7 |
Sources: OpenStax: Introductory Statistics 2e: Measures of the Spread of the Data ↗ · OpenStax: Introductory Statistics 2e: Measures of the Location of the Data ↗
5. Solved example: the same mean can hide different consistency
Illustrative delivery times in minutes are A: 9, 10, 10, 10, 11 and B: 2, 6, 10, 14, 18. Each total is 50, so each mean is 10. Both medians are also 10. Yet A has range 2 and B has range 16. If planning around predictable arrival matters, the individual values reveal a difference that the mean alone conceals.
Another spread measure averages distances from the mean. For A these distances are 1, 0, 0, 0, 1, giving 2/5 = 0.4 minute. For B they are 8, 4, 0, 4, 8, giving 24/5 = 4.8 minutes. This is the mean absolute deviation about the mean; “absolute” means use nonnegative distances. Simply adding signed differences would give zero in both groups and hide their variation.
Sources: OpenStax: Introductory Statistics 2e: Measures of the Spread of the Data ↗
6. Weight groups by their sizes
Three learners average 20 minutes of reading, while seven learners average 30 minutes. The combined mean is not (20 + 30)/2 = 25, because the groups differ in size. Recover the totals: 3 × 20 = 60 minutes and 7 × 30 = 210 minutes. Across ten learners, the mean is (60 + 210)/10 = 27 minutes. The larger group contributes more observations.
Frequency tables apply the same reasoning. If 0, 1, 2 and 3 books were borrowed by 2, 5, 2 and 1 learners respectively, multiply each value by its frequency. Total books are 0 × 2 + 1 × 5 + 2 × 2 + 3 × 1 = 12, over ten learners, so the mean is 1.2 books. Count the frequencies, not just the four different listed values.
Sources: OpenStax: Introductory Statistics 2e: Measures of the Center of the Data ↗
7. Report a summary that the data can support
For the travel example, “median 11.5 minutes, range 8 to 40, eight illustrative journeys” communicates more than “average 15”. Specify that the range interval runs from 8 to 40, while its numerical width is 32. Show a dot plot or the full short list when possible. A sample from one neighbourhood or one time of day does not automatically represent all travellers. More arithmetic cannot repair an unrepresentative selection.
Sources: OpenStax: Introductory Statistics 2e: Measures of the Center of the Data ↗ · OpenStax: Introductory Statistics 2e: Measures of the Spread of the Data ↗ · OpenStax: Introductory Statistics 2e: Data, Sampling, and Variation in Data and Sampling ↗
PUT IT INTO PRACTICE
Compare two study-time summaries
- Use these illustrative daily reading times: 10, 15, 15, 20, 20, 25, 25, 70 minutes. Calculate the mean, median and range, keeping all observations.
- Using the stated median-of-halves convention, find both quartiles and the IQR. Draw one dot per observation on a labelled number line.
- Check: sum 200, mean 25, median 20, range 60, Q₁ = 15, Q₃ = 25, IQR = 10 minutes. Explain why the mean exceeds the median, then state which summary suits a typical day.
Check your understanding
Find the median of 9, 2, 7, 4, 6.
Order them as 2, 4, 6, 7, 9. The middle value is 6.
Four observations have mean 12. What total do they have?
4 × 12 = 48. Multiplying the mean by the count recovers the total.
Every observation increases by 5. What happens to the mean and range?
The mean increases by 5; the range stays unchanged because both extremes increase equally.
Five students average 6 books and fifteen average 2. What is the combined mean?
(5 × 6 + 15 × 2)/20 = 60/20 = 3 books, not 4.
Can a poll of only library visitors estimate every resident's reading habits reliably?
Not automatically. Library visitors may differ from other residents; the sampling method can distort the estimate even when the calculations are correct.