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A complete study example: mean and median
Follow an original source text through a traceable summary, four worked questions and three flashcards. Calculate the mean and median yourself.
This illustrative lesson pack demonstrates a study method. It is not a live file-processing result or real student data. Read the source first, then attempt the questions before opening their answers.
1. The source text
S1 · Understand the data. Imagine that five students take 18, 20, 20, 22 and 70 minutes to travel to school. These numbers were invented for this example. The unit is minutes and there are five observations. This small group cannot establish a result about all students.
S2 · The arithmetic mean. Add the values and divide by the number of observations: 18 + 20 + 20 + 22 + 70 = 150; 150 ÷ 5 = 30 minutes. A mean of 30 does not mean that every student travels for 30 minutes. There is not even a 30-minute observation in this dataset.
S3 · The median. Order the values from smallest to largest. These five values are already ordered, and the third, middle value is 20. The median is 20 minutes. With an even number of observations, we would take the arithmetic mean of the two middle values.
S4 · Interpret a change. If the final journey took 120 rather than 70 minutes, the total would become 200 and the mean would become 40 minutes. The middle value would remain 20, so the median would not change. A large value can affect the mean substantially. However, we do not remove an observation simply because it is large; first we investigate measurement errors and the question we are trying to answer.
2. A summary you can trace
A useful summary preserves both a result and the limits on interpreting it. Each row below points back to the paragraph that supports it.
| Study note | Source |
|---|---|
| There are five observations measured in minutes. The data were invented for teaching. | S1 |
| The mean is 30 minutes. This is not each student's journey time. | S2 |
| The middle value of the ordered group is 20, so the median is 20 minutes. | S3 |
| Changing 70 to 120 increases the mean to 40 but leaves this group's median unchanged. | S4 |
Keep this limit: A result for these five observations does not establish the travel time of all students. Adding a generalisation the source does not support makes a note misleading even when the calculation is correct.
3. Try four questions
Work each question out on paper before opening its explanation. Try to explain why an answer is correct as well as finding it.
1. Why is the mean 30 but the median 20?
The total of 150 divided by five gives a mean of 30. The third of the five ordered values is 20, giving a median of 20. These are two different ways to summarise the same group. See S2 and S3.
2. Is “Every student travels to school in 30 minutes” correct?
No. A mean does not replace the individual observations. The values are 18, 20, 20, 22 and 70; none is 30. A supported statement is: “The arithmetic mean of these five observations is 30 minutes.” See S2.
3. What changes if 70 becomes 120?
The total becomes 200 and the mean is 200 ÷ 5 = 40. The third ordered value is still 20, so the median stays the same. Changing one large value affects the two measures differently in this example. See S4.
4. What is the median of 10, 12, 14 and 60?
There are four observations. The two middle values are 12 and 14, giving a median of (12 + 14) ÷ 2 = 13. You can also check the mean: 96 ÷ 4 = 24. This question applies S3's rule for an even number of observations to new data.
4. Three short flashcards
How do you calculate the arithmetic mean?
Divide the sum of the values by the number of observations. Keep the unit in your answer.
What comes before finding the median?
Order the values. Use the middle value for an odd count, or the mean of the two middle values for an even count.
Should you automatically delete a large observation?
No. Investigate the accuracy of the measurement and the purpose of the analysis; do not delete it just because it changes the result.
Reciting a definition and solving a problem are different tasks. Close the cards and try question four again with numbers of your own.
Apply the method to your own lesson
Choose a definition, an example and a limitation. Put a page number, section title or recording timestamp beside each important summary claim. Then write a definition question and an application question. Check uncertain answers against the source before turning them into cards to learn.
Check your notes · Plan your review
Concept reference
For definitions of the mean and median, see OpenStax, Introductory Statistics 2e, section 2.5. The travel-time data, worked questions and study sequence on this page were created for LectureSift.