The learning goal
Keep all valid observations and sort numerically. With an odd count, select the central value; with an even count, average the two central values. Our lists 101,10,14,12,13 and 100,14,10,12 both have median 13, while their means are 30 and 34 respectively.
Establish the valid count before choosing positions
The NIST location-measures reference defines the median from sorted values: position (N+1)/2 for an odd count, or the average of positions N/2 and N/2+1 for an even count. The mean uses the sum divided by the count. All numbers below are original practice data, not observations from a study.
Confirm consistent units, the meaning of an observation and the treatment of missing values first. Do not fill an unknown with zero merely to complete a list. Repeated values can represent separate valid observations and should remain. Write the valid N explicitly so that position choices follow evidence rather than the visual middle of a worksheet.
| Check | Practice rule | Mistake to avoid |
|---|---|---|
| Units | Use a consistent numeric quantity | Sort mixed units without conversion |
| Repeated observations | Retain every valid observation | Delete equal values as if they were duplicate records |
| Missing values | Keep unresolved values separate | Insert zero and change N |
| Ordering | Sort numerically ascending | Use original row order or text sorting |
Odd example: select the third of five values
Start with 101,10,14,12,13 and N=5. The sorted list is 10,12,13,14,101. Position (5+1)/2 is 3, whose value is 13. Choosing the original third entry would produce 14; the error is missing the ordering step, rather than faulty arithmetic.
Calculate the mean independently: (101+10+14+12+13)/5 = 150/5 = 30. Median 13 and mean 30 describe different summaries, so disagreement does not itself establish an error. For a related check of means and combined totals, try the weighted-average reconstruction exercise.
Leave a traceable calculation beside the answer
Input: 101,10,14,12,13 Original third entry: 14
Sorted: 10,12,13,14,101 Third entry: 13; sum 150; mean 30
Even example: the median need not be an observed value
The second input is 100,14,10,12 with N=4. Sort it to 10,12,14,100. Positions 2 and 3 contain 12 and 14, giving median (12+14)/2 = 13. The mean is 136/4 = 34. Although 13 does not appear in this input, it is the correct median.
Do not average the endpoints instead. (10+100)/2 = 55 calculates the midpoint of the extremes, which is a different quantity. Average only the two central values for this even-sized sample; do not replace that operation with an average of all entries or a convenient-looking pair.
| Check | Five-value example | Four-value example |
|---|---|---|
| Sorted values | 10,12,13,14,101 | 10,12,14,100 |
| Central positions | Position 3 | Positions 2 and 3 |
| Median | 13 | (12+14)/2 = 13 |
| Mean | 150/5 = 30 | 136/4 = 34 |
Change one extreme value and explain the result
Replace 101 in the first list with 1001, leaving the other four entries unchanged. Sorting still places 13 third, so the median remains 13. The new sum is 1050 and the mean is 210. Recalculating positions and totals explains exactly why one summary changed and the other did not.
This variation does not authorize deleting extreme observations or establish that a median is always the preferred summary. Check whether 1001 is valid, then report measures appropriate to the question and show relevant range information. NIST discusses how extreme tails affect location measures; our small arithmetic example is an explanation, rather than an empirical experiment.
Cover the answers and try three variations
Changed value
Replace 101 with 1001: median 13, mean 210. Explain the unchanged central position and changed sum.
Repeated values
For 2,2,9,10, average the central 2 and 9 to get 5.5. Keep both observations equal to 2.
Find the mistake
Why is 14 wrong for 100,14,10,12? The sorted center contains both 12 and 14, not one chosen entry.
Check understanding through explanation
Write five steps: establish N, sort, mark the center, calculate and verify. Shuffle the original order for a second attempt and ask whether another learner can reproduce the answer from your working. Then change whether N is odd or even. The objective is to justify the rule, rather than memorize the recurring answer 13.
Use the worked-example self-explanation method to add a reason at each step: why sorting is required, why equal observations remain, and why an even count needs two positions. Return later with new data and record the step where an error occurred. This is a proposed practice routine, not a measured promise of better grades or memory.
Before accepting an answer
- N counts valid observations, without replacing missing values by zero.
- The ascending list retains every valid observation, including equal values.
- Central positions follow N’s parity; an even count uses the central pair.
- Median and mean are calculated and labeled separately.
- You can explain which step changes when a value or N changes.
Common questions
Must the median occur in the input?
No. For an even count it averages the two central values and can be absent from the original list. The median of 10,12,14,100 is 13.
Should equal values be deduplicated first?
Retain them when they represent separate valid observations. Removing them changes N and the center. Investigate duplicate data entry separately rather than deleting observations solely because their values match.
Should a data set with an extreme value report only the median?
Not automatically. Verify the observation and choose measures appropriate to the question, potentially reporting the mean, median and range together. Do not remove valid observations merely to obtain a more appealing result.
