Tales by Dots and Lines · Lesson 4 of 13
The Median When Data Changes
“Track middle positions when data is inserted, removed, or repeated.”
• Find medians for odd and even numbers of observations. • Reorder data after inserting or removing values. • Explain why an inserted value above the median need not raise it. • Use examples and counterexamples to test claims about the median.
Seven people stand in a line from shortest to tallest. The fourth person has three people on either side. This middle position gives us the median height. If one more person joins the line, there are now two middle positions, so the way we find the median changes.
The mean depends on every value through the total. The median depends on positions after sorting. This makes it useful, but it also means that changing the number of observations can shift the middle in ways that deserve careful attention. We will always write the new ordered list before deciding what happens.
Tinkering with Median
For an odd number of observations, one value occupies the middle position. With 7 values, the middle is the fourth. With 9, it is the fifth. In general, the middle position is (n + 1) ÷ 2 when n is odd.
For an even number, use the two middle positions and take the mean of those two values. With 8 observations, the middle positions are fourth and fifth. Their positions are n ÷ 2 and n ÷ 2 + 1. The median may be a number that does not appear in the original list.
The middle value of ordered data when the count is odd, or the mean of the two middle values when the count is even.
Problem
Find the median of 12, 5, 9, 7, and 10.
- 1.Arrange the values in increasing order: 5, 7, 9, 10, 12.
- 2.There are 5 observations, so the middle position is (5 + 1) ÷ 2 = 3.
- 3.The third value is 9, so the median is 9.
- 4.Choosing the third value from the unsorted list would not be a valid method, even if it happened to give the same answer.
Problem
Find the median of 3, 4, 8, 10, 12, and 17.
- 1.The data is already sorted and contains 6 observations.
- 2.The middle positions are third and fourth, containing 8 and 10.
- 3.Their mean is (8 + 10) ÷ 2 = 9.
- 4.The median is 9, although no observation equals 9.
Adding a Value Above the Median
For 2, 4, and 8, the median is 4. Insert 10 and sort: 2, 4, 8, 10. The two middle values are 4 and 8, so the new median is 6. In this case the median rises.
But consider 2, 4, and 4, also with median 4. Insert 10 to obtain 2, 4, 4, 10. The middle values are both 4, so the new median is still 4. The added value was above the old median, yet the median did not increase. Repeated values can keep the middle unchanged.
| Original data | Inserted value | New ordered data | Old median → new median |
|---|---|---|---|
| 2, 4, 8 | 10 | 2, 4, 8, 10 | 4 → 6 |
| 2, 4, 4 | 10 | 2, 4, 4, 10 | 4 → 4 |
| 2, 6, 10 | 1 | 1, 2, 6, 10 | 6 → 4 |
Inserting a value above the old median cannot lower the median, but it may increase it or leave it unchanged. Inserting a value below it cannot raise the median, but it may lower it or leave it unchanged.
Problem
A learner says, “Adding a value greater than the median always increases the median.” Test the claim.
- 1.Start with 1, 5, 5. The median is 5.
- 2.Insert 9, which is greater than 5. The ordered data is 1, 5, 5, 9.
- 3.The two middle values are 5 and 5, so the new median is 5.
- 4.This single counterexample disproves “always”. The increase happens sometimes, not always.
Removing an Observation
After removal, count and sort again. Removing a low value can move the middle upward; removing a high value can move it downward. Ties can still keep it unchanged. Removing the observation that currently sits at the middle does not mean there is no median: the remaining data has its own middle position or positions.
It helps to think of two tasks separately: identify which positions are central, then read the values in those positions. A central position is a place in the list, not a permanent label attached to one observation.
Problem
Remove 6 from 2, 4, 6, 8, 12. What happens to the median?
- 1.The original five values have middle value 6.
- 2.The remaining ordered values are 2, 4, 8, 12.
- 3.The new median is the mean of 4 and 8: (4 + 8) ÷ 2 = 6.
- 4.The median stays 6 even though the observation equal to 6 has been removed.
Keeping the Median Fixed
There can be many values that preserve a median. Consider 3, 7, 7, and 12. Its median is 7. Inserting a low, middle, or high value still leaves 7 in the third position of the five-value list. The two copies of 7 protect the middle.
A different dataset may allow fewer choices. Inserting one value into 3, 7, 11 must make the new middle pair average to the desired median. To keep median 7, inserting 7 works. Checking the actual positions is more reliable than trying to use the mean’s excess-and-shortage rule.
Always, Sometimes, or Never?
Investigate a claim by searching for both supporting examples and counterexamples.
- Test “The median must be one of the observations” using an odd-count dataset.
- Test the same claim using 2, 6, 10, 14.
- Decide whether one supporting example proves an always claim.
- Explain your classification in one sentence.
The claim is sometimes true. Odd-count data has an observed middle value, but 2, 6, 10, 14 has median 8, which is not observed. One counterexample is enough to disprove an always claim.
Do not say that adding a high value must raise the median. Repeated middle values may hold it fixed. Also, a known mean by itself usually does not determine the median.
Quiz
What is the median of 9, 2, 6, 4, 7?
What is the median of 2, 5, 9, 12?
Insert 20 into 1, 4, 4. What is the new median?
Which must happen before finding a median?
Remove 10 from 2, 4, 6, 10. What is the new median?
Sort to get 2, 4, 6, 7, 9. The third value is 6.
Practice Problems
- Find the median of 15, 8, 12, 6, 10, 4.
- Insert 20 into 1, 3, 7, 9. Compare the old and new medians.
- Give a dataset where removing a below-median value leaves the median unchanged.
- Remove 3 from 3, 5, 8, 10, 12. Find the new median.
- Insert one whole number into 2, 6, 10 so that the median remains 6.
- Two datasets both have mean 6. Must they have the same median?
Step 1: Sort: 4, 6, 8, 10, 12, 15. Step 2: The middle values are 8 and 10. Step 3: Their mean is 9, so the median is 9.
Key Takeaways
• Sort the complete dataset before finding the median. • Odd counts have one middle value; even counts have two middle values to average. • Insertion or removal changes the middle positions. • Ties can keep the median unchanged even after a large value is inserted. • Use counterexamples to challenge statements containing always.