Descriptive Statistics • Topic 4 of 4

Effect of Changing a Data Point

This is the single most GRE-characteristic idea in the whole chapter, and it rewards reasoning over computation. When you add, remove or alter one value in a set, each statistic responds differently, and knowing the pattern lets you answer without recomputing anything. The mean moves in the direction of the change, by (change in total)/(number of values) — so adding a value above the current mean raises it, and below it lowers it; adding a value exactly equal to the mean leaves it unchanged. The median often does not move at all, because a new value on one side of the middle may only shift the position slightly, or not at all. The mode changes only if the frequencies change. The classic Quantitative Comparison here gives you a set, changes one value, and asks you to compare the new mean or median with the old — the answer is usually reachable by pure logic. A powerful shortcut: adding a value equal to the current mean keeps the mean fixed but can still move the median, so "the mean is unchanged" never implies "the median is unchanged".

✅ Solved examples

1. A set of 5 numbers has mean 20. A sixth number, 32, is added. Find the new mean.
Old sum = 20 × 5 = 100. New sum = 132 over 6 values. New mean = 132/6 = 22. The mean rose because 32 exceeds the old mean.
2. In the set 4, 6, 8, 10, 12 the value 12 is changed to 22. What happens to the mean and the median?
Median is the 3rd value, 8, and is untouched by changing the largest value. Mean was 8; the total rises by 10, so the new mean is 8 + 10/5 = 10. Mean up, median flat — the signature GRE contrast.
3. A list of 7 test scores has median 15. One score below the median is increased but stays below 15. Does the median change?
No. With 7 values the median is the 4th when sorted. Raising a value that remains below the middle does not change which value sits 4th, so the median stays 15.
4. Quantitative Comparison. Set has n values with mean m. A new value equal to m is added. Quantity A: the new mean. Quantity B: m. Which is greater?
New sum = nm + m = (n+1)m over (n+1) values, so new mean = m exactly. Adding a value equal to the mean never moves the mean. Quantity A = Quantity B. Answer: the two are equal.

✏️ Practice — try these, take hints as needed

1. A set of 4 numbers has mean 15. A fifth number, 25, is added. New mean?
Old total = 15 × 4 = 60.
New total = 85 over 5 values.
85 / 5.
17
2. In 3, 5, 7, 9, 11 the value 3 is replaced by 3. Does the mean change? By how much?
Nothing about the total changes.
Same values, same count.
Mean depends only on total and count.
No change
3. A set of 6 numbers has mean 30. Removing one value leaves a mean of 32 over 5 values. Find the removed value.
Old total = 30 × 6 = 180.
New total = 32 × 5 = 160.
180 − 160.
20
4. To {2, 4, 6, 8} you add a number equal to the current mean. Give the new median.
Current mean = 5.
New sorted set: 2, 4, 5, 6, 8.
Median of 5 values = the 3rd.
5
5. A set of 5 numbers has mean 12. If each value is increased by 3, the new mean is?
Adding 3 to every value adds 3 to the mean.
No need to recompute the total.
12 + 3.
15

📝 Topic test — 8 questions

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