Mean, Median & Mode
1. Core Idea
Three different answers to "what is a typical value?":
- Mean — the arithmetic average. Sensitive to every value, including outliers.
- Median — the middle value when sorted. Immune to outliers.
- Mode — the most frequent value. A set can have none, one, or several.
The GRE's favourite question is not "compute these" but "what happens to each when the data changes?"
2. Must-Know Rules
| Measure | Definition |
|---|---|
| Mean | sum of values / number of values |
| Sum | mean x count (rearranged — the most useful form) |
| Median (odd n) | the middle value after sorting |
| Median (even n) | the average of the two middle values |
| Mode | the most frequently occurring value |
| Range | largest - smallest |
Sort before finding the median. Always. The single most common careless error in this topic.
Effect of adding a constant k to every value:
| Measure | Effect |
|---|---|
| Mean | increases by k |
| Median | increases by k |
| Mode | increases by k |
| Range | unchanged |
| Standard deviation | unchanged |
Effect of multiplying every value by k:
| Measure | Effect |
|---|---|
| Mean, median, mode | multiplied by k |
| Range, standard deviation | multiplied by |k| |
Mean vs median and skew:
- Symmetric data: mean = median.
- Skewed right (a few very large values): mean > median.
- Skewed left (a few very small values): mean < median.
Evenly spaced sets: mean = median = (first + last)/2. This shortcut appears constantly.
3. Worked Examples
Example 1 — Using sum = mean x count
The average of 6 numbers is 15. When a seventh number is added, the average becomes 17. What is the seventh number?
Original sum = 6 x 15 = 90.
New sum = 7 x 17 = 119.
Seventh number = 119 - 90 = 29.
Example 2 — Median with an even count
Find the median of 12, 3, 19, 7, 15, 8.
Sort: 3, 7, 8, 12, 15, 19.
Two middle values are 8 and 12.
Median = (8 + 12)/2 = 10.
Example 3 — Mean vs median under an outlier
A set is 4, 5, 6, 7, 8. What happens to the mean and median if 8 is replaced by 88?
Original: mean 6, median 6.
New set 4, 5, 6, 7, 88: mean = 110/5 = 22, median still 6.
The mean jumps to 22; the median is unchanged. This is exactly why the median is used for house prices and salaries.
4. GRE Traps
- Finding the median without sorting.
- Assuming the median is a member of the set. For an even count it usually is not.
- Assuming a unique mode. A set can be bimodal, or have no mode at all if every value appears once.
- Assuming the mean is a member of the set. It rarely is.
- Adding a constant changes the range. It does not.
- Confusing "average of the averages" with the overall average. That only works when the groups are the same size — otherwise you need a weighted average.
- Mean = median implies symmetry. It is suggestive but not a proof.
5. Speed Tricks
- Sum = mean x count. Almost every "average" word problem is solved by converting averages into sums.
- For evenly spaced sets, mean = median = (first + last)/2. No addition needed.
- Deviation shortcut: to average numbers near 200, average their differences from 200 and add 200 back.
- To find a missing value, work with sums, not averages.
- On "which changes" questions, test with a tiny concrete set (three or five numbers) rather than reasoning abstractly.
- The median only cares about position — changing an extreme value usually leaves it alone.
6. Self-Check
Q1. The average of 5 numbers is 24. If one number is removed and the average becomes 26, what was removed?
Q2. Find the median of 2, 9, 4, 4, 11, 7, 4.
Q3. Every value in a data set is increased by 10. What happens to the range?
Answers
A1. Original sum 120; new sum = 4 x 26 = 104. Removed = 16.
A2. Sorted: 2, 4, 4, 4, 7, 9, 11. Middle (4th) value = 4.
A3. Unchanged — adding a constant shifts everything equally.
7. One-Line Summary for the Board
Sum = mean x count. Sort before you take the median. Outliers move the mean, not the median.