GRE Statistics: Center and Spread
GRE statistics rewards knowing what each measure is sensitive to. Mean, median, mode, and range are quick once you stop averaging by brute force and start using sum = mean x count and the evenly-spaced shortcut.
Mean: think in sums
The mean (average) is the sum divided by the count, but the version you will actually use is sum = mean x count. Given an average and how many values there are, recover the total, then it is easy to add a value, remove one, or solve for a missing number. The mean is sensitive to outliers: one extreme value drags it.
Median, mode, and range
- Median: sort the values and take the middle one. With an even count, average the two middle values. It ignores how extreme the outliers are.
- Mode: the most frequent value. A set can have no mode or several.
- Range: largest minus smallest. A measure of spread, not center.
A classic comparison question adds one large value to a set: the mean rises, the median barely moves, and the range jumps.
The evenly-spaced shortcut
For any evenly spaced set, consecutive integers, multiples of a number, or any arithmetic sequence, the mean equals the median, and both equal the average of the first and last terms. The average of 1 through 99 is simply (1 + 99) / 2 = 50. No summing required.
Weighted averages
When combining groups of different sizes, you cannot average the averages. Weight by count: total of everything over total count. Ten scores of 90 and twenty of 60 give (900 + 1200) / 30 = 70, which leans toward the bigger group, not the midpoint 75. For spread, see standard deviation.
Frequently asked questions
What is the most useful fact about the mean on the GRE?
Sum = mean x count. Most mean questions are not asking you to average; they give you the mean and the count so you can recover the total, then add or remove a value. If five numbers average 20, their sum is 100.
When is the median more useful than the mean?
When there are outliers. The mean gets pulled toward extreme values; the median (the middle value) does not. If a set has one huge or tiny value, expect the question to hinge on that difference.
What is the shortcut for an evenly spaced set?
For any evenly spaced set (consecutive integers, multiples of 5, an arithmetic sequence), the mean equals the median, and both equal the average of the first and last terms. This skips all the adding.
How is a weighted average different from a normal average?
You cannot just average the group averages unless the groups are the same size. Weight each average by its count: total of all values divided by total count. A class of 10 at 90 and 20 at 60 averages (900 + 1200) / 30 = 70, not 75.
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More GRE Quant: all topics, standard deviation, formula sheet.