Descriptive statistics
Mean median mode questions ask for one summary measure of a small data set. The mean is the total divided by the count, the median is the middle value once the set is sorted, and the mode is the value appearing most often. Sorting first costs a few seconds and makes the median and the range immediate.
These are quick marks that get lost to a single unsorted list. The median of 12, 9, 10, 7, 6, 9, 10 is not 7 — it is 9, and you can only see that after sorting. The same drill also asks for range, interquartile range and standard deviation, where knowing which shifts and which does not saves the entire calculation.
Try it against the clock. Descriptive Statistics runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.
Everything except the mean is easier on a sorted list, and the median is impossible without it. The few seconds spent sorting are recovered immediately.
With an odd count, the median is the middle entry. With an even count it is the average of the two middle entries, which need not be a member of the set.
The mode is the most frequent value, and the range is the last entry minus the first. A set can have more than one mode, or none if every value is distinct.
Adding a constant to every value moves the mean, median and mode by that constant but leaves the range, the interquartile range and the standard deviation unchanged.
| Measure | Effect of adding 3 to all |
|---|---|
| Mean | Increases by 3 |
| Median | Increases by 3 |
| Mode | Increases by 3 |
| Range | Unchanged |
| Standard deviation | Unchanged |
Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
The data set is 0, 0, 10, 10.
5 is added to every value.
Use the population formula (divide by n, not n − 1).
What is the new population standard deviation?
The data set is 4, 4, 4, 6, 7, 3.
What is the mode?
The data set is 0, 0, 10, 10.
2 is added to every value.
Use the population formula (divide by n, not n − 1).
What is the new population standard deviation?
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
The middle position of the list as written is not the median. Sorting is a prerequisite, not an optimisation.
Adding a constant moves every value and the mean together, so every deviation is identical. The standard deviation does not change at all.
Dividing by n and by n − 1 give different answers. The question states which is intended, and both results are usually offered.
The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.
No — 7 sits in the middle of the list as written, not of the sorted list. Sorted, the set is 6, 7, 9, 9, 10, 10, 12, and the fourth of seven values is 9. Reading the middle position before sorting is the most common error in this topic.
It stays exactly the same. Every value moves up by 3 and so does the mean, so every deviation from the mean is untouched. If the original population standard deviation was 2, it is still 2 — and recomputing from scratch wastes most of your time budget.
The range is the largest value minus the smallest, so it uses the two ends of the sorted list. The gap between the middle values relates to the interquartile range instead, which measures the middle half. The two are different measures and both appear as options.
Sort the values, then average the two middle entries. For a set of six numbers that means the third and fourth after sorting. The result need not be one of the original values, which is expected rather than a sign of an error.
Nothing. Adding a constant shifts every value and the mean by the same amount, so each deviation from the mean is exactly as it was. The standard deviation, the range and the interquartile range are all unchanged, while the mean, median and mode all move.
Yes. If two values tie for the highest frequency, the set has two modes, and if every value appears exactly once there is no mode at all. Questions that hinge on this usually construct a deliberate tie, so check frequencies rather than assuming a single winner.
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