Educerie · SAT · Math
Problem-Solving and Data Analysis · PSDA.3 One-variable data: distributions and measures of centre and spread
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Find mean, median, mode and range | From a list, a table of frequencies, or a graph |
| Say what an outlier does to each | The mean moves, the median barely does |
| Compare spread between two sets | Standard deviation, without ever calculating one |
| Read a histogram or a box plot | Where the middle is, where the bulk sits |
| Say which measure fits a skewed distribution | The median, almost always |
1The idea in one paragraph
Centre and spread are two separate questions about the same list. The mean uses every value, so a single extreme number drags it; the median only cares about position, so it barely notices. On this test you are rarely asked to compute a standard deviation and very often asked which of two sets has the bigger one — and that is answered by looking at how tightly the values cluster, not by calculating anything.
Skewed by a few large values: mean above median. Skewed by a few small values: mean below median. Roughly symmetric: the two are close.
2The four measures
| Measure | How | Sensitive to outliers? |
|---|---|---|
| Mean | Total ÷ count | Yes, strongly |
| Median | Middle value once ordered | Barely |
| Mode | Most frequent value | No |
| Range | Largest − smallest | Yes, entirely |
With an even count the median is the average of the two middle values. With a frequency table, remember that each value appears as many times as its frequency says: the median of a table with 40 entries is between the 20th and 21st, not in the middle row.
3Adding a value, and what moves
Seven measurements have a mean of 12. An eighth is added and the mean becomes 13. What was it?
That "total = mean × count" step answers most mean questions on the test, including ones that look like they need the individual values. When a question adds an extreme value, expect the mean to move noticeably and the median to shift by one position at most.
4Standard deviation, comparatively
Standard deviation measures how far values sit from the mean, on average. You will not have to compute one. You will have to say which of two data sets has the larger one, and the answer is always the one whose values are more spread out from their own centre.
Two sets can have the same mean and very different spreads. A set clustered at 49, 50, 51 and a set at 20, 50, 80 both average 50; the second has a far larger standard deviation.
5Histograms and box plots
A histogram shows how many values fall in each interval. The median is in the interval containing the middle count — work along the bars adding frequencies until you pass half the total.
A box plot shows five numbers: minimum, first quartile, median, third quartile, maximum. The box holds the middle half of the data. A long tail on one side means skew in that direction, and skew is what makes the mean and the median part company.
6Which measure to report
When a question asks which measure best represents a set with a few extreme values, the answer is the median. That is the reasoning behind reporting median house prices and median incomes rather than mean ones, and the test expects that reasoning in exactly those words.
Where points are lost
- Forgetting to order the list before taking the median.
- Ignoring frequencies and taking the middle row of a table.
- Averaging the two middle values — or forgetting to, with an even count.
- Assuming an outlier moves the median as much as the mean.
- Computing a standard deviation when a comparison was wanted.
- Reading a histogram's tallest bar as the median rather than the mode.
Work it right
- Read which measure is asked for.
- Order the data, or read the frequencies properly.
- For a mean, use total = mean × count.
- For spread, compare clustering by eye.
- If the data is skewed and the question asks what best represents it, say the median.
Try it
Q1. The values 4, 7, 7, 9, 13, 28 are recorded. Which statement is true?
A) The mean is less than the median. B) The mean is greater than the median. C) The mean and the median are equal. D) The median is less than the mode.
Q2. Six numbers have a mean of 15. When a seventh number is included, the mean becomes 16. What is the seventh number? (Type your answer.)
Q3. Data set P is 48, 50, 50, 52. Data set Q is 20, 50, 50, 80. Which statement is true?
A) P has the larger standard deviation. B) Q has the larger standard deviation. C) They have equal standard deviations. D) Standard deviation cannot be compared without the mean.
Q4. A town's house prices include a small number of very expensive properties. Which measure best represents a typical price?
A) Mean B) Median C) Range D) Mode
Q5. The frequency table shows scores in a quiz.
| Score | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| Frequency | 5 | 9 | 4 | 2 |
What is the median score? (Type your answer.)
In one breath
The mean uses every value, so one extreme number drags it; the median only cares about position, so it hardly moves — which is why skewed data is described by its median and why the mean sits above the median when a few values are very large. Use total = mean × count for anything about means, read frequencies properly before finding a median, and when you are asked to compare spread, look at how tightly each set clusters rather than calculating anything.
Answers
Q1. B. the value 28 drags the mean above the median
One large value moves the mean and leaves the median where it was. A reverses the effect. C would require a roughly symmetric set. D is false — the median is 8 and the mode is 7.
Q2. 22. totals, not averages
Q3. B. Q's values sit far from their mean Both sets average 50. P's values are within 2 of it; Q's are 30 away. D is the trap for a student who thinks a calculation is required.
Q4. B — Median. a few extreme values pull the mean This is why published house prices are medians. A is dragged upward by the expensive properties; C and D describe spread and frequency, not a typical value.
Q5. 4. twenty values, so average the 10th and 11th
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Problem-Solving and Data Analysis, skill/knowledge testing point "One-variable data: distributions and measures of center and spread"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
Check your understanding
The main ideas of this note. Tick each one you could do now, in an exam, without looking back up. Anything you cannot tick yet is the part to read again.