How to solve SAT mean, median, and spread questions

One-variable data questions give you a single set of numbers, as a list, a frequency table, a dot plot, a histogram, or a box plot, and ask about its center and spread. On the Digital SAT that means mean, median, range, and a conceptual understanding of standard deviation, plus how those measures change when data points are added, removed, or shifted. Problem-Solving and Data Analysis is about 15% of the Math section, or 5 to 7 of the 44 questions. Those questions are split across seven skills, so expect about one question on this skill, sometimes two.

What the questions look like

Compute a center. You find the mean or median from a frequency table or a dot plot, or compare the two. Frequency tables are where most students lose points.

Effect of a change. A value is added, removed, or corrected, and you decide how the mean, median, range, or standard deviation changes. An outlier is the usual suspect.

Compare distributions. Two dot plots, histograms, or box plots are shown, and you decide which data set has the greater mean, median, range, or standard deviation. You never compute a standard deviation by hand on the SAT. You judge it from how spread out the data look.

Step-by-step method

For the mean, divide the sum by the count. With a frequency table, multiply each value by its frequency, add those products, and divide by the total frequency.
For the median, put the data in order and find the middle position, $\frac{n + 1}{2}$ . If $n$ is even, average the two middle values. With a frequency table, count through the frequencies until you reach that position.
For spread, range is maximum minus minimum. The interquartile range is $Q_3 - Q_1$ , which a box plot shows as the width of the box. Standard deviation measures how far values typically are from the mean. Data bunched near the center have a small standard deviation.
Judge changes with these facts.
- An extreme value pulls the mean toward it but barely moves the median.
- Adding the same number to every value shifts the mean and median by that number and leaves range and standard deviation unchanged.
- Multiplying every value by a number multiplies the mean, median, range, and standard deviation by that number, using its absolute value for the spread measures.
For a missing value, use totals. Sum equals mean times count.

Worked example

Example 1. The table shows the number of books 25 students read over the summer.

Books read	0	1	2	3	4
Number of students	2	8	6	4	5

(a) Find the mean and the median. (b) A 26th student who read 12 books joins the group. How do the mean and median change?

(a) Multiply each value by its frequency and add:

$0(2) + 1(8) + 2(6) + 3(4) + 4(5) = 0 + 8 + 12 + 12 + 20 = 52$

The mean is $\frac{52}{25} = 2.08$ books.

For the median, the middle position is $\frac{25 + 1}{2} = 13$ . Count through the table: positions 1 to 2 are 0 books, positions 3 to 10 are 1 book, and positions 11 to 16 are 2 books. The 13th value is 2, so the median is 2.

(b) The new sum is $52 + 12 = 64$ and the new count is 26, so the mean is $\frac{64}{26} \approx 2.46$ . The mean increased. With 26 values, the median is the average of the 13th and 14th values. Both are still 2, so the median stays 2.

Example 2. The mean of six numbers is 14. When a seventh number is added, the mean of all seven is 15. What is the seventh number?

The six numbers add to $6 \times 14 = 84$ . The seven numbers add to $7 \times 15 = 105$ . The seventh number is $105 - 84 = 21$ .

Common traps

Finding the middle of the value column instead of the middle observation. In Example 1, the value column runs 0 to 4, and its middle entry is 2 by coincidence. In general the median depends on the frequencies, so always count through them.
Averaging the values without weighting. The mean of 0, 1, 2, 3, and 4 is 2, but that ignores how many students read each amount. Multiply by frequency first.
Expecting an outlier to move the median a lot. Adding one extreme value moves the median at most to a neighboring value in the ordered list. The mean is the measure that shifts noticeably.
Thinking standard deviation measures size. A data set of 100, 101, and 102 has a smaller standard deviation than one of 1, 10, and 20, even though its values are much larger. Spread is about distance from the mean.
Reading a histogram's bar heights as data values. The height of a bar is a frequency. The values are on the horizontal axis.

Desmos tip

Type a list as L = [3, 7, 7, 9, 12], then type mean(L), median(L), or stdev(L) on new lines to get each value. For a frequency table, enter the values as V = [0,1,2,3,4] and the counts as F = [2,8,6,4,5], then type total(V*F)/total(F) for the mean. Desmos multiplies the two lists term by term, so this matches the hand method above and returns 2.08.