StudyPebble

One-variable data and statistics

Compare mean, median, range, and spread for a data set, and predict how a change affects each one.

Problem-Solving and Data AnalysisMath
Practice one-variable data and statistics free

What the SAT actually asks

You work with a single list of values, a frequency table, a dot plot, or a histogram. Questions ask for the mean, median, mode, or range, or ask how those measures change when a value is added, removed, or altered. Comparison questions give two data sets and ask which has the larger standard deviation, which is about how spread out the values are rather than how large they are. Frequency tables require weighting each value by how many times it occurs.

One-variable statistics usually appear roughly 1 to 3 times out of the 44 math questions, within the roughly 15 percent data share.

Where students lose points

How to attack these questions

  1. 1Write out or mentally expand a frequency table into the actual list before finding a median or a range.
  2. 2For mean questions, work with the total sum rather than the average, since adding or removing a value changes the sum predictably.
  3. 3For standard deviation comparisons, look at how tightly values cluster around the center, not at the center itself.

Example questions

Written for this skill, with the answer explained. 133 more are in the app.

Try one
One-variable data and statistics
Real question

What is the mean (average) of the following data set: 6, 16, 18, 23, 27?

Easier

A transit agency asked 14 riders at one stop how many minutes they had waited for a bus. The dot plot shows the results. What is the range of the wait times, in minutes?

  • A) 12 ✓
  • B) 14
  • C) 16
  • D) 24

Why: The range is the largest value minus the smallest. The longest wait shown is 14 minutes and the shortest is 2 minutes, so the range is 14 - 2 = 12 minutes. The value "14" is the largest wait itself, and it is also the number of riders surveyed, but neither is a spread. Adding the two ends instead of subtracting them, 14 + 2, gives "16". The value "24" is larger than the longest wait recorded, so no two of these times can differ by that much.

Medium

A travel planner recorded 5 hotel prices, in dollars: 20, 25, 30, 35, 40. The value equal to the mean, 30.0, is removed from the list. What happens to the mean of the remaining values?

  • A) The mean decreases, because there are fewer values to average.
  • B) The mean increases, because removing any value raises what remains.
  • C) It cannot be determined without recalculating from the remaining four values.
  • D) The mean stays exactly the same. ✓

Why: The five prices total 150, so the mean is 150/5 = 30 dollars. Removing that 30 leaves 20 + 25 + 35 + 40 = 120 over 4 values, and 120/4 = 30. The mean stays exactly the same, because the removed value was neither above nor below the mean. Saying "The mean decreases, because there are fewer values to average." ties the mean to the count, and what matters is how the removed value compared with the mean. Saying "The mean increases, because removing any value raises what remains." expects any removal to raise the mean, and removing a high value would lower it. Saying "It cannot be determined without recalculating from the remaining four values." calls it undeterminable, and the recomputation above settles it at 30 dollars.

Medium

A data set of 9 distinct values has a median of 40. The largest value in the set is then increased by 100, and no other value changes. Which statement is true about the new data set?

  • A) Both the median and the mean increase.
  • B) The median is unchanged and the mean increases. ✓
  • C) Both the median and the mean are unchanged.
  • D) The median increases and the mean is unchanged.

Why: With 9 values the median is the 5th in order, and it is 40. Increasing the largest value by 100 keeps it the largest, so the order of the other values does not change and the 5th value is still 40. The sum grows by 100, so the mean rises by 100/9, about 11.1. So the median is unchanged and the mean increases. "Both the median and the mean increase." is wrong because nothing crossed the middle position. "Both the median and the mean are unchanged." ignores that the total grew by 100. "The median increases and the mean is unchanged." reverses which measure responds to a change in one extreme value.

Questions students ask

What does standard deviation measure on the SAT?

It measures how spread out the values are around the mean. A data set with values bunched close to the average has a small standard deviation. You are never asked to calculate it by hand, only to compare two sets and say which is more spread out.

How does an outlier affect the mean and the median?

The mean gets pulled toward the outlier because it uses every value in the sum. The median barely moves, since it depends only on position in the ordered list. That is why questions often ask which measure better represents a skewed data set.

How do I find the median from a frequency table?

Add the frequencies to get the total number of values, then find the position of the middle value. Count down the frequency column until you reach that position. The value in that row is the median.

Drill this skill until it is automatic

Questions on one-variable data and statistics alone, with explanations on every one, and the app tracks which ones you keep missing.

Start practicing

Question types inside this skill

Guides that cover this

Other Problem-Solving and Data Analysis skills