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Math: Problem-Solving and Data Analysis, 3 original questions

SAT One-Variable Data Practice: Questions and Answers

Practice frequency-table means, medians, outliers and spread in three original SAT-style questions. Compare data transformations with complete answer explanations.

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The short answer

One-variable data describe one measured quantity across observations. Decide whether the question asks about center, spread or the shape of a distribution. A frequency table repeats each value the stated number of times, so calculate a weighted total before dividing by the observation count. Sort values to find the median. Outliers can pull the mean more than the median. Adding the same constant shifts the center but leaves distances between values unchanged, while multiplying changes both location and spread.

Method at a glance

  1. 1Count the observations
  2. 2Choose the requested statistic
  3. 3Inspect spread and outliers
  4. 4Track a transformation

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Count the observations

    In a frequency table, add the frequencies rather than counting only the distinct listed values.

  2. Step 2: Choose the requested statistic

    For mean, divide the weighted total by the count. For median, order observations and locate the middle value or average of the two middle values.

  3. Step 3: Inspect spread and outliers

    Compare distances, range or standard deviation as requested. A far-away observation can influence the mean without moving the median much.

  4. Step 4: Track a transformation

    Adding c shifts the mean and median by c but preserves spread. Multiplying by k scales standard deviation by |k|.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

A data set contains three observations of 2, two observations of 4 and one observation of 7. What is its mean?

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Correct answer

Choice B: 3.5

Explanation

There are 3 + 2 + 1 = 6 observations. Their total is 3 × 2 + 2 × 4 + 1 × 7 = 21. The mean is 21/6 = 3.5. The frequencies must be included.

Why each choice is right or wrong

  1. Choice A

    3 is the median of the ordered data, not the mean.

  2. Choice B

    Correct answer

    Correct: the weighted total 21 divided by 6 is 3.5.

  3. Choice C

    This does not use the weighted total and observation count correctly.

  4. Choice D

    7 is the largest observation, not the mean.

Question 2 of 3

A new data set is made by adding 12 to every observation in an original data set. Which statement is true?

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Correct answer

Choice C: The mean increases by 12 and standard deviation is unchanged

Explanation

Every observation and the mean shift upward by 12. Each observation's distance from the new mean equals its old distance from the old mean, so the standard deviation is unchanged. This holds whether the original spread is summarized using a sample or population standard deviation.

Why each choice is right or wrong

  1. Choice A

    The mean increases by 12, but spread does not change under an equal shift.

  2. Choice B

    The mean changes, while standard deviation stays the same.

  3. Choice C

    Correct answer

    Correct: location changes by 12 but distances from the mean do not.

  4. Choice D

    The standard deviation is unchanged, but the mean does increase.

Question 3 of 3

A data set is 10, 11, 12, 13 and 54. Which choice gives its mean and median?

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View worked answer for question 3

Correct answer

Choice D: Mean 20; median 12

Explanation

The sum is 100, so the mean is 100/5 = 20. The ordered middle observation is 12, so the median is 12. The large value 54 pulls the mean upward more than it affects the median.

Why each choice is right or wrong

  1. Choice A

    This swaps the mean and median.

  2. Choice B

    The mean is correct, but the third observation is 12, not 13.

  3. Choice C

    12 is the median, but the total gives mean 20.

  4. Choice D

    Correct answer

    Correct: the mean is 20 and the median is 12.

Common mistakes

  • Averaging distinct values without their frequencies. Each occurrence contributes to the total and the observation count.

  • Finding a median from unsorted data. Put the observations in order first, including repeated values.

  • Assuming a shifted data set has a larger standard deviation. Equal additions change location, not distances from the mean.

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Common questions

How do I find the median when there are an even number of values?

Order all observations and average the two middle values. Repeated observations count separately.

Which center is less sensitive to a large outlier?

The median is generally less sensitive because it depends on order and middle position rather than the size of every observation.

What happens to standard deviation when all values double?

It doubles: every distance from the mean doubles. More generally, multiplication by k scales standard deviation by |k|.

Sources

  1. College Board: Math section overviewsatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides