Center, Spread, and Data Transformations

Adding the same constant shifts a data set without changing its spread. Multiplying every observation scales both its center and its spread, with spread multiplied by the absolute scale factor.

2 worked examples and 3 practice questions, with explained answers. No sign-in required.

Jump to practice

A reliable method

  1. Decide whether a change is additive or multiplicative.
  2. Apply shifts to mean and median but not to standard deviation.
  3. Multiply standard deviation by the absolute scale factor.
  4. Keep sample and population standard-deviation conventions distinct.

Example 1

Find mean, median, mode, and range for 2, 4, 4, 6, 9.

  1. Sort the observations: 2, 4, 4, 6, 9.
  2. The sum is \(25\) across \(5\) observations, so mean = \(5\).
  3. The middle value(s) give median \(4\). The most frequent repeated value(s) give the mode; if none repeats, report no mode.
  4. Subtract minimum from maximum for range. Mean \(5\); median \(4\); mode 4; range \(7\).

Answer: Mean \(5\); median \(4\); mode 4; range \(7\)

Example 2

A data set has mean 12 and standard deviation 3. Add 5 to every observation. Find the new mean and standard deviation.

  1. Adding 5 shifts every observation equally.
  2. The mean rises from 12 to 17.
  3. Distances from the mean do not change, so standard deviation remains 3.

Answer: Mean 17; standard deviation 3

Common mistakes

  • A larger mean does not necessarily mean larger spread.
  • A negative scale factor reverses order, but standard deviation cannot become negative.

Practice without the answer first

  1. A data set has mean 8 and standard deviation 2. Multiply each observation by -3. Find the new mean and standard deviation.
  2. Find the population variance and standard deviation of 2, 4, 6.
  3. Find mean, median, mode, and range for 5, 7, 9, 11, 13.
Answers and step-by-step explanations

Practice 1

  1. The mean scales by -3: -24.
  2. Deviations scale by -3, but their magnitude scales by 3.
  3. Standard deviation becomes 3 times 2, or 6.

Answer: Mean -24; standard deviation 6

Practice 2

  1. The mean is 4.
  2. Squared deviations are 4, 0, and 4; their sum is 8.
  3. Divide by the population count 3 for variance 8/3, then take the nonnegative square root. A sample variance would use divisor 2 instead.

Answer: Variance \(8/3\); standard deviation \(sqrt(8/3)\)

Practice 3

  1. Sort the observations: 5, 7, 9, 11, 13.
  2. The sum is \(45\) across \(5\) observations, so mean = \(9\).
  3. The middle value(s) give median \(9\). The most frequent repeated value(s) give the mode; if none repeats, report no mode.
  4. Subtract minimum from maximum for range. Mean \(9\); median \(9\); mode none; range \(8\).

Answer: Mean \(9\); median \(9\); mode none; range \(8\)

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.

Use this skill in your study plan

This is shared skill instruction, not an official question set or a prediction of your exam score. Choose your exam for its coverage notes and provider requirements.

Original Testinar instructional content. Updated November 11, 2026. Editorial standards ยท Report a correction

Page UP