Computing the mean and standard deviation
Add up a small data set to find its mean (average), then measure how spread out the values are with the standard deviation.
- Adding values and dividing: The mean is the sum of the values divided by how many there are, so basic addition and division come first.
- Squaring numbers and square roots: Standard deviation squares each deviation and then takes a square root at the end, so both operations are needed.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
Compute the mean first, then measure spread: square how far each value sits from the mean, average those squares, and take the square root for the standard deviation.
For the data set {2, 4, 6, 8, 10}, find the population standard deviation. (Mean is 6; divide the squared deviations by n = 5.)
Reviewed- A.√10 ≈ 3.16
- B.√8 ≈ 2.83
- C.8
- D.6
Show the worked solution ▾
Answer: B. √8 ≈ 2.83
- Step 1: Find the deviations: Values minus the mean of 6: -4, -2, 0, 2, 4.
- Step 2: Square and add them: 16 + 4 + 0 + 4 + 16 = 40.
- Step 3: Divide by n and take the root: 40 ÷ 5 = 8, and √8 ≈ 2.83.
Why it's right: The squared deviations total 40; for a population we divide by n = 5 to get 8, and √8 ≈ 2.83.
- A: √10 comes from dividing 40 by 4 (the sample formula, n − 1); the prompt asks for the population, dividing by 5.
- C: 8 is the variance (40 ÷ 5) before taking the square root.
- D: 6 is the mean, not the spread.
Aligned to BI 2.1: standard deviation (population) · reading level ~grade 9
- A teacher reports the mean and standard deviation of a class's reaction-time data so readers see both the typical value and how much students varied.
Fill these in as you work through the lesson.
- Mean (the average; add then divide):
- Deviation (how far one value is from the mean):
- Standard deviation (the typical spread around the mean):
- Population standard deviation (the spread when your data covers every member of the group, not a sample):
To find the mean, add every value and divide by . For a population standard deviation, square each value's distance from the mean, average those by dividing by , and take the .
- For the data set {3, 5, 7, 9, 11}, add the values and divide to find the mean. Show your steps.
- Two data sets have the same mean, but one has values bunched close together and one has values far apart. Which has the larger standard deviation?
- Why do we square each deviation before averaging instead of just adding the raw distances?
For the data set {2, 4, 6, 8, 10}, find the mean, then find each value's distance from the mean, square those distances, average them by dividing by 5 (population), and take the square root to get the population standard deviation.