Standard Deviation & Spread
Two data sets can share the same average yet look nothing alike. Spread tells you how scattered the values are, and standard deviation puts a single number on it.
The mean tells you the center of a data set, but it hides how the values are scattered around that center. Spread is the missing half of the story. Standard deviation (SD) turns spread into one number: roughly the typical distance of a value from the mean. A small SD means the data is tight and consistent, a large SD means it is spread out. Epidemiologists use spread to see whether an outbreak is hitting one age group tightly or striking evenly across a city. Biostatisticians report the mean plus or minus the SD so readers know whether a drug's effect was steady or all over the place. Lab scientists check the SD of repeated measurements to decide if an instrument is precise enough to trust. Doctors reading a study lean on SD to judge whether an average result would likely hold for their own patient. Learn to compute and read spread and you stop being fooled by an average that hides wild variation underneath.
- Common Core · HSS-ID.A.2Use statistics that fit the data (mean and standard deviation) to compare the center and spread of two or more different data sets.
- Common Core · HSS-ID.A.3Interpret differences in shape, center, and spread of data sets in context, accounting for possible effects of extreme values.
- Ohio · Ohio HS S.ID.2Compare the center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- NGSS · SEP-4Analyzing and Interpreting Data: use measures of variability such as range and standard deviation to describe and compare data sets.
- AP · AP Bio SP 6 (Statistics)Describe and compare data using measures of center and spread, including standard deviation, when analyzing biological data.
- Compute the mean (average): Standard deviation is measured from the mean, so students must be able to add the values and divide by how many there are before they can find any deviation.
- Order data and find the median: Range and IQR both depend on sorting the data first and locating the median to split it into halves for the quartiles.
- Square a number and take a square root: Variance squares each deviation and standard deviation takes the square root, so students need to be comfortable with both operations.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
Standard deviation is roughly the typical distance of a value from the mean, and you build it in five clear steps: find the mean, find each deviation, square the deviations, average the squares (the variance), then take the square root. Work it on a tiny data set once and the pattern sticks.
| Value | Deviation (value - 6) | Squared deviation |
|---|---|---|
| 2 | -4 | 16 |
| 4 | -2 | 4 |
| 6 | 0 | 0 |
| 8 | +2 | 4 |
| 10 | +4 | 16 |
| Sum | 0 | 40 |
Using the table, the squared deviations of 2, 4, 6, 8, 10 (mean 6) add up to 40. There are 5 values. What is the variance?
Reviewed| Value | Deviation | Squared deviation |
|---|---|---|
| 2 | -4 | 16 |
| 4 | -2 | 4 |
| 6 | 0 | 0 |
| 8 | +2 | 4 |
| 10 | +4 | 16 |
| Sum | 0 | 40 |
- A.5
- B.8
- C.40
- D.200
Show the worked solution ▾
Answer: B. 8
- Step 1: Recall the rule: Variance is the average of the squared deviations: the sum of the squares divided by how many values there are.
- Step 2: Divide: 40 divided by 5 equals 8.
Why it's right: Variance is the sum of squared deviations (40) divided by the count (5), which equals 8.
- A: This is the count of values, not the variance.
- C: This is the sum of the squared deviations, before dividing by 5.
- D: This multiplies 40 by 5 instead of dividing.
Aligned to Common Core HSS-ID.A.2: compute variance · reading level ~grade 9
The variance of a data set is 8. What is the standard deviation (rounded to two decimal places)?
Reviewed- A.4.00
- B.2.83
- C.64.00
- D.1.60
Show the worked solution ▾
Answer: B. 2.83
- Step 1: Recall the link: Standard deviation is the square root of the variance.
- Step 2: Take the square root: The square root of 8 is about 2.83 (since 2.83 times 2.83 is about 8).
Why it's right: Standard deviation is the square root of the variance: the square root of 8 is about 2.83.
- A: This halves 8; the standard deviation is the square root of 8, not half of it.
- C: This squares 8 instead of taking its square root.
- D: This divides 8 by 5; the SD is the square root of the variance, not the variance over the count.
Aligned to Common Core HSS-ID.A.2: compute standard deviation · reading level ~grade 9
Two samples both have a mean of 6. Sample A is 5, 6, 7. Sample B is 2, 6, 10. Which sample has the larger standard deviation?
Reviewed- A.Sample A, because its values are closer together
- B.Sample B, because its values are farther from the mean
- C.They are equal, because they have the same mean
- D.You cannot tell without more data
Show the worked solution ▾
Answer: B. Sample B, because its values are farther from the mean
- Step 1: Compare the deviations: In A the deviations are minus 1, 0, plus 1 (small). In B they are minus 4, 0, plus 4 (large).
- Step 2: Bigger deviations mean bigger SD: Standard deviation grows as values sit farther from the mean, so B has the larger SD even though both means are 6.
Why it's right: Both means are 6, but Sample B's values sit farther from the mean (deviations of 4 versus 1), so B has the larger standard deviation.
- A: A's values are closer together, which gives it the smaller SD, not the larger one.
- C: Equal means do not force equal spread; the SDs differ because the deviations differ.
- D: You can tell directly: both data sets are fully shown, and B's values are clearly more spread out.
Aligned to Common Core HSS-ID.A.3: interpret spread · reading level ~grade 9
- A lab computes the SD of five repeated absorbance readings to report how precise the measurement was.
- A coach compares two athletes with the same average sprint time by their SD to see who is more consistent.
- A student reports a class score as a mean plus or minus the SD so readers know how tightly the scores clustered.
Fill these in as you work through the lesson.
- Deviation (value minus the mean):
- Variance (the average of the squared deviations):
- Standard deviation (the square root of the variance):
- IQR (Q3 minus Q1, the middle-half spread):
To find the standard deviation, first find the , then each deviation (value minus mean), then each deviation, average those to get the variance, and finally take the of the variance.
- For the set 2, 4, 6, 8, 10, list the five deviations from the mean of 6 and show that they add to zero.
- Explain why we square the deviations before averaging them instead of just averaging the deviations.
- Two samples have the same mean but different SDs. What does the larger SD tell you about that sample?
For 2, 4, 6, 8, 10 the mean is 6. The squared deviations are 16, 4, 0, 4, 16, which sum to 40. Divide by 5 to get the variance: 40 divided by 5 equals ____. Then take the square root to get the standard deviation, which is about ____.
The vocabulary of this topic, shown in the way you will meet it.