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Standard deviation
In a survey of 1,000 adult women, haemoglobin is normally distributed with a mean of 12.0 g/dL and a standard deviation of 1.5 g/dL. Between which two values would you expect approximately 95% of the women's haemoglobin values to lie?
- correctA. 9.0 and 15.0 g/dL
- B. 10.5 and 13.5 g/dL
- C. 7.5 and 16.5 g/dL
- D. 11.9 and 12.1 g/dL
Explanation
Key fact
In a normal distribution, mean plus or minus 2 standard deviations covers about 95% of observations. One covers 68%, three cover 99.7%.
9.0 and 15.0 g/dL. Two standard deviations is 3.0, so the range is 12 minus 3 to 12 plus 3.
How it happens
- The question asks about individual observations, so use the standard deviation.
- For 95% coverage, take the mean plus or minus 2 standard deviations.
- Two standard deviations = 2 x 1.5 = 3.0 g/dL.
- Lower limit = 12.0 - 3.0 = 9.0.
- Upper limit = 12.0 + 3.0 = 15.0.
How much of a normal distribution lies within each band
| Range | Proportion covered |
|---|---|
| Mean +/- 1 SD | About 68% |
| Mean +/- 2 SD | About 95% |
| Mean +/- 3 SD | About 99.7% |
Mean plus or minus two standard deviations covers 95%
Why not the others
- B. 10.5 and 13.5 g/dL — is mean plus or minus one standard deviation, which covers about 68%.
- C. 7.5 and 16.5 g/dL — is mean plus or minus three standard deviations, covering about 99.7%.
- D. 11.9 and 12.1 g/dL — uses the standard error of the mean, about 0.047 here. That describes where the true mean lies, not where individuals lie.
Remember
Standard deviation describes people; standard error describes the mean. The question tells you which by asking about individuals or about an estimate.
Go deeper
Standard error = standard deviation divided by the square root of the sample size. It shrinks as the sample grows; the standard deviation does not.
The strict multiplier for 95% is 1.96, not 2, but 2 is used for mental arithmetic and is what examination questions expect.
Park K. Park's Textbook of Preventive and Social Medicine, 28th ed. (Bhanot, 2025).