The normal curve
z-scores, the 68–95–99.7 rule, and margins of error.
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the idea
Many measurements pile up in a bell shape, the normal curve, centred on the mean. A z-score says how many standard deviations (SDs) a value sits from the mean: positive above, negative below.
The 68–95–99.7 rule: about 68% of values fall within 1 SD of the mean, 95% within 2 and 99.7% within 3. The rest splits evenly between the two tails.
Sample averages vary less than single values. Their SD, the standard error, is the SD divided by the square root of the sample size. A margin of error is z standard errors.
techniques
Distance over the SD
- Subtract the mean from the value, keeping the sign.
- Divide by the SD.
worked example
Delivery times have mean 40 minutes and standard deviation 6 minutes. What is the z-score of a delivery that takes 31 minutes?
- 31 − 40 = −9.
- −9 ÷ 6 = −1.5, so 1.5 SDs below the mean.
Answer: −1.5
Tails from the rule
- Within 1, 2 or 3 SDs: 68%, 95% or 99.7%.
- Beyond that on one side: half of what is left.
- Everything except one tail: 100% minus that tail.
worked example
Battery lives are normal with mean 40 hours and standard deviation 5 hours. Using the 68–95–99.7 rule, about what percent of batteries last longer than 35 hours?
- 35 is 1 SD below the mean.
- 68% are within 1 SD, leaving 16% in each tail.
- All but the low tail: 100% − 16% = 84%.
Answer: 84%
Divide by the root of n
- Take the square root of the sample size.
- Standard error: the SD divided by that root.
- Margin of error: z × the standard error. Round only at the end.
worked example
A population has standard deviation 15. For a sample of size 25, using z = 1.96 for 95% confidence, what is the margin of error for the mean? Round to 2 decimal places.
- The square root of 25 is 5.
- Standard error: 15 ÷ 5 = 3.
- Margin: 1.96 × 3 = 5.88.
Answer: 5.88
watch out for
- Giving the raw distance from the mean. A z-score divides it by the SD.
- Counting both tails when the question asks about one side. Beyond 2 SDs above the mean is 2.5%, not 5%.
- Dividing by the sample size instead of its square root.
practice
z-score
worked example
Delivery times have mean 46 minutes and standard deviation 3 minutes. What is the z-score of a delivery that takes 43 minutes?
Answer: -1
- z = (43 − 46) ÷ 3 = −3 ÷ 3 = −1.
- That is 1 standard deviation below the mean.
68–95–99.7 rule
worked example
Adult heights are normal with mean 179 cm and standard deviation 10 cm. Using the 68–95–99.7 rule, about what percent of adults are between 159 cm and 199 cm tall?
Answer: 95%
- 159 and 199 are 2 standard deviations either side of the mean 179.
- The rule says about 95% fall within 2 SD.
Standard error
worked example
A population has standard deviation 4. What is the standard error of the mean for samples of size 400?
Answer: 0.2
- Standard error = σ ÷ √n = 4 ÷ √400 = 4 ÷ 20 = 0.2.
Margin of error
worked example
A population has standard deviation 19. For a sample of size 121, using z = 1.96 for 95% confidence, what is the margin of error for the mean? Round to 2 decimal places.
Answer: 3.39
- Margin = z × σ ÷ √n = 1.96 × 19 ÷ 11
- = 37.24 ÷ 11 ≈ 3.3855, which rounds to 3.39.