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The normal curve

level 34 course

z-scores, the 68–95–99.7 rule, and margins of error.

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

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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

A z-score.

  1. Subtract the mean from the value, keeping the sign.
  2. Divide by the SD.
worked example

Example: Delivery times have mean 40 minutes and standard deviation 6 minutes. What is the z-score of a delivery that takes 31 minutes?

  1. 31 − 40 = −9.
  2. −9 ÷ 6 = −1.5, so 1.5 SDs below the mean.

Answer: −1.5

Tails from the rule

A cutoff 1, 2 or 3 SDs from the mean of normal data.

  1. Within 1, 2 or 3 SDs: 68%, 95% or 99.7%.
  2. Beyond that on one side: half of what is left.
  3. Everything except one tail: 100% minus that tail.
worked example

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?

  1. 35 is 1 SD below the mean.
  2. 68% are within 1 SD, leaving 16% in each tail.
  3. All but the low tail: 100% − 16% = 84%.

Answer: 84%

Divide by the root of n

The standard error or the margin of error for a mean.

  1. Take the square root of the sample size.
  2. Standard error: the SD divided by that root.
  3. Margin of error: z × the standard error. Round only at the end.
worked example

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.

  1. The square root of 25 is 5.
  2. Standard error: 15 ÷ 5 = 3.
  3. Margin: 1.96 × 3 = 5.88.

Answer: 5.88

Tips by skill

  • Tipz-score: Subtract the mean from the value, then divide by the SD. Below the mean, the z-score is negative.
  • Tip68–95–99.7 rule: 68%, 95% and 99.7% sit within 1, 2 and 3 SDs. What is left splits evenly between the two tails.
  • TipStandard error: Take the square root of the sample size, then divide the SD by it.
  • TipMargin of error: z × SD ÷ the square root of n. Take the root first and round only at the end.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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