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๐Ÿ“ˆ Intro to Statistics ยท Lesson 4 of 7

The normal curve and z-scores

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The 68โ€“95โ€“99.7 rule and comparing values on different scales.

The bell curve

Many measurements โ€” heights, measurement errors, many test scores โ€” follow a roughly normal distribution: symmetric, bell-shaped, described by its mean ฮผ and standard deviation ฯƒ.

The 68โ€“95โ€“99.7 rule

Within 1 ฯƒ of the meanAbout 68% of values
Within 2 ฯƒAbout 95%
Within 3 ฯƒAbout 99.7%

Example

SAT Math scores are roughly normal with mean โ‰ˆ 520 and ฯƒ โ‰ˆ 115 (illustrative numbers). About 68% of students score between 405 and 635, and about 95% between 290 and 750. A score of 750 is about 2 ฯƒ above the mean โ€” better than about 97.5% of test-takers.

z-scores

z = (x โˆ’ mean) / standard deviation

Exam A: you scored 82, class mean 70, SD 6  โ†’ z = (82 โˆ’ 70)/6 = 2.0
Exam B: you scored 90, class mean 85, SD 10 โ†’ z = (90 โˆ’ 85)/10 = 0.5

The 82 was the more impressive score relative to the class, even though 90 is higher.

Tip

๐Ÿ’ก A z-score says how many standard deviations a value is from the mean. Positive = above, negative = below. |z| > 2 is unusual; |z| > 3 is very rare.

โœ๏ธ Practice: your turn

  1. Use the 68โ€“95โ€“99.7 rule on heights: mean 69 in, SD 3 in โ€” what range holds 95%?
  2. Compute z-scores for two of your own exam scores (estimate class mean and SD)
  3. Find what percent of a normal distribution is above z = 1
  4. Sketch a bell curve and mark ฮผ, ยฑ1ฯƒ, ยฑ2ฯƒ

โœ๏ธ Scientific calculator with a statistics tab (mean, median, standard deviation)

๐Ÿง  Quick quiz

Answer all 4 questions correctly to mark this lesson as done.

1. About what percent of normal data fall within 2 SD of the mean?
2. Mean 70, SD 6, score 82: z =
3. A negative z-score meansโ€ฆ
4. Which is most unusual?

0 of 4 correct

All lessons in Intro to Statistics

  1. ๐ŸŽฒ 1. Data, variables and samplingCategorical vs. quantitative, populations, samples and bias
  2. ๐ŸŽฏ 2. Mean, median and modeMeasures of center, and which one to use when data are skewed
  3. โ†”๏ธ 3. Spread: range, IQR and standard deviationHow spread out the data are, and how to read a box plot
  4. ๐Ÿ”” 4. The normal curve and z-scoresThe 68โ€“95โ€“99.7 rule and comparing values on different scales
  5. ๐Ÿ“‰ 5. Correlation and regressionScatterplots, r, the line of best fit โ€” and causation
  6. ๐ŸŽฏ 6. Confidence intervalsEstimate ยฑ margin of error, and what โ€œ95% confidentโ€ means
  7. โš—๏ธ 7. Hypothesis tests and p-valuesNull and alternative, the p-value, significance and errors