Skills

The Normal Curve and Percentiles

In a normal distribution, scores form a symmetric bell curve around the mean.

Practice 3: Data Interpretation

Part 1 · Hook

Why this matters

You score 72 on a biology test and 85 on a history test. Which went better? It depends on how everyone else did. If the class averaged 60 in biology and 80 in history, your 72 might be the more impressive score. The normal curve and the z score let you compare scores from different tests on the same scale.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A test has a mean of 75 and a standard deviation of 15. What does the standard deviation describe?

  1. How spread out the scores are around the mean
  2. The most common score on the test
  3. The middle score when they are put in order
  4. The difference between the highest and lowest scores
Show the answer

The standard deviation is roughly the typical distance of scores from the mean.

  • Correct: How spread out the scores are around the mean:
  • The most common score on the test:
  • The middle score when they are put in order:
  • The difference between the highest and lowest scores:

2. What is the mean of 6, 8 and 10?

  1. 8
  2. 24
  3. 6
  4. 10
Show the answer

The total is 24, and 24 divided by 3 values is 8.

  • Correct: 8:
  • 24:
  • 6:
  • 10:

Part 4 · See it

See it first

A bell-shaped normal curve with vertical lines at the mean and at 1, 2 and 3 standard deviations on each side. The bands hold 34% of scores between the mean and 1 SD on each side, 13.5% between 1 and 2 SD, 2.35% between 2 and 3 SD, and 0.15% beyond 3 SD. Below the axis, z scores run from minus 3 to plus 3, with percentiles 0.15, 2.5, 16, 50, 84, 97.5 and 99.85. Brackets show 68% of scores within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
The normal curve: 68% of scores within one standard deviation of the mean, 95% within two and 99.7% within three. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Many traits and test scores are shaped by many small influencestheir frequency distribution is bell-shaped and symmetric: the normal curve
  2. Mark the mean and each standard deviation on the curvethe 68-95-99.7 rule gives the share of scores between the marks
  3. Subtract the mean from a score and divide by the standard deviationthe z score says how many standard deviations the score is from the mean
  4. Add up the share of scores below a markthe percentile rank: 50 at the mean, 84 at one SD above, 16 at one SD below

Part 6 · Key ideas

Key ideas

  • A normal curve is symmetric and bell-shaped: mean, median and mode are equal, in the middle.
  • The 68-95-99.7 rule: about 68% of scores lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Bands: 34%, 13.5%, 2.35% on each side.
  • A z score: z=x−x¯SDz = \frac{x - \bar{x}}{\text{SD}}. Above the mean, positive; below, negative.
  • A percentile rank is the percent of scores below a score, not the percent correct. One SD above the mean is the 84th percentile.

Part 7 · Misconception

A common mistake

The wrong idea: The 90th percentile means getting 90% of the questions right, and 68% means a score at the 68th percentile.

What actually happens: A percentile is a rank among test takers: the 90th percentile beats about 90% of them, whatever the percent correct. 68% is the share of scores within one standard deviation on both sides, not a percentile.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. Scores on a memory test are normally distributed with a mean of 50 and a standard deviation of 10. What is the z score of a student who scored 70?

Type a number. No calculator needed: work it out by hand, as on the exam.

Show the answer

zz=70−5010= \frac{70 - 50}{10}=2010= \frac{20}{10}=2= 2: two standard deviations above the mean.

  • Answer: 2

2. In a normal distribution, what percent of scores fall more than one standard deviation above the mean?

Type a number (%). No calculator needed: work it out by hand, as on the exam.

Show the answer

Half the scores are above the mean (50%), and 34% are between the mean and z=1z = 1. So 50−34=16%50 - 34 = 16\% are above z=1z = 1.

  • Answer: 16 %

3. Maya scored at the 90th percentile on a reading test. What does that mean?

  1. She scored higher than about 90% of test takers
  2. She answered about 90% of the test questions correctly
  3. Her score is 90 points above the mean
  4. She is in the top 90% of the test takers
Show the answer

A percentile rank is the percent of scores that fall below a given score. Percent correct is a different number.

  • Correct: She scored higher than about 90% of test takers: Right: a percentile rank is the percent of scores below hers.
  • She answered about 90% of the test questions correctly: Percentile is about rank among test takers, not the percent correct.
  • Her score is 90 points above the mean: A percentile is a rank, not a distance in points.
  • She is in the top 90% of the test takers: The top 90% would include nearly everyone; she is in the top 10%.

Figure

A normal curve of test scores

557085100115130145Reading test score34%34%13.5%13.5%2.35%2.35%

4. By the 68-95-99.7 rule, what percent of test takers scored between 85 and 115?

Type a number (%). No calculator needed: work it out by hand, as on the exam.

Show the answer

85 and 115 are one standard deviation below and above the mean of 100: 34+34=68%34 + 34 = 68\%.

  • Answer: 68 %

5. What percent of test takers scored above 130?

Type a number (%). No calculator needed: work it out by hand, as on the exam.

Show the answer

130 is two standard deviations above the mean. Above it: 2.35+0.15=2.5%2.35 + 0.15 = 2.5\%.

  • Answer: 2.5 %

6. About what percentile rank does a score of 115 have?

  1. The 34th percentile
  2. The 68th percentile
  3. The 84th percentile
  4. The 115th percentile
Show the answer

Percentile rank is the percent of scores below a score: 50+34=8450 + 34 = 84.

  • The 34th percentile: 34% is the band from 100 to 115, not the percent below 115.
  • The 68th percentile: 68% lies between 85 and 115; it is not the percent below 115.
  • Correct: The 84th percentile: Right: 50% below the mean plus 34% from 100 to 115.
  • The 115th percentile: Percentile ranks run from 0 to 100; 115 is the score.

Part 9 · Summary

Summary

In a normal distribution, scores form a symmetric bell curve around the mean. By the 68-95-99.7 rule, about 68% of scores lie within one standard deviation of the mean, 95% within two and 99.7% within three. A z score gives a score’s distance from the mean in standard deviations, which lets you compare scores from different tests, and a percentile rank gives the percent of scores below it.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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