Lesson 5: Normal Curve and Z-Scores

September 14, 2026

Review:

• Exam 1 Results

Presentation:

• Normal Curves
  • Normal Curve explained in 1 min
  • Density Curves
    • Height of curve indicates proportion of values
    • Area under the curve = 1.0
    • Any sub-area under the curve is then a proportion (% of values)
  • Normal Curves
    • A special case of density curves
    • Bell shaped and symmetrical
    • Mean and Standard deviation
    • 68 – 95 – 99.7 Empirical Rule
    • Video – Normal Curves
    • Demonstrate with student height data
• Standard Normal Distribution
  • Mean = 0
  • Standard Deviation = 1
  • Calculating Z-Scores
    • (x – mean)/std dev
    • Example
      • Young women heights normally distributed
      • Mean = 64.5 in
      • Standard Deviation = 2.5 in
      • Z-Score for woman 68 inches tall
        • Z = (68 – 64.5)/2.5 = 1.40
      • Z-Score for woman 60 inches tall
        • Z = (60 – 64.5)/2.5 = -1.80
  • 68 – 95 – 99.7 Empirical Rule
    • Add or subtract multiples of Standard Deviation from the mean to find specific heights
    • For example, 64.5 + 5 = 69.5 is 2.00 standard deviations above the mean
    • And, 64.5 – 5 = 59.5 is 2.00 standard deviations below the mean
    • Using the Empirical Rule, 95% of women are between 59.5 and 69.5 inches tall.

Activity:

Problem 1. Normal Curves and the 68–95–99.7 Rule

The Indiana Statewide Testing for Educational Progress (ISTEP) is a program for assessing the skills of students in various grades. In a recent year, 76,531 tenth grade Indiana students took the English language arts exam. The mean score was 572 and the standard deviation was 51. Assume the ISTEP scores are approximately Normally distributed to answer the following questions.

1. Draw a Normal curve and label the mean.
2. Label the values one, two, and three standard deviations above and below the mean.
3. Give the interval containing approximately 68% of scores.
4. Give the interval containing approximately 95% of scores.
5. Give the interval containing approximately 99.7% of scores.
6. Approximately what percentage of students scored above 674?
7. Approximately what percentage scored below 521?

Problem 2. Heights and Z-Scores

The heights of young women ages 18–24 are approximately Normally distributed with mean 64.5 inches and standard deviation 2.5 inches.

1. Draw a Normal curve and label the mean.
2. Label the values one, two, and three standard deviations above and below the mean.
3. A 20-year-old woman is 6 feet tall. Calculate her z-score.
4. Interpret the z-score in words.
5. Approximately what percentage of young women are taller than 69.5 inches?
6. Approximately what percentage are between 59.5 and 69.5 inches?

Problem 3. Standard Normal Areas

For a standard Normal distribution, find the percentage of observations that satisfy each condition. For each one, draw a Normal curve and shade the appropriate area.

1. $z < -1$
2. $z < 1$
3. $z > 1$
4. $-1 < z < 1$
5. $z > 2$
6. $-2 < z < 2$

Problem 4. Comparing Scores with Z-Scores

Scores on the SAT Math exam are approximately Normally distributed with mean 500 and standard deviation 100. ACT Math scores are approximately Normally distributed with mean 18 and standard deviation 6.

Julie earned a 630 on SAT Math. John earned a 22 on ACT Math.

1. Calculate Julie’s z-score.
2. Calculate John’s z-score.
3. Interpret each z-score in words.
4. Assuming the two exams measure similar mathematical ability, who performed better relative to other test takers?
5. What percentage of SAT Math scores are above 600?