Virginia SOL Math / Algebra II / Statistics

A2.ST.1

The Data Cycle & Normal Curves

What you'll learn

  • Describe the shape of a normal distribution
  • Apply the 68-95-99.7 rule using the mean and standard deviation
  • Estimate percentages within given intervals

Words to know

Normal distribution —
a symmetric, bell-shaped distribution
Mean —
the balance point, at the center of the curve
Standard deviation —
a measure of spread from the mean
Empirical rule —
the 68-95-99.7 percentages
Symmetric —
the two halves are mirror images

Why it matters Many natural measurements, such as heights and test scores, follow a bell curve, so the empirical rule quickly estimates how common a value is.

Worked examples

Model ItI do

Test scores are normal with mean 70 and standard deviation 8. What percent score between 62 and 78?

The interval 62 to 78 is one standard deviation below to one above the mean of 70, since 70 - 8 = 62 and 70 + 8 = 78. By the empirical rule, about 68% of scores fall within one standard deviation of the mean.

Try It TogetherWe do

Using the same distribution, what percent score between 54 and 86?

This interval is two standard deviations on each side (70 - 16 = 54 and 70 + 16 = 86). By the empirical rule, about 95% of scores fall there.

Now YouYou do

What percent of scores are above the mean of 70?

Use the symmetry of the bell curve about the mean. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.About what percent of a normal distribution lies within 1 standard deviation of the mean?

  2. 2.About what percent lies within 2 standard deviations of the mean?

  3. 3.What percent of a normal distribution is below the mean?

On Level

The core Quick Check from the book.

  1. 1.About what percent lies within 1 standard deviation of the mean?

  2. 2.About what percent lies within 2 standard deviations?

  3. 3.About what percent lies within 3 standard deviations?

  4. 4.Where is the mean located on a normal curve?

  5. 5.What percent of a normal distribution is below the mean?

Challenge

Stretch your thinking.

  1. 1.Heights are normal with mean 64 in and standard deviation 3 in. About what percent are between 61 and 67 in?

  2. 2.Scores are normal with mean 500 and standard deviation 100. About what percent score above 700?

  3. 3.For a normal distribution, about what percent lies between the mean and one standard deviation above it?