Unit 4: Normal Distributions

Part 1StatisticsMr. Evans

Focus Points

• Graph a normal curve and summarize its important properties

• Apply the empirical rule to solve real-world problems.

Normal Distribution

• The normal distribution of a set of data is a very important class of statistical distributions.

Properties of Normal Curve

• The curve is bell-shaped with the highest point over the mean μ• It is symmetrical about the vertical line through the mean μ• The curve approaches the horizontal axis but never touches or

crosses it• The transition points between cupping upward and downward

occur above μ + σ and μ – σ

Empirical Rule

• For a distribution that is symmetrical and bell-shaped• Approximately 68% of the data values will lie within one

standard deviation on each side of the mean• Approximately 95% of the data values will lie within two

standard deviation on each side of the mean• Approximately 99.7% of the data values will lie within three standard deviations on each side of the mean

Guided Exercise # 1

• Sketch the graph with mean μ = 35 and standard deviation σ = 8.

11 19 27 35 43 51 59

Guided Exercise # 1

• Sketch the graph with mean μ = 19.3 and standard deviation σ = 4.7.

5.2 9.9 14.6 19.3 24.0 28.7 33.4

Empirical Rule Graphs

Empirical Rule Graphs

Empirical Rule Graphs

Empirical Rule Graphs

Empirical Rule Graphs

Empirical Rule Graphs

Guided Exercise # 2

The yearly wheat yield per acre on a particular farm is normally distributed with mean μ = 35 bushels and standard deviation σ = 8.a) Shade the area under the curve that represents between

19 and 35 bushels. b) Find the percentage of area over the interval between

19 and 35.

11 19 27 35 43 51 59

If we don’t match any specific graph we can use the 4th graph

x x 13.5+34=47.5%

Guided Exercise # 2

Assuming the heights of college women are normally distributed with mean μ = 65 in and standard deviation σ = 2.5 in.a) Shade the area under the curve that represents shorter

than 67.5 in.b) Find the percentage of area under 67.5 in.x x

xxx

57.5 60 62.5 65 67.5 70 72.5

Matches the right area of the to right graph on the back

84%