Normal Distributions Normal Distributions and the Empirical Ruleand the Empirical Rule

Learning Target: Learning Target: I can use percentiles and the I can use percentiles and the Empirical rule to determine relative standing of data on Empirical rule to determine relative standing of data on

the standard normal curve.the standard normal curve.

Notes: on your own paperNotes: on your own paper

Normal distributions: N (Normal distributions: N (μ, σ)μ, σ)

SymmetricSymmetric, single peaked and bell shaped., single peaked and bell shaped.

Center of the curve are Center of the curve are μμ, , MMd d & Mode& Mode..

Standard deviation Standard deviation σ controls the spread σ controls the spread (variability) (variability) of the curve.of the curve.

Normal curves are a good description Normal curves are a good description of some real data:of some real data:

test scorestest scores biological measurements biological measurements also approximate chance outcomes like also approximate chance outcomes like

tossing coinstossing coins

The Empirical rule The Empirical rule (68-95-99.7 rule)(68-95-99.7 rule)

In the normal dist. with mean μ and In the normal dist. with mean μ and standard deviation σ.standard deviation σ.

68%68% of the observations fall within of of the observations fall within of the mean.the mean.

95%95% of the observations fall within of of the observations fall within of the mean.the mean.

99.7%99.7% of the observations fall within of of the observations fall within of the mean.the mean.

1σ

2σ

3σ

Normal distributions: N (Normal distributions: N (μ, σ)μ, σ)

Inflection points:Inflection points: points where change of points where change of curvature takes place is located a distance curvature takes place is located a distance σ on either side of μ.σ on either side of μ.

Create a Normal Distribution Create a Normal Distribution Curve, N (Curve, N (μ, σ)μ, σ)

Create a Normal Distribution Create a Normal Distribution Curve, N (Curve, N (μ, σ)μ, σ)

Create a Normal Distribution Create a Normal Distribution Curve, N (Curve, N (μ, σ)μ, σ)

99.7%

95%

9

68%

Example: MenExample: Men’’s Heightss HeightsThe distribution of adult American men is The distribution of adult American men is approximately normal with mean approximately normal with mean 6969 inches and inches and standard deviation standard deviation 2.5 inches2.5 inches. Draw the curve . Draw the curve and and mark points if inflectionmark points if inflection. .

69 71.5 74 76.566.56461.5

N(69, 2.5) Inflection Points

Example: MenExample: Men’’s Heightss HeightsThe distribution of adult American men is approximately normal The distribution of adult American men is approximately normal with mean with mean 6969 inches and standard deviation inches and standard deviation 2.5 inches2.5 inches. .

a) What percent of men are taller than 74inches?a) What percent of men are taller than 74inches?

69 71.5 74 76.566.56461.5

.0235 + .0015 = .025 = 2.5%

Example: MenExample: Men’’s Heightss HeightsThe distribution of adult American men is approximately normal The distribution of adult American men is approximately normal with mean with mean 6969 inches and standard deviation inches and standard deviation 2.5 inches2.5 inches. .

b) Between what heights do the middle 95% of men fall?b) Between what heights do the middle 95% of men fall?

69 71.5 74 76.566.56461.5

95%

Example: MenExample: Men’’s Heightss HeightsThe distribution of adult American men is approximately normal The distribution of adult American men is approximately normal with mean with mean 6969 inches and standard deviation inches and standard deviation 2.5 inches2.5 inches. .

c) c) What percent of men are shorter than 66.5 inchesWhat percent of men are shorter than 66.5 inches??

69 71.5 74 76.566.56461.5

.135

.0235.0015

.0015 +.0235 +.135 = .16 = 16%

Homework:Homework:Textbook: Textbook:

Page 89, problems 2.8 and 2.9Page 89, problems 2.8 and 2.9

For each problem, For each problem, Draw the normal distribution curveDraw the normal distribution curve Mark the points of inflectionMark the points of inflection Label the x-values for each standard Label the x-values for each standard

deviationdeviation AnswAnswer all parts of each questioner all parts of each question