Empirical Rule MM3D3

Normal Distributions Normal distributions are based on two parameters Mean If you have population data use 𝜇 If you have sample data use 𝑥 Standard Deviation If you have population data use 𝜎 If you have sample data use s When a distribution is normal we use shorthand to show the mean and standard deviation Sample: N 𝑥 , 𝑠 Population: N 𝜇, 𝜎

Normal Curves Use the parameters to find the inflection points on the curve. Where the curve changes concavity

Normal Curves The mean is in the exact middle Add and subtract the standard deviation to find the inflection points 𝑥 −3𝑠 𝑥 −2𝑠 𝑥 −𝑠 𝑥 𝑥 +𝑠 𝑥 +2𝑠 𝑥 +3𝑠

Normal Curves IQ scores are normally distributed with a mean of 110 and standard deviation 25. 𝑥 −3𝑠 𝑥 −2𝑠 𝑥 −𝑠 𝑥 𝑥 +𝑠 𝑥 +2𝑠 𝑥 +3𝑠 35 60 85 110 135 160 185

Empirical Rule 68% of the data is within one standard deviation of the mean 95% of the data is within two standard deviations of the mean 99.7% of the data is within three standard deviations of the mean 99.7% 95% 68% 𝑥 −3𝑠 𝑥 −2𝑠 𝑥 −𝑠 𝑥 𝑥 +𝑠 𝑥 +2𝑠 𝑥 +3𝑠

Applying the Empirical Rule Often the empirical rule is used to determine the percent of data that falls above or below a point of inflection. For example what percent of people score lower than 110 on the IQ test It is helpful to know the percent of the curve that is represented by each section of the distribution.

The middle 68 % ? ? 34% 34%

The middle 95 % 34% 34% 13.5% ? ? 13.5%

The middle 99.7 % 34% 34% 13.5% 13.5% ? 2.35% 2.35% ?

The Tails 34% 34% 13.5% 13.5% 0.15% ? ? 0.15% 2.35% 2.35%

IQ Scores What is the 68% range? 85-135 N (110, 25) 35 60 85 110 135 160 185 What is the 68% range? 85-135

IQ Scores What is the 95% range? 60-160 N (110, 25) 35 60 85 110 135 185 What is the 95% range? 60-160

IQ Scores What is the 99.7% range? 35-185 N (110, 25) 35 60 85 110 135 160 185 What is the 99.7% range? 35-185

IQ Scores What percent falls between 85 and 160? 81.5 N (110, 25) 35 135 160 185 What percent falls between 85 and 160? 81.5

IQ Scores What percent falls between 35 and 135? 83.85 N (110, 25) 35 60 85 110 135 160 185 What percent falls between 35 and 135? 83.85

IQ Scores What percent falls below 85? 16 N (110, 25) 35 60 85 110 135 160 185 What percent falls below 85? 16

IQ Scores What percent falls below 185? 99.85 N (110, 25) 35 60 85 110 135 160 185 What percent falls below 185? 99.85

IQ Scores What percent is above 185? 0.15 N (110, 25) 35 60 85 110 135 160 185 What percent is above 185? 0.15

IQ Scores What percent is above 60? 97.5 N (110, 25) 35 60 85 110 135 160 185 What percent is above 60? 97.5