Unit 5 Data Analysis

MM3D3 Empirical Rule

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

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

Empirical Rule Expanded Sometimes, 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%

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: IQ scores are normally distributed with a mean of 110 and standard deviation 25. What percent of people score lower than 110 on the IQ test

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

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

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

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

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

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

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

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

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

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

Z-Scores

Recall: 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

Example IQ Scores are Normally Distributed with N(110, 25) Complete the axis for the curve 99.7% 95% 68% 35 60 85 110 135 160 185

Example 16% What percent of the population scores lower than 85? 99.7% 95% 68% 35 60 85 110 135 160 185

Example What percent of the population scores lower than 100? 99.7% 95% 68% 35 60 85 100 110 135 160 185

Z Scores Allow you to get percentages that don’t fall on the boundaries for the empirical rule Convert observations (x’s) into standardized scores (z’s) using the formula: 𝑧= 𝑥−𝜇 𝜎

Z Scores The z score tells you how many standard deviations the x value is from the mean The axis for the Standard Normal Curve: -3 -2 -1 1 2 3

Z Score Table: The table will tell you the proportion of the population that falls BELOW a given z-score. The left column gives the ones and tenths place The top row gives the hundredths place What percent of the population is below .56? .7123 or 71.23%

Z Score Table: The table will tell you the proportion of the population that falls BELOW a given z-score. The left column gives the ones and tenths place The top row gives the hundredths place What percent of the population is below .4? .6554 or 65.54%

Using the z score table You can also find the proportion that is above a z score Subtract the table value from 1 or 100% Find the percent of the population that is above a z score of 2.59 1-.9952 .0048 or .48% Find the percent of the population that is above a z score of -1.91 1-.0281 .9719 or 97.19%

Using the z score table You can also find the proportion that is between two z scores Subtract the table values from each other Find the percent of the population that is between .27 and 1.34 .9099-.6064 .3035 or 30.35% Find the percent of the population that is between -2.01 and 1.89 .9706-.0222 .9484 or 94.84%

Application 1 IQ Scores are Normally Distributed with N(110, 25) What percent of the population scores below 100? Convert the x value to a z score 𝑧= 𝑥−𝜇 𝜎 Use the z score table .3446 or 34.46% = 100−110 25 =−.4

Application 2 IQ Scores are Normally Distributed with N(110, 25) What percent of the population scores above 115? Convert the x value to a z score 𝑧= 𝑥−𝜇 𝜎 Use the z score table .5793 fall below This question is asking for above, so you have to subtract from 1. 1-.5793 .4207 or 42.07% = 115−110 25 =.2

Application 3 IQ Scores are Normally Distributed with N(110, 25) What percent of the population score between 50 and 150? Convert the x values to z scores 𝑧= 𝑥−𝜇 𝜎 Use the z score table .9452 and .0082 This question is asking for between, so you have to subtract from each other. .9452-.0082 .9370 or 93.7% = 150−110 25 =1.6 = 50−110 25 =−2.4