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### Unit 5 Data Analysis

Empirical Rule

Normal Distributions

- Normal distributions are based on two parameters
- Mean
- If you have population data use
- If you have sample data use

- Standard Deviation
- If you have population data use
- If you have sample data use s

- Mean
- When a distribution is normal we use shorthand to show the mean and standard deviation
- Sample: N
- Population: N

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

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%

Empirical Rule Expanded

- Sometimes, it is helpful to know the percent of the curve that is represented by each section of the distribution.

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

- For example: IQ scores are normally distributed with a mean of 110 and standard deviation 25.

Normal Curves

- IQ scores are normally distributed with a mean of 110 and standard deviation 25.

60

110

85

135

160

185

35

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%

Example

- IQ Scores are Normally Distributed with N(110, 25)
- Complete the axis for the curve

99.7%

95%

68%

35

60

85

160

135

185

110

Example

- What percent of the population scores lower than 85?

16%

99.7%

95%

68%

35

60

85

160

135

185

110

Example

- What percent of the population scores lower than 100?

99.7%

95%

68%

35

60

85

100

160

135

185

110

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 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

1

0

3

-2

2

-1

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%

- Subtract the table value from 1 or 100%

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%

- Subtract the table values from each other

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
- Use the z score table
- .3446 or 34.46%

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
- 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%

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
- 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%

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