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Unit 5 Data Analysis

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Unit 5 Data Analysis

MM3D3

- 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

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

- The mean is in the exact middle
- Add and subtract the standard deviation to find the inflection points

- 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%

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

?

?

34%

34%

34%

34%

13.5%

?

?

13.5%

34%

34%

13.5%

13.5%

?

?

2.35%

2.35%

34%

34%

13.5%

13.5%

0.15%

0.15%

?

?

2.35%

2.35%

- 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.

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

60

110

85

135

160

185

35

- N (110, 25)

What is the 68% range?

85-135

110

135

160

85

185

35

60

- N (110, 25)

What is the 95% range?

60-160

110

135

160

85

185

35

60

- N (110, 25)

What is the 99.7% range?

35-185

110

135

160

85

185

35

60

- N (110, 25)

What percent falls between 85 and 160?

81.5

110

135

160

85

185

35

60

- N (110, 25)

What percent falls between 35 and 135?

83.85

110

135

160

85

185

35

60

- N (110, 25)

What percent falls below 85?

16

110

135

160

85

185

35

60

- N (110, 25)

What percent falls below 185?

99.85

110

135

160

85

185

35

60

- N (110, 25)

What percent is above 185?

0.15

110

135

160

85

185

35

60

- N (110, 25)

What percent is above 60?

97.5

110

135

160

85

185

35

60

- 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%

- 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

- What percent of the population scores lower than 85?

16%

99.7%

95%

68%

35

60

85

160

135

185

110

- What percent of the population scores lower than 100?

99.7%

95%

68%

35

60

85

100

160

135

185

110

- 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:

- 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

- 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%

- 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%

- 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%

- 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

- 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%

- 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%

- 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%