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Unit 5 Data Analysis

Unit 5 Data Analysis. MM3D3. 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

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Unit 5 Data Analysis

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  1. Unit 5 Data Analysis

  2. MM3D3 Empirical Rule

  3. 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 • When a distribution is normal we use shorthand to show the mean and standard deviation • Sample: N • Population: N

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

  5. Normal Curves • The mean is in the exact middle • Add and subtract the standard deviation to find the inflection points

  6. 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%

  7. Empirical Rule Expanded • Sometimes, it is helpful to know the percent of the curve that is represented by each section of the distribution.

  8. The middle 68 % ? ? 34% 34%

  9. The middle 95 % 34% 34% 13.5% ? ? 13.5%

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

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

  12. 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

  13. Normal Curves • IQ scores are normally distributed with a mean of 110 and standard deviation 25. 60 110 85 135 160 185 35

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

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

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

  17. IQ Scores • N (110, 25) What percent falls between 85 and 160? 81.5 110 135 160 85 185 35 60

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

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

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

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

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

  23. Z-Scores

  24. 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%

  25. 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

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

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

  28. 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:

  29. 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

  30. 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%

  31. 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%

  32. 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%

  33. 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%

  34. 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%

  35. 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%

  36. 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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