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Do you think you are normal?

Do you think you are normal?. Yes No I’m not average, but I’m probably within 2 standard deviations. Upcoming In Class. Part 1 of the Data Project due today Homework #3 due Sunday at 10:00 pm Quiz #2 in class 9/11 (open book). Chapter 6.

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Do you think you are normal?

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  1. Do you think you are normal? • Yes • No • I’m not average, but I’m probably within 2 standard deviations.

  2. Upcoming In Class • Part 1 of the Data Project due today • Homework #3 due Sunday at 10:00 pm • Quiz #2 in class 9/11 (open book)

  3. Chapter 6 The Standard Deviation as a Ruler and the Normal Model

  4. The standard deviation, s,is just the square root of the variance and is measured in the same units as the original data. What About Spread? The Standard Deviation (cont.)

  5. The Standard Deviation as a Ruler • The trick in comparing very different-looking values is to use standard deviations as our rulers. • The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group. • As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

  6. Normal Model • The following shows what the 68-95-99.7 Rule tells us:

  7. We compare individual data values to their mean, relative to their standard deviation using the following formula: We call the resulting values standardized values, denoted as z. They can also be called z-scores. Standardizing with z-scores

  8. Standardizing Data • By calculating z-scores for each observation, we change the distribution of the data by • Shifting the data • Rescaling the data

  9. Shifting Data • Shifting data: • Adding (or subtracting) a constant to every data value adds (or subtracts) the same constant to measures of position. • Adding (or subtracting) a constant to each value will increase (or decrease) measures of position: center, percentiles, max or min by the same constant. • Its shape and spread - range, IQR, standard deviation - remain unchanged.

  10. Shifting Data • The following histograms show a shift from men’s actual weights to kilograms above recommended weight:

  11. Rescaling Data • Rescaling data: • When we multiply (or divide) all the data values by any constant, all measures of position (such as the mean, median, and percentiles) and measures of spread (such as the range, the IQR, and the standard deviation) are multiplied (or divided) by that same constant.

  12. Rescaling Data (cont.) • The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data:

  13. Two standardized tests, A and B use very different scales of scores. The formula A=50*B+200 approximates the relationship between scores on the two two test. Use the summary statistics who took test B to determine the summary statistics for equivalent scores on test A. • Lowest = 18 Mean = 26 St. Dev=5 • Median=28 Q3=30 IQR = 6

  14. What is the lowest score for test A? • 18 • 50 • 200 • 250 • 1100 • 1500 • 2000

  15. What is the mean for test A? • 26 • 50 • 200 • 250 • 1100 • 1500 • 2000

  16. What is the IQR for test A? • 200 • 250 • 300 • 500 • 1000

  17. What is the Q3 for test A? • 1000 • 1400 • 1500 • 1600 • 1700

  18. Back to z-scores • Standardizing data into z-scores shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation. • Standardizing into z-scoresdoes not change the shape of the distribution. • Standardizing into z-scoreschanges the center by making the mean 0. • Standardizing into z-scoreschanges the spread by making the standard deviation 1.

  19. Standardizing with z-scores • Standardized values have no units. • z-scores measure the distance of each data value from the mean in standard deviations. Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean. • We can compare values that are measured on different scales, with different units, or from different populations. • A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

  20. Two students take language exams • Anna score 87 on both • Megan scores 76 on first, and 91 on the second • Overall student scores on the first exam • Mean=83 • St. Dev. 5 • Second exam • Mean = 70 • St. Dev. 14

  21. Who performed better overall? • Anna • Megan

  22. Three types of questions • What’s the probability of getting X or greater? • What’s the probability of getting X or less? • What’s the probability of X falling within in the range Y1 and Y2?

  23. Example: IQ – Categorizes • Over 140 - Genius or near genius • 120 - 140 - Very superior intelligence • 110 - 119 - Superior intelligence • 90 - 109 - Normal or average intelligence • 80 - 89 - Dullness • 70 - 79 - Borderline deficiency • Under 70 - Definite feeble-mindedness

  24. Normal Model • The following shows what the 68-95-99.7 Rule tells us:

  25. Asking Questions of a Dataset • What is the probability that someone has an IQ over 100? • What is the probability that someone has an IQ between 85 and 115? • What is the probability that someone has an IQ between 85 and 130?

  26. About what percent of people should have IQ scores above 145? • .3% • .15% • 3% • 1.5% • 5% • 2.5%

  27. Finding Normal Percentiles by Hand • When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles. • Table Z in Appendix E provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.

  28. Finding Normal Percentiles • Use the table in Appendix E • Excel • =NORMDIST(z-stat, mean, stdev, 1) • Online • http://davidmlane.com/hyperstat/z_table.html

  29. What percent of the population has an IQ of 50 or less? • 0.0001% • 0.0000% • 0.0004% • 0.04%

  30. What percent of the population has an IQ of 120 or more? • 1.333 • .9082 • .0918 • 90.82% • 9.18%

  31. From Percentiles to Scores: z in Reverse • Sometimes we start with areas and need to find the corresponding z-score or even the original data value. • Example: What z-score represents the first quartile in a Normal model?

  32. IQ Problem • At what IQ does a quarter of people fall below? • At what IQ does a quarter of people fall above?

  33. Upcoming In Class • Homework #3 due Sunday at 10:00 pm • Quiz #2 in class 9/11 (open book)

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