1. D O YOU THINK YOU ARE NORMAL ? Slide 1- 1 1. Yes 2. No 3. I’m not average, but I’m probably within 2 standard deviations.

2. U PCOMING I N C LASS Part 1 of the Data Project due (9/5) Homework #3 due Sunday (9/9) at 11:59 pm Quiz #2 next Wed. (9/12) in class (open book) Slide 4- 2

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

4. W HAT A BOUT S PREAD ? T HE S TANDARD D EVIATION ( CONT.) The variance, notated by s 2, is found by summing the squared deviations and (almost) averaging them: The variance will play a role later in our study, but it is problematic as a measure of spread—it is measured in squared units!

5. W HAT A BOUT S PREAD ? T HE S TANDARD D EVIATION ( CONT.) The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data.

6. T HE S TANDARD D EVIATION AS A R ULER 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.

7. S TANDARDIZING WITH Z - SCORES Slide 1- 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.

8. S TANDARDIZING WITH Z - SCORES ( CONT.) Standardized values have no units. z -scores measure the distance of each data value from the mean in standard deviations. 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.

9. B ENEFITS OF S TANDARDIZING Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean. Thus, we can compare values that are measured on different scales, with different units, or from different populations.

10. S TANDARDIZING D ATA By calculating z-scores for each observation, we change the distribution of the data by Shifting the data Rescaling the data

11. S HIFTING D ATA 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.

12. S HIFTING D ATA The following histograms show a shift from men’s actual weights to kilograms above recommended weight:

13. R ESCALING D ATA 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.

14. R ESCALING D ATA ( 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: Slide 1- 14

15. T WO STANDARDIZED TESTS, A AND B USE VERY DIFFERENT SCALES OF SCORES. T HE FORMULA A=50*B+200 APPROXIMATES THE RELATIONSHIP BETWEEN SCORES ON THE TWO TWO TEST. U SE THE SUMMARY STATISTICS WHO TOOK TEST B TO DETERMINE THE SUMMARY STATISTICS FOR EQUIVALENT SCORES ON TEST A. Lowest = 18Mean = 26 St. Dev=5 Q3=28Median=30IQR = 6 Slide 1- 15

16. W HAT IS THE LOWEST SCORE FOR TEST A? Slide 1- 16 1. 18 2. 50 3. 200 4. 250 5. 1100 6. 1500 7. 2000

17. W HAT IS THE MEAN FOR TEST A? Slide 1- 17 1. 26 2. 50 3. 200 4. 250 5. 1100 6. 1500 7. 2000

18. W HAT IS THE STANDARD DEVIATION ? Slide 1- 18 1. 200 2. 250 3. 450 4. 500 5. 1100 6. 1500

19. W HAT IS THE Q3 FOR TEST A? Slide 1- 19 1. 1000 2. 1400 3. 1500 4. 1600 5. 2000

20. W HAT IS THE MEDIAN FOR TEST A? Slide 1- 20 1. 1000 2. 1400 3. 1500 4. 1700 5. 2000

21. W HAT IS THE IQR FOR TEST A? Slide 1- 21 1. 200 2. 250 3. 300 4. 500 5. 1000

22. B ACK 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- scores does not change the shape of the distribution. Standardizing into z- scores changes the center by making the mean 0. Standardizing into z- scores changes the spread by making the standard deviation 1. Slide 1- 22

23. W HEN I S A Z - SCORE B IG ? A z -score gives us an indication of how unusual a value is because it tells us how far it is from the mean. The larger a z -score is (negative or positive), the more unusual it is. We use the theory of the Normal Model to see.

24. N ORMAL M ODEL The following shows what the 68-95-99.7 Rule tells us: Slide 1- 24

25. N ORMAL M ODEL There is a Normal model for every possible combination of mean and standard deviation. We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ. We use Greek letters because this mean and standard deviation do not come from data—they are numbers (called parameters) that specify the model. Slide 1- 25

26. W HEN I S A Z - SCORE B IG ? ( CONT.) Summaries of data, like the sample mean and standard deviation, are written with Latin letters. Such summaries of data are called statistics. When we standardize Normal data, we still call the standardized value a z- score, and we write

27. T WO 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

28. T O QUALIFY FOR LANGUAGE HONORS, A STUDENT MUST AVERAGE AT LEAST 85 ACROSS ALL COURSE. D O A NNA AND M EGAN QUALIFY ? Slide 1- 28 1. Only Anna qualifies 2. Both qualify 3. Neither qualify 4. Only Megan

29. W HO PERFORMED BETTER OVERALL ? Slide 1- 29 1. Anna 2. Megan

30. A SSUMING N ORMALITY Once we have standardized, we need only one model: The N (0,1) model is called the standard Normal model (or the standard Normal distribution). Be careful—don’t use a Normal model for just any data set, since standardizing does not change the shape of the distribution. Slide 1- 30

31. A SSUMING N ORMALITY When we use the Normal model, we are assuming the distribution is Normal. We cannot check this assumption in practice, so we check the following condition: Nearly Normal Condition: The shape of the data’s distribution is unimodal and symmetric. This condition can be checked with a histogram or a Normal probability plot (to be explained Thursday).

32. N ORMAL P ROBABILITY P LOTS When you actually have your own data, you must check to see whether a Normal model is reasonable. Looking at a histogram of the data is a good way to check that the underlying distribution is roughly unimodal and symmetric. A more specialized graphical display that can help you decide is the Normal probability plot. If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal. Slide 1- 32

33. N ORMAL P ROBABILITY P LOTS ( CONT.) Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example: Slide 1- 33

34. N ORMAL P ROBABILITY P LOTS ( CONT.) A skewed distribution might have a histogram and Normal probability plot like this: Slide 1- 34

35. T HE 68-95-99.7 R ULE ( CONT.) The following shows what the 68-95-99.7 Rule tells us: Slide 1- 35

36. T HREE 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 Y 1 and Y 2 ? Slide 1- 36

37. IQ – C ATEGORIZES 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 Slide 1- 37

38. W HAT DOES THE DISTRIBUTION LOOK LIKE ? 1. A 2. B 3. C

39. A SKING Q UESTIONS OF A D ATASET What is the probability that someone has an IQ over 100? What is the probability that someone has an IQ lower than 85? What is the probability that someone has an IQ between 85 and 130?

40. A BOUT WHAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES ABOVE 145? Slide 1- 40 1..3% 2..15% 3. 3% 4. 1.5% 5. 5% 6. 2.5%

41. W HAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES BELOW 130? Slide 1- 41 1. 95% 2. 5% 3. 2.5% 4. 97.5%

42. F INDING N ORMAL P ERCENTILES BY H AND 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. Slide 1- 42

43. F INDING N ORMAL P ERCENTILES BY H AND ( CONT.) Table Z is the standard Normal table. We have to convert our data to z -scores before using the table. Figure 6.7 shows us how to find the area to the left when we have a z -score of 1.80:

45. Slide 1- 45

46. F INDING N ORMAL P ERCENTILES Use the table in Appendix E Excel =NORMDIST(z-stat, mean, stdev, 1) Online http://davidmlane.com/hyperstat/z_table.html

47. C ATEGORIES OF R ETARDATION Severity of mental retardation can be broken into 4 levels: 50-70 - Mild mental retardation 35-50 - Moderate mental retardation 20-35 - Severe mental retardation IQ < 20 - Profound mental retardation Slide 1- 47

48. W HAT PERCENT OF THE POPULATION HAS AN IQ OF 20 OR LESS ? 1. 0.0001% 2. 0.0000% 3. 0.0004% 4. 0.04%

49. W HAT PERCENT OF THE POPULATION HAS AN IQ OF 50 OR LESS ? 1. 0.0001% 2. 0.0000% 3. 0.0004% 4. 0.04%

50. F ROM P ERCENTILES TO S CORES : Z IN R EVERSE 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? Slide 1- 50

51. F ROM P ERCENTILES TO S CORES : Z IN R EVERSE ( CONT.) Look in Table Z for an area of 0.2500. The exact area is not there, but 0.2514 is pretty close. This figure is associated with z = -0.67, so the first quartile is 0.67 standard deviations below the mean. Slide 1- 51

52. H EIGHT P ROBLEM At what height does a quarter of men fall below? At what height does a quarter of women fall below? Slide 1- 52

53. Z S CORE CALCULATORS Excel =NORMINV(prob, mean, stdev) =NORMINV(0.25, 0, 1) Online Calculator TI – 83/84 TI-89 Slide 1- 53

54. TI- 83/84 Slide 1- 54

55. TI - 89 Slide 1- 55

56. R ECOVERING THE M EAN AND S TANDARD D EV. 17.5% 18 and under 7.6% 65 and over What is the mean and the standard deviation of the population? Slide 1- 56