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DO YOU THINK YOU ARE NORMAL? 1. 2. 3. Yes 17% 17% 17% 17% No I’m not average, but I’m probably within 2 standard deviations. 17% 17% Slide 1- 1 1 2 3 4 5 6 CHAPTER 6 The Standard Deviation as a Ruler and the Normal Model NORMAL PROBABILITY PLOTS 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. Slide 1- 3 NORMAL PROBABILITY PLOTS (CONT.) A more specialized graphical display that can help you decide whether a Normal model is appropriate 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- 4 NORMAL PROBABILITY PLOTS (CONT.) Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example: Slide 1- 5 NORMAL PROBABILITY PLOTS (CONT.) A skewed distribution might have a histogram and Normal probability plot like this: Slide 1- 6 THE 68-95-99.7 RULE (CONT.) The following shows what the 68-95-99.7 Rule tells us: Slide 1- 7 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? Slide 1- 8 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 Slide 1- 9 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 lower than 85? What is the probability that someone has an IQ between 85 and 130? PROBLEM 8 A B C ABOUT WHAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES ABOVE 145? .3% .15% 3% 1.5% 5% 2.5% Slide 1- 12 WHAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES BELOW 130? 95% 5% 2.5% 97.5% Slide 1- 13 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. Slide 1- 14 FINDING NORMAL PERCENTILES Use the table in Appendix E Excel =NORMDIST(z-stat, mean, stdev, 1) Online http://davidmlane.com/hyperstat/z_table.html FINDING NORMAL PERCENTILES BY HAND (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: Slide 1- 18 CATEGORIES OF RETARDATION 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- 19 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 20 OR LESS? 0.0001% 0.0000% 0.0004% 0.04% WHAT PERCENT OF THE POPULATION HAS AN IQ OF 50 OR LESS? 0.0001% 0.0000% 0.0004% 0.04% IQ - CATEGORIES 115-124 - Above average (e.g., university students) 125-134 - Gifted (e.g., post-graduate students) 135-144 - Highly gifted (e.g., intellectuals) 145-154 - Genius (e.g., professors) 155-164 - Genius (e.g., Nobel Prize winners) 165-179 - High genius 180-200 - Highest genius >200 - "Unmeasurable genius" Slide 1- 22 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 155 OR MORE? 99.99% .01% .9999 .0001 Slide 1- 23 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 120 OR MORE? 1.333 .9082 .0918 90.82% 9.18% Slide 1- 24 AVERAGE HEIGHT IN INCHES WHAT FRACTION OF MEN ARE LESS THAN 5’9 FOOT TALL? 50% .1027 54.09% 45.91% WHAT FRACTION OF WOMEN ARE LESS THAN 5’9 FOOT TALL? 1.78 96.25% 3.75% 45.91% Slide 1- 27 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? Slide 1- 28 HEIGHT PROBLEM At what height does a quarter of men fall below? At what height does a quarter of women fall below? Slide 1- 29 FROM PERCENTILES TO SCORES: Z IN REVERSE (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- 30 Slide 1- 31 Z SCORE CALCULATORS Excel =NORMINV(prob, mean, stdev) =NORMINV(0.25, 0, 1) Online Calculator TI – 83/84 TI-89 Slide 1- 32 TI- 83/84 Slide 1- 33 TI - 89 Slide 1- 34 RECOVERING THE MEAN AND STANDARD DEV. 17.5% 18 and under 7.6% 65 and over What is the mean and the standard deviation of the population? Slide 1- 35 FOR NEXT WEEK… Monday HW3 Data Project Step 2 – Due Tuesday. Thursday Quiz 2, covers HW2, HW3 and HW4 of the material learned in class.