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DO YOU THINK YOU ARE NORMAL? 1. 2. 3. Yes 33% 33% No I’m not average, but I’m probably within 2 standard deviations. 33% Slide 1- 1 1 2 3 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 *RE-EXPRESSING SKEWED DATA TO IMPROVE SYMMETRY Slide 1- 7 TRANSFORMING DATA y=Log(x) To get original data back x=10^y =10y y=Sqrt(x) To get original data back x=y^2 = y*y Slide 1- 8 *RE-EXPRESSING SKEWED DATA TO IMPROVE SYMMETRY (CONT.) One way to make a skewed distribution more symmetric is to re-express or transform the data by applying a simple function (e.g., logarithmic function). Note the change in skewness from the raw data (previous slide) to the transformed data (right): THE 68-95-99.7 RULE (CONT.) The following shows what the 68-95-99.7 Rule tells us: Slide 1- 10 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- 11 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- 12 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- 15 FINDING NORMAL PERCENTILES Use the table in Appendix E Excel =NORMDIST(z-stat, mean, stdev, 1) Online http://davidmlane.com/hyperstat/z_table.html 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- 17 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 20 OR LESS? 1. 2. 3. 4. 0.0001% 0.0000% 0.0004% 0.04% 25% 1 25% 25% 2 3 25% 4 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 50 OR LESS? 1. 2. 3. 4. 0.0001% 0.0000% 0.0004% 0.04% 25% 1 25% 25% 2 3 25% 4 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- 20 WHAT PERCENT OF THE POPULATION HAS AN IQ OF 155 OR MORE? 1. 2. 3. 4. 99.99% .01% .9999 .0001 25% 25% 25% 25% Slide 1- 21 1 2 3. 4. WHAT PERCENT OF THE POPULATION HAS AN IQ OF 120 OR MORE? 1. 2. 3. 4. 5. 1.333 .9082 .0918 90.82% 9.18% 20% 20% 20% 20% 20% Slide 1- 22 1. 2. 3. 4 5 AVERAGE HEIGHT IN INCHES WHAT FRACTION OF MEN ARE LESS THAN 5’9 FOOT TALL? 1. 2. 3. 4. 50% .1027 54.09% 45.91% 25% 1 25% 25% 2. 3 25% 4 WHAT FRACTION OF WOMEN ARE LESS THAN 5’9 FOOT TALL? 1. 2. 3. 4. 1.78 96.25% 3.75% 45.91% 25% 25% 25% 25% Slide 1- 25 1. 2 3 4 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- 26 HEIGHT PROBLEM At what height does a quarter of men fall below? At what height does a quarter of women fall below? Slide 1- 27 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- 28 Z SCORE CALCULATORS Excel =NORMINV(prob, mean, stdev) =NORMINV(0.25, 0, 1) Online http://stattrek.com/online-calculator/normal.aspx Calculator TI – 83/84 TI-89 Slide 1- 29 TI- 83/84 Slide 1- 30 TI - 89 Slide 1- 31 RECOVERING THE MEAN AND STANDARD DEV. 17.5% 18 and under 7.6% 65 and over What is the mean age and the standard deviation of the population? Slide 1- 32 FOR NEXT WEEK… Sunday HW3 (2/3) by midnight Thursday Quiz 2, covers HW3 and HW4 of the material learned in class. Data Project Step 2 – Due Tuesday (2/12).