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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 UPCOMING IN CLASS 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 CHAPTER 6 The Standard Deviation as a Ruler and the Normal Model WHAT ABOUT SPREAD? THE STANDARD DEVIATION (CONT.) The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them: y y 2 s 2 n 1 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! WHAT ABOUT SPREAD? THE STANDARD DEVIATION (CONT.) The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data. y y 2 s n 1 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. STANDARDIZING WITH Z-SCORES We compare individual data values to their mean, relative to their standard deviation using the following formula: y y z s We call the resulting values standardized values, denoted as z. They can also be called z-scores. Slide 1- 7 STANDARDIZING 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. BENEFITS OF STANDARDIZING 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. STANDARDIZING DATA By calculating z-scores for each observation, we change the distribution of the data by Shifting the data Rescaling the data 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. SHIFTING DATA The following histograms show a shift from men’s actual weights to kilograms above recommended weight: 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. 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: Slide 1- 14 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 Q3=28 Median=30 IQR = 6 Slide 1- 15 WHAT IS THE LOWEST SCORE FOR TEST A? 1. 2. 3. 4. 5. 6. 7. 18 50 200 250 1100 1500 2000 14% 14% 14% 14% 14% 14% 14% Slide 1- 16 1 2 3 4 5 6 7 WHAT IS THE MEAN FOR TEST A? 1. 2. 3. 4. 5. 6. 7. 26 50 200 250 1100 1500 2000 14% 14% 14% 14% 14% 14% 14% Slide 1- 17 1 2 3 4 5 6 7 WHAT IS THE STANDARD DEVIATION? 1. 2. 3. 4. 5. 6. 200 250 450 500 1100 1500 17% 17% 17% 17% 17% 17% Slide 1- 18 1 2 3 4 5 6 WHAT IS THE Q3 FOR TEST A? 1. 2. 3. 4. 5. 1000 1400 1500 1600 2000 20% 20% 20% 20% 20% Slide 1- 19 1 2 3 4 5 WHAT IS THE MEDIAN FOR TEST A? 1. 2. 3. 4. 5. 1000 1400 1500 1700 2000 20% 20% 20% 20% 20% Slide 1- 20 1 2 3 4 5 WHAT IS THE IQR FOR TEST A? 1. 2. 3. 4. 5. 200 250 300 500 1000 20% 20% 20% 20% 20% Slide 1- 21 1 2 3 4 5 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-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 WHEN IS A Z-SCORE BIG? 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. NORMAL MODEL The following shows what the 68-95-99.7 Rule tells us: Slide 1- 24 NORMAL MODEL 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 WHEN IS A Z-SCORE BIG? (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 y y z s 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 TO QUALIFY FOR LANGUAGE HONORS, A STUDENT MUST AVERAGE AT LEAST 85 ACROSS 25% 25% 25% 25% ALL COURSE. DO ANNA AND MEGAN QUALIFY? 1. 2. 3. 4. Only Anna qualifies Both qualify Neither qualify Only Megan Slide 1- 28 1 2 3 4 WHO PERFORMED BETTER OVERALL? 1. 2. Anna Megan 50% 50% Slide 1- 29 1 2 ASSUMING NORMALITY 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 ASSUMING NORMALITY 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). 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. 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 NORMAL PROBABILITY PLOTS (CONT.) Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example: Slide 1- 33 NORMAL PROBABILITY PLOTS (CONT.) A skewed distribution might have a histogram and Normal probability plot like this: Slide 1- 34 THE 68-95-99.7 RULE (CONT.) The following shows what the 68-95-99.7 Rule tells us: Slide 1- 35 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- 36 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- 37 WHAT DOES THE DISTRIBUTION LOOK LIKE? 33% 1. 2. 3. A B C 1 33% 2 33% 3 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? ABOUT WHAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES ABOVE 145? 1. 2. 3. 4. 5. 6. .3% .15% 3% 1.5% 5% 2.5% 17% 17% 17% 17% 17% 17% Slide 1- 40 1 2 3 4 5 6 WHAT PERCENT OF PEOPLE SHOULD HAVE IQ SCORES BELOW 130? 1. 2. 3. 4. 95% 5% 2.5% 97.5% 25% 25% 25% 25% Slide 1- 41 1 2 3 4 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- 42 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- 45 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- 47 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 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- 50 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- 51 HEIGHT PROBLEM At what height does a quarter of men fall below? At what height does a quarter of women fall below? Slide 1- 52 Z SCORE CALCULATORS Excel =NORMINV(prob, mean, stdev) =NORMINV(0.25, 0, 1) Online Calculator TI – 83/84 TI-89 Slide 1- 53 TI- 83/84 Slide 1- 54 TI - 89 Slide 1- 55 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- 56