Standardized scores and the Normal Model Chapter 6 standardizing with z-scores Walrus weights The mean weight for an adult male walrus is 1215 kg, with a standard deviation of 82 kg. Chuck (pictured here) weighs 1321 kg. Find Chuck’s standardized score (z-score), then interpret this in context. Standardizing with z-scores or “exp” obs mean z st dev z x We call the resulting values standardized scores, or z-scores. more walrus weights The mean weight for an adult FEMALE walrus is 812 kg, with a standard deviation of 67 kg. Delilah weighs 680 kg (she’s been watching her figure!). Find Delilah’s standardized score (z-score), then interpret this in context. comparing walrus weights… Based on the z-scores that we calculated, who’s weight is MORE UNUSUAL for their Delilah? z = 1.29 gender – Chuck or z = -1.97 In an AP Comic Design class… • Melody scored 84 on a test where the class mean = 80 and a standard deviation of 4. • Josh scored 90 on another test where the mean = 87 with a standard deviation of 3. Who scored better relative to the other students in their AP Comic Design class? the Normal Model (back to your own notes ) There is a model that shows up over and over in Statistics, called the Normal model We use the Normal model to APPROXIMATE actual distributions that are unimodal and roughly symmetric. (it’s not exact) Adult FEMALE walrus weights are APPROXIMATELY NORMALLY DISTRIBUTED, with a mean of 812 kg, and a standard deviation of 67 kg. Draw the Normal model for these female weights. 68% of observations fall within 1 of 95% of observations fall within 2 of 99.7% of observations fall within 3 of The 68-95-99.7 Rule (a.k.a. “the empirical rule”) Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. What proportion of walrus weights is… a) between 812 and 879 kg? Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. What proportion of walrus weights is… b) less than 879 kg? Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. What proportion of walrus weights is… c) between 745 and 946kg? d) greater than 946kg? e) less than 611kg? Quick facts about the Normal model 1) The total area under the standard normal curve is 1.0 (or 100%) 2) In theory, the normal curve extends FOREVER in both directions (the height never reaches zero) Once we have standardized… (converted everything into z-scores) The N(0,1) model is called the standard Normal model when we can’t use the 68-95-99.7 rule…? one option is to use CALCULUS… 1 P( x) e 2 ( x ) 2 (2 ) Gasp! 2 …take the integral under the curve… so instead, we’ll use the z-table! walruses revisited Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. If we select an adult female walrus at random, what is the probability that her weight is… a) less than 879 kg? You are expected to communicate your process. (what about MORE than 879 kg?) Your work and drawings are an important part of that Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. If we select an adult female walrus at random, what is the probability that her weight is… b) more than 780 kg? You are expected to communicate your process. Your work and drawings are an important part of that Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. If we select an adult female walrus at random, what is the probability that her weight is… c) less than 600 kg? You are expected to communicate your process. Your work and drawings are an important part of that Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. If we select an adult female walrus at random, what is the probability that her weight is… d) between 700 and 800 kg? You are expected to communicate your process. Your work and drawings are an important part of that Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. If we select an adult female walrus at random, what is the probability that her weight is… e) between 720 kg and 785 kg? You are expected to communicate your process. Your work and drawings are an important part of that using the z-table in reverse (working backwards with the Normal model) What z-score corresponds with the 60th percentile? What about the 10th percentile? Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. f) Approximately what weight represents the cutoff for the TOP 5% of adult female walrus weights? Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. g) Approximately what weight represents the cutoff for the BOTTOM 20% of adult female walrus weights? Adult FEMALE walrus weights are approximately normally distributed, with a mean of 812 kg, and a standard deviation of 67 kg. *h) What is the IQR for adult female walrus weights? Would the Normal model be appropriate for this distribution? No – to use the Normal model, the distribution should be unimodal and approximately symmetric. (standardizing DOES NOT change the shape of the distribution) Hair Lengths Students zscore 1 32.0 0.378557 2 2.0 -1.01631 3 1.0 -1.0628 4 11.0 -0.597848 5 1.0 -1.0628 6 2.0 -1.01631 7 4.0 -0.923317 8 6.0 -0.830326 9 4.0 -0.923317 10 35.0 0.518044 11 2.0 -1.01631 12 40.0 0.750521 13 60.0 1.68043 14 47.0 1.07599 15 31.0 0.332062 16 6.0 -0.830326 17 17.0 -0.318875 18 7.0 -0.78383 19 33.0 0.425053 20 57.0 1.54095 21 34.0 0.471548 <new> If a distribution is NOT approximately normal… • We CAN calculate a z-score (and it still represents a # of SD’s from the MEAN) • We CANNOT use the normal model (or the z-table) to find a probability with that zscore. California condors have a mean wingspan of 9.1 feet, with a standard deviation of 0.63 feet. If the distribution of these wingspans is approximately normal, what is the probability that a randomly selected condor has a a) less than 8 feet wingspan of… b) at least 9.9 feet c) between 8 feet and 10 feet d) Find the cut-off (in feet) for the largest 25% of wingspans. a) 0.0404 b) 0.1021 c) 0.8830 d) about 9.53 feet 6-month old male babies have a mean weight of 16.5 pounds. My little nephew weighs 20 pounds, which places him at the 95TH PERCENTILE for babies (meanwhile he was at the 50 PERCENTILE for height…) his age. TH What is the standard deviation of weights for male babies at 6 months of age? ANSWER: about 2.13 pounds the end for now… (homework #14 is VERY IMPORTANT, and due next time)