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standard deviations standard deviation spread distance difference Sample Data Consider the following test scores for a small class: 79 81 80 77 73 83 74 93 78 80 75 67 77 83 86 90 79 85 83 89 84 82 77 72 73 Jenny’s score is noted in red. How did she perform on this test relative to her peers? 6| 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 6| 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Her score is “above average”... but how far above average is it? a standardized value a z-score no comparable change the SHAPE the center….your new mean is ZERO the spread….your new standard deviation is ONE N (0, 1) Calculating z-scores Consider the test data and Jenny’s score. 79 81 80 77 73 83 74 93 78 80 75 67 77 83 86 90 79 85 83 89 84 82 77 72 According to Minitab, the mean test score was 80 while the standard deviation was 6.07 points. Jenny’s score was above average. Her standardized z-score is: x 80 86 80 z 0.99 6.07 6.07 Jenny’s score was almost one full standard deviation above the mean. What about Kevin: x= 73 Calculating z-scores 79 81 80 77 73 83 74 93 78 80 75 67 77 83 86 90 79 85 83 89 84 82 77 72 6| 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 569 9 | 03 Jenny: z=(86-80)/6.07 z= 0.99 {above average = +z} Kevin: z=(72-80)/6.07 z= -1.32 {below average = -z} Katie: z=(80-80)/6.07 z= 0 {average z = 0} 73 Comparing Scores Standardized values can be used to compare scores from two different distributions. Statistics Test: mean = 80, std dev = 6.07 Chemistry Test: mean = 76, std dev = 4 Jenny got an 86 in Statistics and 82 in Chemistry. On which test did she perform better? Statistics 86 80 z 0.99 6.07 Chemistry 82 76 z 1.5 4 Although she had a lower score, she performed relatively better in Chemistry. -∞ = -EE99 ∞=EE99 2:Normalcdf (lower bound, upper bound) N(500,100) What percentile will you be in if you scored a 720 on the SAT?? N(500,100) What percent of students scored between 550 and 630 on the SAT?? under the normal curve ONE SYMMETRIC!!! One of the side effects of flooding a lake in northern boreal forest areas (e.g. for a hydro-electric project) is that mercury is leached from the soil, enters the food chain, and eventually contaminates the fish. The concentration in fish will vary among individual fish because of differences in eating patterns, movements around the lake, etc. Suppose that the concentrations of mercury in individual fish follows an approximate normal distribution with a mean of 0.25 ppm and a standard deviation of 0.08 ppm. Fish are safe to eat if the mercury level is below 0.30 ppm. What proportion of fish are safe to eat? Marks on a Chemistry test follow a normal distribution with a mean of 65 and a standard deviation of 12. Approximately what percentage of the students have scores below 50? inverse % or percentile Next Question… What is your test score above if you are in the top 20%? 2.1 Summary We can describe the overall pattern of a distribution using a density curve. The area under any density curve = 1. This represents 100% of observations. Areas on a density curve represent % of observations over certain regions. An individual observation’s relative standing can be described using a zscore or percentile rank. x mean z standard deviation END OF NOTES Homework: Packet p. 7 #1-8, omit 4 Make sure you understand #2.2!! Warm Up – Day 8 1) 2) 3) 4) What is the z-score formula? Can you say and spell the Greek symbols in the z score formula? The average number of parents that attend Meet the Teacher night at GHHS is 323 with a standard deviation of 5.6. Interpret the standard deviation in context. (*This stat was created by your teacher!) What is wrong with this statement: I have a z score of -4.3 points. Work on Matching Activity Answers on BB END OF NOTES • • Start HOMEWORK: • Packet page 6 #1 - 8 Remember packet 6 is due Thursday!