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The normal distribution and standard score

The normal distribution and standard score. Standardizing a score. Let’s say the SAT has a mean of 500 and a standard deviation of 100 while the ACT has a mean of 24 and a standard deviation of 3. John Marshall scores a 660 on the SAT and Susan Marshall scores 28 on the ACT.

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The normal distribution and standard score

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  1. The normal distributionand standard score

  2. Standardizing a score • Let’s say the SAT has a mean of 500 and a standard deviation of 100 while the ACT has a mean of 24 and a standard deviation of 3. • John Marshall scores a 660 on the SAT and Susan Marshall scores 28 on the ACT. • Who had the better score? • It’s hard to tell unless we standardize the scores…

  3. We call a standard score a Z-score • Here’s how we do it: • x is our observed score (your book uses letter y, but it doesn’t matter) • the Greek letter “mu” is the mean • and small-case “sigma” is the population standard deviation. • What are John’s and Susan’s standard scores?

  4. 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. • A data value that sits right at the mean, has a z-score equal to 0. • A z-score of 1 means the data value is 1 standard deviation above the mean. • A z-score of –1 means the data value is 1 standard deviation below the mean.

  5. When Is a z-score BIG? • How far from 0 does a z-score have to be to be interesting or unusual? • There is no universal standard, but the larger a z-score is (negative or positive), the more unusual it is. • Often (but not always!), we consider a z-score greater than 2 or less than -2 to be roughly an indication of unusualness. But every data set is different, so be careful!

  6. Being “normal”… • The Normal distribution function is central to the study of Statistics. • Many natural phenomena produce data that fall into a roughly Normal shape. • We even have a mathematical function that models this shape…

  7. This is the standard Normal curve

  8. The Empirical Rule • The values in the Normal distribution approximately follow this pattern: 68-95-99.7 • 68% of the values fall within one standard deviation of the mean. • 95% of the values fall within two standard deviations of the mean. • 99.7% of the values fall within three standard deviations of the mean.

  9. Scoring the decathlon • Three competitors have completed three events. Here are their marks with overall means and st. devs. • Who gets the gold medal? • Whose individual performance was the most extraordinary?

  10. Sketch a Normal model • The standard Normal distribution N(0, 1) • Birthweights of babies are N(7.6, 1.3) • ACT scores at College of Bob are N(21.2, 4.4)

  11. The Normal model applied (1) • Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg. • What percent of all cars get less than 12 mpg? • What percent of all cars get between 18 and 30 mpg?

  12. Calculator skills Look only at 2: and 3: for now. normalcdf( takes up to four inputs: lower bound, upper bound, mean, st. dev.

  13. Calculator skills Look only at 2: and 3: for now. invNorm( takes only one. Proportion, mean, st. dev.

  14. Normal models with the calculator What percent of a standard normal model is found in each area? a) b) c) In the standard Normal model, what value of z cuts off the region described? d) The lowest 12% e) The highest 30% f) The middle 50%

  15. The Normal model applied (2) • Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg. • What percent of all cars get less than 15 mpg? • What percent of all cars get between 20 and 32?

  16. The Normal model applied (3) • Suppose cars in California have fuel efficiency that roughly fits a Normal model with mean 24 mpg and standard deviation of 6 mpg. • Describe the fuel efficiency of the lowest 20% of cars. • What gas mileage represents the third quartile?

  17. The Normal model applied (4) • An environmental group is lobbying for a national goal of no more than 10% of cars having mileage under 20 mpg. If the st. dev. does not change, what should the mean rise to? • Auto makers say that they can only raise the mean to 26 mpg. What standard deviation would allow them to still make the goal of only 10% under 20 mpg?

  18. The normal model applied (5) • At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes. • What percent of employees are late on a typical work day?

  19. The normal model applied (6) • At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes. • A typical worker needs 5 minutes to adjust to their surroundings before beginning duties. What percent of this company’s employees arrive early enough to make this adjustment?

  20. The normal model applied (7) • At a large business, employees must report to work at 7:30am. The actual arrival times of employees can be described by a Normal model with a mean at 7:22am and a standard deviation of 4 minutes. • If the mean arrival time does not change, what standard deviation would we need to make sure that virtually all employees are on time to work?

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