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Interpreting Center & Variability

Interpreting Center & Variability. through Percentiles. Suppose you take the SAT test and the ACT test. Not using the chart they provide, can you directly compare your SAT Math score to your ACT math score? Why or why not? We need to standardized these scores so that we can compare them.

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Interpreting Center & Variability

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  1. Interpreting Center & Variability throughPercentiles

  2. Suppose you take the SAT test and the ACT test. Not using the chart they provide, can you directly compare your SAT Math score to your ACT math score? Why or why not? We need to standardized these scores so that we can compare them.

  3. z score Standardized score Has m = 0 & s = 1

  4. Let’s explore . . . So what does the z-score tell you? Suppose the mean and standard deviation of a distribution are m = 50 & s = 5. If the x-value is 55, what is the z-score? If the x-value is 45, what is the z-score? If the x-value is 60, what is the z-score? 1 -1 2

  5. What do these z scores mean? -2.3 1.8 6.1 -4.3 2.3 s below the mean 1.8 s above the mean 6.1 s above the mean 4.3 s below the mean

  6. Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with m = 45 and s = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations (z score) below the mean. Jonathan scores 35 on this test. Does he get the job? No, he scored 2.78 SD below the mean

  7. Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of 2.8. Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best? She did better on test A.

  8. SAT Scores • Let’s start by talking about your SAT score…. • Say you scored a 600 on the math section of your SAT, and we know the mean is a 500 with a standard deviation of 100. • Well that’s simple – you are 1 SD above the mean, which means that 84% of all scores are below you. • But, let’s say you take it again, and now you score a 680. Where do you stand now? • After some calculations we can determine that you have a z-score of 1.80 • So you are between 1 SD and 2 SD, so what percentile is that…..

  9. There are 2 ways to figure it out • Method 1: • You can utilize the table of Normal percentiles that is given to us on the AP exam. • So, let’s look to see what our percentile is. • Our score is above _______________% of all scores

  10. Method 2: • Technology is a wonderful thing! • Click 2nd VARS button on your calculator (will only work on a 84 and up) • Locate the normalcdf - hit enter • We will need to input 2 values • The farthest left value we want to measure, and the farthest right. • Since we want what percentile we are in, we want all values up to our score. So the farthest left. • We will always use -99 for our lowest, and +99 for the highest • So we input normalcdf(-99,1.8) Enter……and we get___________

  11. Example 1 • Recall our cattle problem…..the mean weight of the steers was 1152 lbs with a standard deviation of 84 lbs. If your steer weighs 1276 lbs, in what percentile is your steer? • Before we can find the percentile, we must first find the z-score. • = 1.48 SD • Using the table we get .93056, which means: 93.056% • Let’s check it with the calculator • Normalcdf(-99, 1.48) • = 93.056%

  12. Example 2 • Let’s return to the SAT scores….. • You scored a 720 on the math section, and your best friend scored a 590. The mean score was a 500 with a standard deviation of 100. What percentage of the scores exist between your scores? • Start by finding each z-score • Yours: 2.20 Friends: .9 • Manually, you can find the percent's on the table, and subtract • Using the calculator, enter normalCDF(______ , ______)

  13. Now we go in reverse! • Sometimes we are asked to find the z-score from the percentage • Suppose a college says that they will only admit students with a SAT math score amongst the top 10%. How high of a score would it take to be elligible? • If you are using the table…..you must find a value as close to .900 as possible from the interior of the table. • If you are using your calculator…… • We go back to DIST to get the invNorm • We enter invNorm(.90) • 1.28 SD away • 500 + 1.28*100 = 628

  14. Let’s do an Example…. • The average weight of eggs laid by young hens is 50.9 grams, and only 25% of their eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the z-score be for the minimum weight? • .674

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