Reminders

• Email me and the TAs today if you have anyquestions/concerns about grading of quiz 4 or HW 4

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Warm Up

• Final grades in BIO 180 at UW are Normally distributedwith a mean of 3.0 and a SD of 0.3. Final grades inSTAT 566 are Normally distributed with a mean of 3.7and a SD of 0.1. If you get a 3.4 in BIO 180 and a 3.8 inSTAT 566, in which course is your standard score higher?

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Chapter 13 part 2: Normal calculations

Aaron ZimmermanSTAT 220 - Summer 2014

Department of StatisticsUniversity of Washington - Seattle

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Percentiles

• The cth percentile of a distribution is the value suchthat c percent of the observations lie below it and therest lie above it

? If you score at the 65th percentile on a test, then 65%of the students who took the exam scored lower than youand the other 35% scored higher

? If your height is at the 20th percentile within yourgender, then 20% of all people in your gender are shorterthan you and the rest are taller

Key idea #1: Each percentile in a normal distributioncorresponds with a standard score

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Percentiles

• Table B in the back of your book allows you to“convert” between standard scores and percentiles

? For example, a standard score of -2.0 corresponds withthe 2.27th percentile (2.27% of observations in Normallydistributed data have a standard score less than -2.0)

? We’ve already seen this one, but the table will allow youto do it for percents other than 68-95-99.7

Key idea #2: Percentiles can be interpreted as the areaunder the standard normal curve to the left of the stan-dard score

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How to use a Standard Score Table

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How to use a Standard Score Table

Check how the table works!

You can either:

• (1) Find a percentile given a score

? e.g. What percentile is associated with a score of -0.5?

• (2) Find a score given a percentile

? What standard score has 30% of standard scores lessthan it?

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How to use a Standard Score Table

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ACT scores

• Remember thatACT math scoresare normallydistributed with amean of 18 and aSD of 6

• Grainne had astandard score of1.5 while Karthikhad a standardscore of -0.5 on theACT math test

• What were theirmath scores?

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ACT scores

• The percentile ofKarthik’s score isthe area to the leftof his standard scorein the standardnormal distribution

? If you look up -0.5in Table B, you seethat Karthik scoredat the 30.85thpercentile on theACT math test.

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ACT scores

• The percentile ofGrainne’s score isthe area to the leftof her standardscore in thestandard normaldistribution

? If you look up 1.5in Table B, you seethat Grainne scoredat the 93.32thpercentile on theACT math test.

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What percentile corresponds to a standard score of -1.2?

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What percentile corresponds to a standard score of -1.2?

Answer: 11.51 percentile 13

What standard score corresponds to the 90th percentile?

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What standard score corresponds to the 90th percentile?

Answer: Approximately a score of 1.3 15

What percent of standard scores are between 0.5 and 1?

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What percent of standard scores are between 0.5 and 1?

Answer: 84.13− 69.15 = 14.98 % 17

What % of scores are larger than a standard score of 0.3?

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What % of scores are larger than a standard score of 0.3?

Answer: 1− 61.79 = 38.21 % 19

Strategy for problems involving percentiles of

Normal distributions

• Find the relevant mean

• Find the relevant standard deviation

• If you have an observation and want to calculate thepercentile (the percent below that value):

? Calculate the standard score

? Use Table B to convert to a percentile

? If you need the percent above the observation, calculatethe percentile and subtract from 100

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Distribution of heightsThe heights of men are Normally distributed with a mean of70 in. and a SD of 2.5 in. The heights of women are Normallydistributed with a mean of 66 in. and a SD of 3 in.

• If Karthik is 75 in. tall,what percentile is hisheight?

? Relevant mean: 70 in.

? Relevant SD: 2.5 in.

? Standard score:75−702.5

= 2

• Percentile: 97.73

• Conclusion: 97.73% ofmen are shorter thanKarthik

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Distribution of heightsThe heights of men are Normally distributed with a mean of70 in. and a SD of 2.5 in. The heights of women are Normallydistributed with a mean of 66 in. and a SD of 3 in.

• If my little sister is 58.5in. tall, what percent ofwomen are taller than her?

? Relevant mean: 66 in.

? Relevant SD: 3.0 in.

? Standard score:58.5−66

3.0= −2.5

• Percentile: 0.62

• Conclusion: 1-0.62 =99.38% of women aretaller than Eliana

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Heights of mapletrees are Normallydistributed with amean of 34 ft. and astandard deviation of9 ft., while theheights of elm treesare Normallydistributed with amean of 19 ft. and astandard deviation of3 ft.What percentile isan elm tree that is20.5 ft. tall?

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Heights of mapletrees are Normallydistributed with amean of 34 ft. and astandard deviation of9 ft., while theheights of elm treesare Normallydistributed with amean of 19 ft. and astandard deviation of3 ft.What percent ofmaple trees aretaller than 25 ft.?

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Strategy for problems involving percentiles of

Normal distributions• Find the relevant mean• Find the relevant standard deviation• If you have an observation and want to calculate thepercentile (the percent below that value):

? Calculate the standard score

? Use Table B to convert to a percentile

? If you need the percent above the observation, calculatethe percentile and subtract from 100

• If you have a percentile and need the observation value:

? Find the standard score in Table B for that percentile

? Work backwards:observation = (standard score)× SD +mean

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Distribution of heights

The heights of men are Normally distributed with a mean of 70 in. and aSD of 2.5 in. The heights of women are Normally distributed with amean of 66 in. and a SD of 3 in.

• If you are a woman, how tallmust you be to be at the 90th

percentile?

? Relevant mean: 66 in.

? Relevant SD: 3.0 in.

? Percentile: 90

? Standard score: about 1.3

• obs = (1.3)× 3 + 66 = 69.9

• Conclusion: A woman must be69.9 in. tall to be at the 90thpercentile

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Distribution of heights

The heights of men are Normally distributed with a mean of 70 in. and aSD of 2.5 in. The heights of women are Normally distributed with amean of 66 in. and a SD of 3 in.

• If you are a man, how tallmust you be to be in the top35% of men’s heights?

? Relevant mean: 70 in.

? Relevant SD: 2.5 in.

? Percentile: 65

? Standard score: about 0.4

• obs = (0.4)× 2.5 + 70 = 71

• Conclusion: A man must be 71in. tall to be in the top 35% ofmen’s heights

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Heights of mapletrees are Normallydistributed with amean of 34 ft. and astandard deviation of9 ft., while theheights of elm treesare Normallydistributed with amean of 19 ft. and astandard deviation of3 ft.How tall is an elmtree at the 10th

percentile?

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Heights of mapletrees are Normallydistributed with amean of 34 ft. and astandard deviation of9 ft., while theheights of elm treesare Normallydistributed with amean of 19 ft. and astandard deviation of3 ft.How tall does amaple tree need tobe to be in the top1 % of all mapletrees?

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Homework #5

• Finish reading Chapter 13 if you haven’t already

• Do problems 13.21, 13.22, 13.23, 13.24, 13.25, 13.27,13.29, 13.30

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