Percentiles and the Normal Curve

1
Percentiles and the Normal Curve
2
A Normal Curve
3
The Normal Approximation
4
Fathers Height

• The height of the men in the sample was on
  average 69 inches the SD was 3 inches.
• Estimate the percentage of these men with heights
  between 63 inches and 72 inches.
• 63 inch 2 SD 72 inch 1 SD.
• The area between 2 and 1 is 47.5, between 0
  and 1 is 34.5.
• Approximately 82 of the men will be between 63
  and 72 inches.

5
Financial Hardship

• The average income for families was about
  45,000.
• The standard deviation was about 32,000.
• The normal table tells us that 8 of the area
  lies to the left of 1.4 standard deviation.
• This suggests that about 8 of the families have
  a negative income.

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An Income Histogram
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Percentiles

• When the normal curve is not a good approximation
  to a histogram, we often use percentiles.
• We have seen this idea before the median is the
  50th percentile.
• The kth percentile is that number such that k
  percent of the data is less than that number.

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Percentiles for Family Income
1 1,300
10 10,200
25 20,100
50 36,800
75 58,100
90 85,000
99 151,800
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Quartiles

• The 25th percentile is often called the lower
  quartile.
• The 75th percentile is often called the upper
  quartile.
• The interquartile range is the
• 75th quartile 25th quartile.
• A five number summary of a data set minimum,
  lower quartile, median, upper quartile, maximum.

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Percentiles and the Normal Curve

• When a histogram does not follow the normal
  curve, we often use percentiles instead.
• When a histogram does follow a normal curve, we
  can use the normal table to estimate the
  percentiles.

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Example

• Among all applicants to a university one year,
  the Math SAT scores averaged 535, the SD was 100,
  and the scores followed the normal curve.
• Estimate the 95th percentile.

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Example

• This score is above average, by some number of
  standard deviations, z.
• We cannot use the normal table directly, because
  this only gives us values for the area between z
  and z.
• The area to the right of the z is 5. Hence the
  area to the left of z is 5, too. The area
  between z and z is 90.

13
Example

• We can look this up in the normal table z
  1.65.
• You have to score 1.65 SD above average to be in
  the 95th percentile.
• This is a score 1.65 x 100 165 points above
  average.
• The 95th percentile of the score distribution is
  535 165 700.

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Percentile vs. Percentile Rank

• A percentile on a test is a score
  the 95th percentile is a score of 700.
• A percentile rank is a percentile
  if you score 700, then your percentile rank is
  95.

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Freshman GPA

• Suppose that the average freshmen GPA is around
  3.0, and the SD is about 0.5. Assume that the
  histogram follows the normal curve. Estimate the
  GPA of someone in the 30th percentile.
• This GPA is below average, say -z SD.
• The area to the left is 30. Thus, the area
  outside z and z is 60, inside 40.
• Look up in the table z is between 0.5 and 0.55.
• GPA is about 2.75.