Transcript Percentiles and the Normal Curve

Percentiles and
the Normal Curve
A Normal Curve
The Normal Approximation
Father’s 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.
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.
An Income Histogram
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.
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
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.
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.
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.
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%.
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.
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%.
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.