Download ppt#1 Density Curve and The Normal Distribution

2.1
Density Curves &
The Normal Distribution
Density Curves
• Sometimes it is
easier to work with a
smooth curve than a
histogram.
• Curves describe
proportions versus
the counts that
histograms describe.
• Areas under the
curve represent
proportions of
observations.
Density Curves (continued)
• Total area under
ANY density
curve is 1 (100%)
• Density curves
ALWAYS lie on or
above the
horizontal axis
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Density Curves (continued)
12% of the observations take values between 7 and 8
Mean & Median of a Density Curve
• Median is equal-areas point, the point with
half the area to the left and half to the right
– Same is true for the quartiles
• Mean is point at which the curve would
balance if it were made of solid material
– Mean of skewed distribution is pulled toward
the direction of the long tail
Mean & Median of a Density Curve
The mean is the balance point of a density curve.
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Density Curves (continued)
A symmetric density curve
median = mean
A skewed right density curve
median < mean
Notable Notations
Sample
Population
Actual observations
Idealized distribution
Histogram
Density curve


x
s

Try This!
1. Sketch a density curve that is
symmetric but has a different
shape from the actual
symmetric density curve.
2. Sketch a density curve that is
strongly skewed to the left and
estimate the median and mean.
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Try This!
3. This is the density curve for a uniform distribution.
(a)
What percent of the observations lie above 0.8?
(b)
What percent of the observations lie below 0.6?
(c)
What percent of the observations lies between 0.25 and 0.75?
Normal Curves
• Symmetric, unimodal,
bell-shaped curves
• Described by the mean
and standard deviation
• Mean and median are
always located at the
center of distribution
• The larger the standard
deviation, the more
spread out the curve
N (, )
Normal Curves
• Symmetric, unimodal,
bell-shaped curves
• Described by the mean
and standard deviation
• Mean and median are
always located at the
center of distribution
• The larger the standard
deviation, the more
spread out the curve
N (, )
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The 68-95-99.7 Rule
The Empirical Rule
Approx. 68% of the data will fall within 1 standard deviation of the mean
(  1 ,  1 )
The 68-95-99.7 Rule
The Empirical Rule
Approx. 95% of the data will fall within 2 standard deviation of the mean
(   2,   2)
The 68-95-99.7 Rule
The Empirical Rule
Approx. 99.7% of the data will fall within 3 standard deviation of the mean
(   3 ,   3 )
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Try This!
4. Suppose it takes you 20 minutes, on average, to drive to school,
with a standard deviation of 2 minutes. Assume the driving time
distribution is approximately normal.
(a) What percent of the time will you arrive
to school in less than 22 minutes?
Try This!
4. Suppose it takes you 20 minutes, on average, to drive to school,
with a standard deviation of 2 minutes. Assume the driving time
distribution is approximately normal.
(b) What percent of the time will it take
you more than 24 minutes?
Try This!
5. The distribution of heights of adult American men is approx. normal
with a mean of 69 inches and a standard deviation of 2.5 inches.
(a) What percent are taller than 74 in.?
(b) Between what heights do the middle
95% of men fall?
(c) What percent are shorter than 66.5 in.?
(d) A height of 71.5 inches corresponds to
what percentile of adult male American
heights?
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