Transcript The Normal Curve

The Normal Curve
It’s a frequency distribution that often occurs when
there is a large number of values in a data set.
 The graph is a symmetric, bell-shaped curve known
as the normal curve.
X  3
X  2
X  1
X
X  1
X  2
 Most of the data occurs around the mean.
 A small portion of the data occurs in the tails.
X  3
Properties of a Normal Distribution
•The maximum point of the curve is at the mean.
•The curve extends indefinitely far to the left and right and approaches the x-axis.
•With a large standard deviation the curve will be flat.
•With a small standard deviation the curve will be tall.
•The total area under the curve is 1.
•The curve is symmetric about the mean.
•About 68% of the data are within 1 standard deviations from the mean.
•About 95% of the data are within 2 standard deviations from the mean.
•About 99.7% of the data are within 3 standard deviations from the mean.
No matter the shape of the bell curve, the area
under it is the same.
Example
Suppose the mean of a set of data is 60 and the standard deviation is 6.
Boundaries:
x 1
to
x  1
68%
60  1(6) to 60  1(6)
54
to
66
68% of the values in this set of data lie
within 1 standard deviation of 60, that is,
between 54 and 66.
54
60
66
If you randomly select one item from the sample, the probability
that the one you pick will be between 54 and 66 is 0.68.
If you repeat this process 1000 times, approximately 68% of those selected will be
between 54 and 66.
Ex. 1
The lifetimes of 10,000 watch batteries are normally distributed.
The standard deviation is 60 days. The mean is 500 days.
Sketch a normal curve that represents the frequency
distribution of lifetimes of the batteries.
Ex. 2
Suppose the scores of 500 college freshmen taking
Psychology 101 are normally distributed.
The standard deviation is 10. The mean score is 60%
Sketch a normal curve that represents the frequency distribution
of scores.
Ex. 3
Find the upper and lower limits of the interval about the
mean in which 68%, 95%, and 99.7% of the values of a
set of normally distributed data can be found if the
mean is 124 and σ is 16.
Ex. 4
The lifetimes of 10,000 watch batteries are normally distributed.
The standard deviation is 60 days. The mean is 500 days.
How many batteries will last between 440-560 days?
 How many batteries will last between 380-620 days?
 How many batteries will last between 320-680 days?
More Examples
• What percent of batteries will last between
500 and 560 days?
• What percent of batteries will last longer
than 620 days?
• What percent of batteries last 320 days at
the most?