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Normal Distribution
Normal Distribution: has data that vary randomly from the mean. The graph of a normal distribution is a normal curve.
one standard deviation
­ 68% of data fall within of the mean
two standard deviations
­ 95% of data fall within of the mean
three standard deviations
­ 99.7% of data fall within of the mean
­ A normal distribution has a symmetric bell shape centered on the mean.
­ Many common statistics (such as human height, weight, or blood pressure) garthered from samples in the natural world tend to have a normal distribution about their mean.
Sometimes an extraordinary factor affects data that would otherwise be normally distributed. In such a case, the data set could have a distribution that is skewed, an asymmetric curve where one end stretches out further than the other end.
(When the majority of the data values fall to the right of the mean the distribution is negatively or left­skewed b/c the mean is to the left of the majority of the data.)
(When the majority of the data values fall to the left of the mean the distribution is positively or right­skewed b/c the mean is to the right of the majority of the data.)
The "tail" of the curve indicates the direction of skewness (right is positive and left is negative)
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The shape and position of a nromal distribution curve depend on two factors: the mean and the standard deviation
The bigger the standard deviation the wider (more spread out) the distribution (curve) is.
34.1%
34.1%
13.59%
13.59%
2.28%
2.28%
≈68%
≈95%
≈99.7%
"All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score:
z = value ­ mean standard deviation
The z score is the number of standard deviations that a particular X value is away from the mean" (Elementary Statistics A step by step approach eighth edition by Bluman) To use the z score to find an area under the standard normal distribution curve:
You must first determine where the values lie with respect to the z score.
1. If you're looking for the area to the left of any z value:
look up the z score on the z table and that's your answer
z=1.06 = .8554 ­­­85.54% of the data falls to the left of z = 1.06
2. If you're looking for the area to the right of any z value:
look up the z score on the z table and subtract that value from 1.0000
z=1.06 = .8554­­­­­­­­ 1.0000 ­ .8554 = .1446
which means that 14.46% of the data falls
to the right of z = 1.06
3. If you're looking for the area between any two z values:
look up the z score on the z table and subtract the one on the left from the one on the right
z= ­1.58
z=1.06
z ­1.58 = .0571 z 1.06 = .8554
so rt ­ lf ­­­ .8554 ­ .0571 = .7983 so
79.83% of the data falls b/w the z's
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of the drinks fall within this SD
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