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Statistics I
Statistics are important in all aspects of life. They play a critical role in determining average measures,
spread of data and how different values compare to one another. This lecture will focus on the most
common and probably the most important probability distribution called the normal distribution. The
normal distribution is represented by the curve shown below.


x
Let’s see how different normal distributions compare to one another. Given the three plots below:
A
B
A
x
x
C
x
Which one has the larger ?
Which one has the larger ?
Which two have equal ?
Which two have equal ?
It is important to keep in mind that the probability of a normal curve (equivalent to the area under the
curve) must always equal 1. This means that the more variability you have in your data the lower the
peak in the bell curve and the more spread out it becomes. Likewise, the more similar your data is the
higher the peak in the bell curve and the less spread out the bell shape is.
Draw examples of these curves below:
Large Data Spread
Small Data Spread
The Z-statistic is one means that we can use to calculate the area under the normal curve. The Z statistic
is given by the following equation where Z is meant to represent a normal random variable with a mean
of zero and a variance of 1.
𝑍=
𝑋−𝜇
𝜎
When approaching a statistics problem where we are looking at the probability that a random value x is
between two particular data points you would do the following:





Calculate the z value at the higher data point value
Look up its probability in the z-statistic table
Calculate the z value at the lower data point value
Look up its probability in the z-statistic table
Subtract the probability for the higher data point value from that of the lower value to get your
answer
Let’s put your knowledge of normal distributions to the test by playing the normal distribution card
game.
References:
1. Walpole, R.E., Myers, R.H., Myers, S.L. (1998). Probability and Statistics for Engineers and
Scientists. Prentice Hall, Upper Saddle River, NJ.