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Statistics: The Normal Distribution
A Student Academic Learning Services Guide
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The Normal Distribution: things to remember
A normal distribution is a continuous probability distribution…

that is bell shaped
o
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that is symmetrical
o
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the right and left halves are identical (mirror images of each other)
that has tails (ends) that approach the bottom axis (but never touch it)
o
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it looks like a bell with a single peak in the middle
“asymptotic”
where the mean, median, and mode are equal
o
represented by the symbol μ
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Points to Ponder

Each point on the curve represents the measurement (x) of an individual in the
population
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The peak in the middle is the average/mean of the population, μ
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To the left of the peak are measurements less than μ
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To the right of the peak are measurements greater than μ
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Because the curve is symmetrical, ½ the curve (or 50%) is to the left of the peak, and ½
the curve (or 50%) is to the right of the peak.
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When choosing a point on the curve randomly, there is a 50% chance of selecting a
point to the left of the peak (a measurement less than μ) , and a 50% chance of
selecting a point to the right of the peak (a measurement greater than μ).
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
Any point (not just the peak) on the curve can divide the distribution into different
parts.
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We can say that the proportion of the curve to the left of any point is equal to the
probability of randomly selecting an individual with less than that measurement.
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We can also say that the proportion of the curve to the right of any point is equal to the
probability of randomly selecting an individual with more than that measurement.
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So how do we determine the relative proportions (left of/right of) created by selecting a
random point (x) on the curve?
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Fortunately, there is a characteristic of the normal curve that can help us (and someone
else has already done the really hard math).
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The normal distribution follows the empirical rule, so we know that about 68% of the
measurements are between one standard deviation (σ) to the left of the mean and one
standard deviation to the right of the mean, 95% are between two standard deviations,
and 99.7% are between three standard deviations.
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The Standard Normal Distribution

If we convert a measurement (x) to the number of standard deviations it is away from
the mean, we can look up the probability that goes with that number.
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The number of standard deviations a measurement is away from the mean is called the
z-score.
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The formula used to calculate a z-score is:
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The numerator of this formula (x – μ) determines how far away a measurement is away
from the mean and in which direction (positive when x is larger than μ, negative when x
is less than µ)
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When we divide the result by the standard deviation (σ), the result is how many standard
deviations that measurement is from the mean. This is the z-score.
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For example, let us say that a population has a mean of 100 with a standard deviation of
15. What is the z-score for a measurement of 125?
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Calculating the numerator (x – μ) we have: 125-100=25.
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We then divide the result (25) by the standard deviation (15) and have: 25÷15=1.67
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The measurement of 125 has a z-score of 1.67. This means the measurement is 1.67
standard deviations to the right of the mean (we know it’s to the right because it is a
positive number)
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z=
x µ

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
There are a number of ways to look up the probabilities that are associated with a
z-score of 1.67. For example, your textbook may have an appendix located in the back,
or you may be using computer software (Microsoft Excel).
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Whatever method is used, a z-score of +1.67 divides the distribution, with 95.22% of
the distribution to the left of the score and 4.78% to the right of the score.
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We can now say that (when chosen randomly) there is a 95.22% chance of choosing an
individual with a measurement less than 125, or a 4.78% chance of choosing an
individual with a measurement greater than 125.
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Student Services Building, SSB 204
905.721.2000 ext. 2491
This document last updated: 7/21/2011