Week 2 - Lesson 5.1 The Standard Normal Probability Distribution Flashcards
Most high schools have a set amount of time in-between classes during which students must get to their next class. If
you were to stand at the door of your statistics class and watch the students coming in, think about how the students
would enter. Usually, one or two students enter early, then more students come in, then a large group of students
enter, and finally, the number of students entering decreases again, with one or two students barely making it on
time, or perhaps even coming in late!
Now consider this. Have you ever popped popcorn in a microwave? Think about what happens in terms of the
rate at which the kernels pop. For the first few minutes, nothing happens, and then, after a while, a few kernels
start popping. This rate increases to the point at which you hear most of the kernels popping, and then it gradually
decreases again until just a kernel or two pops.
Here’s something else to think about. Try measuring the height, shoe size, or the width of the hands of the students in
your class. In most situations, you will probably find that there are a couple of students with very low measurements
and a couple with very high measurements, with the majority of students centered on a particular value.
All of these examples show a typical pattern that seems to be a part of many real-life phenomena. In statistics,
because this pattern is so pervasive, it seems to fit to call it normal, or more formally, the normal distribution. The
normal distribution is an extremely important concept, because it occurs so often in the data we collect from the
natural world, as well as in many of the more theoretical ideas that are the foundation of statistics. This chapter
explores the details of the normal distribution.
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Characteristics of a Normal Distribution
When graphing the data from each of the examples in the introduction, the distributions from each of these situations
would be mound-shaped and mostly symmetric. A normal distribution is a perfectly symmetric, mound-shaped
distribution. It is commonly referred to the as a normal curve, or bell curve.
Because so many real data sets closely approximate a normal distribution, we can use the idealized normal curve to
learn a great deal about such data. With a practical data collection, the distribution will never be exactly symmetric,
so just like situations involving probability, a true normal distribution only results from an infinite collection of data.
Also, it is important to note that the normal distribution describes a continuous random variable
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Due to the exact symmetry of a normal curve, the center of a normal distribution, or a data set that approximates a
normal distribution, is located at the highest point of the distribution, and all the statistical measures of center we
have already studied (the mean, median, and mode) are equal
It is also important to realize that this center peak divides the data into two equal parts
Center
Let’s go back to our popcorn example. The bag advertises a certain time, beyond which you risk burning the popcorn.
From experience, the manufacturers know when most of the popcorn will stop popping, but there is still a chance that
there are those rare kernels that will require more (or less) time to pop than the time advertised by the manufacturer.
The directions usually tell you to stop when the time between popping is a few seconds, but aren’t you tempted to
keep going so you don’t end up with a bag full of un-popped kernels? Because this is a real, and not theoretical,
situation, there will be a time when the popcorn will stop popping and start burning, but there is always a chance, no
matter how small, that one more kernel will pop if you keep the microwave going. In an idealized normal distribution
of a continuous random variable, the distribution continues infinitely in both directions.
Spread
Because of this infinite spread, the range would not be a useful statistical measure of spread. The most common way
to measure the spread of a normal distribution is with the standard deviation, or the typical distance away from the
mean. Because of the symmetry of a normal distribution, the standard deviation indicates how far away from the
maximum peak the data will be. Here are two normal distributions with the same center (mean):
The first distribution pictured above has a smaller standard deviation, and so more of the data are heavily concentrated
around the mean than in the second distribution. Also, in the first distribution, there are fewer data values at the
extremes than in the second distribution. Because the second distribution has a larger standard deviation, the data
are spread farther from the mean value, with more of the data appearing in the tails.
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Technology Note: Investigating the Normal Distribution on a TI-83/84 Graphing Calculator
We can graph a normal curve for a probability distribution on the TI-83/84 calculator. To do so, first press [Y=].
To create a normal distribution, we will draw an idealized curve using something called a density function. The
command is called ’normalpdf(’, and it is found by pressing [2nd][DISTR][1]. Enter an X to represent the random
variable, followed by the mean and the standard deviation, all separated by commas. For this example, choose a
mean of 5 and a standard deviation of 1.
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Because of the similar shape of all normal distributions, we can measure the percentage of data that is a certain
distance from the mean no matter what the standard deviation of the data set is. The following graph shows a normal
distribution with µ = 0 and s = 1. This curve is called a standard normal curve. In this case, the values of x represent
the number of standard deviations away from the mean.
The Empirical Rule
Notice that vertical lines are drawn at points that are exactly one standard deviation to the left and right of the
mean. We have consistently described standard deviation as a measure of the typical distance away from the mean.
How much of the data is actually within one standard deviation of the mean? To answer this question, think about
the space, or area, under the curve. The entire data set, or 100% of it, is contained under the whole curve. What
percentage would you estimate is between the two lines? To help estimate the answer, we can use a graphing
calculator. Graph a standard normal distribution over an appropriate window.
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Technology Note: Investigating the Normal Distribution on a TI-83/84 Graphing Calculator
We can graph a normal curve for a probability distribution on the TI-83/84 calculator. To do so, first press [Y=].
To create a normal distribution, we will draw an idealized curve using something called a density function. The
command is called ’normalpdf(’, and it is found by pressing [2nd][DISTR][1]. Enter an X to represent the random
variable, followed by the mean and the standard deviation, all separated by commas. For this example, choose a
mean of 5 and a standard deviation of 1
Adjust your window to match the following settings and press [GRAPH]
Press [2ND][QUIT] to go to the home screen. We can draw a vertical line at the mean to show it is in the center of the
distribution by pressing [2ND][DRAW] and choosing ’Vertical’. Enter the mean, which is 5, and press [ENTER]
Remember that even though the graph appears to touch the x-axis, it is actually just very close to it.
In your Y= Menu, enter the following to graph 3 different normal distributions, each with a different standard
deviation:
This makes it easy to see the change in spread when the standard deviation changes
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The calculator also gives a very accurate estimate of the area. We can see from the rightmost screenshot above that
approximately 68% of the area is within one standard deviation of the mean. If we venture to 2 standard deviations
away from the mean, how much of the data should we expect to capture? Make the following changes to the
’ShadeNorm(’ command to find out:
Notice from the shading that almost all of the distribution is shaded, and the percentage of data is close to 95%. If
you were to venture to 3 standard deviations from the mean, 99.7%, or virtually all of the data, is captured, which
tells us that very little of the data in a normal distribution is more than 3 standard deviations from the mean.
Notice that the calculator actually makes it look like the entire distribution is shaded because of the limitations of the
screen resolution, but as we have already discovered, there is still some area under the curve further out than that.
These three approximate percentages, 68%, 95%, and 99.7%, are extremely important and are part of what is called
the Empirical Rule.
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The Empirical Rule states that the percentages of data in a normal distribution within 1, 2, and 3 standard deviations
of the mean are approximately 68%, 95%, and 99.7%, respectively
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A z-score is a measure of the number of standard deviations a particular data point is away from the mean. For
example, let’s say the mean score on a test for your statistics class was an 82, with a standard deviation of 7 points.
If your score was an 89, it is exactly one standard deviation to the right of the mean; therefore, your z-score would
be 1. If, on the other hand, you scored a 75, your score would be exactly one standard deviation below the mean,
and your z-score would be 1. All values that are below the mean have negative z-scores, while all values that are
above the mean have positive z-scores. A z-score of 2 would represent a value that is exactly 2 standard deviations
below the mean, so in this case, the value would be 8214 = 68.
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To calculate a z-score for which the numbers are not so obvious, you take the deviation and divide it by the standard
deviation.
You may recall that deviation is the mean value of the variable subtracted from the observed value, so in symbolic
terms, the z-score would be:
As previously stated, since s is always positive, z will be positive when x is greater than µ and negative when x is
less than µ. A z-score of zero means that the term has the same value as the mean. The value of z represents the
number of standard deviations the given value of x is above or below the mean.
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Example: What is the z-score for an A on the test described above, which has a mean score of 82? (Assume that an
A is a 93.)
The z-score can be calculated as follows:
