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Standard Deviation
Z Scores
Learning Objectives
By the end of this lecture, you should be able to:
– Describe the importance that variation plays in interpreting a
distribution
– Understand how the standard deviation is calculated
– Describe what is meant by a z-score and be able to calculate them
– Start learning Greek: Be comfortable with the Greek letters for mean
and standard deviation.
Another way of describing variation
•
•
•
Recall that we usually describe a distribution in terms of its shape, center, and
variation.
One method of describing the variation has already been discussed: Quartiles and
the related 5-number summary.
Another extremely important method of describing variation is the ‘variance’ or
‘standard deviation’.
– Variance and SD are essentially the same thing. The standard deviation is simply the
square root of the variance.
– Eg: If Variance is 25, then SD = 5. If SD = 7, then Variance = 49.
•
Generally, you should only use SD with data that has a symmetrical distribution (ie
doesn’t work well with skewed data).
– This is because in order to calculate the SD, we need to use the mean of that data.
– To review a term: We say that the SD is not resistant to skewed data.
Variation
In every set of values (“dataset”—know this term!) e.g. ages, heights, incomes, crop yields,
spitball-distances, etc there will always be variation. The question is, how do we describe just
how varied our data is? Are all the points closely clustered around some value or are they all over
the map?
Having a sense of how much the various datapoints are spread out around the center (mean or
median) is an important piece of information. Consider employee pay at DePaul University: if we
looked at every income from student workers to the Provost/President, we would see a
tremendous variation around the center as some people make a great deal more than the center,
and some make much less. However, if we focused only on student employees, we would find
that the variation around the center is considerably less.
Suppose we were looking at a widget used in the construction of a high-performance laser.
Suppose this widget is intended to be exactly 2.3 inches in diameter. A manufacturer must make
their parts extremely close to this diameter in order for them to fit properly. We would assume that
the average diameter of, say, 10000 parts would indeed be very close to 2.3. However, the
question from a manufacturing perspective would be: How much variation is there among the
parts? The manufacturer would hope that the variation would be extremely small, that is, nearly all
the parts are very close in size to the mean. If there is a large variation, it would mean that there
are several parts that are inappropriately sized, and many of our very expensive lasers are going
to be faulty.
The most common statistic we used for measuring variability (spread) is called the standard
deviation.
How to calculate the SD
(This is for exaplanatory
purposes only - you will not have
to do this by hand!)
1. Find the mean (e.g. 63.5)
2. Look at one datapoint (e.g.
63) and calculate its
difference from the mean.
(e.g. -0.5) Square it that
value (0.25).
3. Repeat for all datapoints.
4. Then add up all values
from step 3.
5. Divide by by n-1
6. You now have the
variance. To get from
variance to ‘s’, simply take
the square root of the
variance.
Calculating standard deviation: ‘s’
Women height (inches)
i
xi
x
(xi-x)
(xi-x)2
1
59
63.4
-4.4
19.0
2
60
63.4
-3.4
11.3
3
61
63.4
-2.4
5.6
4
62
63.4
-1.4
1.8
5
62
63.4
-1.4
1.8
6
63
63.4
-0.4
0.1
7
63
63.4
-0.4
0.1
8
63
63.4
-0.4
0.1
9
64
63.4
0.6
0.4
10
64
63.4
0.6
0.4
11
65
63.4
1.6
2.7
Degrees freedom (df) = (n − 1) = 13
12
66
63.4
2.6
7.0
s2 = variance = 85.2/13 = 6.55 inches squared
13
67
63.4
3.6
13.3
14
68
63.4
4.6
21.6
Sum
0.0
Sum
85.2
n
1
2
s
(
x

x
)
 i
(n  1) 1
Mean = 63.4
Sum of squared deviations from mean = 85.2
s = standard deviation = √6.55 = 2.56 inches
Mean
63.4
Calculating these by hand can be tedious. We will do a couple of examples,
but will quickly switch over to doing them using a computer.
Review: Why do we care so much about Normal distributions?
• Much of the data we examine in the real world turns out to
have a normal distribution (exam scores, survival length for
cancer patients, crop yields, SAT results, people’s heights,
birth rate of rabbits, games of chance, etc, etc, etc)
• As a result, a great deal of study and research has gone into
the properties of Normal distributions/curves, as a result of
which, we have many tools that we can use to come up with
statistics for analysis of our data.
• In fact, the various tools discussed in this lecture (standard
deviation, z-score, z-tables) usually apply only to Normal
distributions.
Calculating areas under the Normal curve
• In addition to being a measure of spread, the
standard deviation is a number we can use to
accurately determine the area under any
segment of the Normal curve. We will discuss
this over the next couple of lectures.
Overview of determining the area under the curve
A fairly straight-forward process:
1. Using the standard deviation, convert the
value you are interested in (e.g. a Grade
equivalent score of 6.0) into a z-score.
2. Look up your z-score in a normal
probability table (also called a z-table)
3. The value you find on the z-table, is the area
under the curve to the left of your score (e.g.
6.0).
Whiile you should spend a few minutes thinking
about this slide, don’t worry too much about it for
now. We will revisit it later.
The
z-score
Know the following two facts:
1. A z-score measures the number of standard deviations that
some value ‘x’ is from the mean.
•
E.g. a z-score of +1.0 means the value lies 1 standard deviation
above the mean.
•
E.g. a z-score of +0.5 means the value lies one-half sd above the
mean.
•
E.g. a z-score of -1.23 means the value lies 1.23 standard deviation
below the mean.
2. As long as you know the SD, any value within Normal distribution
can be converted to a z-score.
•
That is, any value along the x-axis, can be converted to a z-score
It’s all Greek to me
… And it’s gonna get a little worse.
But we’ll start slow…
• In the math world, there are numerous ‘shortcuts’ that are used
to represent certain concepts.
• Such symbols are both widespread and useful (though it’s true
that sometimes geeks are simply trying to look impressive).
However, you must get comfortable with them as they are
introduced.
• Here are two to begin with:
• σ (sigma)  standard deviation
• μ (mu)  mean
Practice: z-score using the GEQ (Grade Equivalent Score) dataset
If I tell you that: Mean ( μ ) = 7, Standard Deviation (σ ) is 1
• Example: Your score is 6. How many standard deviations does
your score lie above or below the mean?
– Answer: z = -1. Your score is exactly 1 standard deviation below the
mean.
• Example: Your score is 8. How many standard deviations does
your score lie above or below the mean?
– Answer: z = +1. Your score is exactly 1 standard deviation above the
mean.
• Example: Your score is 5.5. How many standard deviations
does your score lie above or below the mean?
– Answer: z = -1.5. Your score is 1.5 standard deviations below the
mean.
z-score = number of SDs
• Recall that z-score is simply a value that tells
us the number of standard deviations above
or below the mean.
• So if I ask you “How many SDs above or below
the mean?” – that is exactly the same thing as
asking you for “the z-score”.
• If you are not clear on this concept, be sure to
practice and review the previous slide.
– Then try your luck again on the next slide.
Practice: z-score using the GEQ (Grade Equivalent Score) dataset
Given: Mean ( μ ) = 7, Standard Deviation ( σ ) is 0.5
• Example:
x = 6. How many standard deviations does this observation lie
above or below the mean? I.e. What is the z-score?
– Answer: z = -2. Your score is exactly 2 standard deviations below the mean.
• Example:
x = 8.5. What is the z-score?
– Answer: z = +3. Your score is 3 standard deviations above the mean.
• Example:
x =5. What is the z-score?
– Answer: z = -4. Your score is 4 standard deviations below the mean.
• Example:
x = 5.432. What is the z-score?
– Answer: Uh-oh, this one is a bit trickier… See next slide
Formula for calculating the z-score
You will find this relatively simple formula quite useful:
Recall that a z-score measures the number of standard deviations that
a data value x is from the mean .
z
(x   )

• x
the observation we are asking about
• μ
the mean of the population
• σ
the standard deviation of the population
Practice: z-score using the GEQ (Grade Equivalent Score) dataset
Example: Your score is 5.423. The distribution has μ = 7, σ = 0.5.
What is your z-score?
– Answer: Use the formula:
– So
• x
• μ
• σ
= 5.423
=7
= 0.5
• Therefore:
z
(x   )

z = (5.423 – 7) / 0.5
z = -3.154
– In other words, our value lies 3.154 standard deviations below
the mean.
• Example: Calculate the z-score for x = 200. μ = 197 and σ
= 342.
• Answer: Because this is not a Normal distribution, the
mean and standard deviation are NOT good choices of
statistics to use in an analysis. And without a mean and
standard deviation, we can not calculate a z-score!
– Take-home point #1: Standard deviations should be avoided for
distributions that are skewed and/or have outliers.
• Punching numbers into a calculator (or statistical software) is easy!!
• It is ALWAYS possible to calculate a mean (and sd, and other statistics)
for a dataset. However, just because you can punch numbers into a
calculator does not mean that those numbers mean anything useful!!!!
• One of the most important things I want you to take away from this
entire course is the ability to distinguish the right tool for the job from
the wrong tool for the job! This takes a little bit of review and practice of
concepts, but it is absolutely doable!!
• Recall from our first lectures that the mean is usually the WRONG tool
for the center of a distribution if the distribution is skewed or has
significant outliers.