Transcript Standard Deviation

Understanding
Standard Deviation
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Introduction to Variation

There is variation in EVERYTHING!
– If you order 25 pepperoni pizzas, does each
slice have the same amount of pepperoni,
cheese and sauce?
– If you try on 10 pairs of jeans, the same brand,
the same style, do they all fit the same?
– When students take a physics exam, do they all
get the same grade?

Variation in the pepperoni count isn’t a
big deal, but what happens if there is a
lot of variation in
– Potency of medicine?
– Airplanes performance during landings?
– Concrete quality in bridges and buildings?
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Introduction to Variation

In order to understand a process or
product’s performance, you have to
understand the variation in the process
or products

Once you understand the variation, then
you can focus on reducing it, which can
save time, materials, and / or money
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Reminder: Mean

One way to understand a data set is by
examining the mean

The mean is often called the average of
the data set. Think about your grade
point average in school. Some grades
are higher, some are lower, and the mean
is somewhere in between, based on the
following calculation:
Mean = X1 + X2 + …. XN
Where N = the # of grades
contributing to the average
N
So, if you had class grades of: 78, 95, 100, 67 (what?!), 82, 90, and 89,
the mean = 78 + 95 + 100 + 67 + 82 + 90 + 89 = 85.9
7
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Reminder: Median

Another way to understand a data set is by
examining the median

The median is the central value of the data
set. So half of the data is greater than the
median and half of the data is less than the
median value.

So, if you had class grades of: 78, 95, 100, 67, 82, 90, and
89,
the median would be: 89
67, 78, 82, 89, 90, 95, 100
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We need more than the Mean and Median!

While the mean and median give useful insight into the data,
they do not give us the complete picture. Two sets of data
can have the same mean and median, but be dramatically
different.
Minutes to get to school, option A
Frequency
5
0-1

2
2
1-2
2-4
6
6
5
2
4-6
6-8
2
8-10 10-12 12-14 14-16 16-18
Frequency
6
Minutes to get to school, option B
5
4-6
6
5
6-8
8-10 10-12
If the two histograms above represented the time it took
using two different routes to get to school in the morning, and
you absolutely had to be on time for the free breakfast pizza
(yes, pizza), which process would you prefer to use and
why? The means and medians are the same.
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Standard Deviation

In order to understand the distribution or spread of the data
around the mean, the Standard Deviation is calculated.
Product Quality, option A

Product Quality, option B
Data Set A has a smaller Standard Deviation than Data Set
B, as it is grouped tighter around the mean of the data set
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Standard Deviation

Standard Deviation is thought of as the average distance of
data points from the mean, and is calculated as follows:
N
1
Standard Deviation =
( xi – x )2
N
i=1
Where N = the number of elements in the data set, and
X = the mean of the data set
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Standard Deviation

Standard Deviation for our set of grades is:
N
Standard Deviation =
1
( xi – x )2
N
i=1
=
((67 – 85.9)2 + (78 – 85.9)2 + (82 – 85.9)2 + (89 – 85.9)2 + (90 – 85.9)2 + (95 – 85.9)2 + (100 – 85.9)2)
7
= 10.3
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Standard Deviation

If you have an understanding teacher, who lets you retake
the test where you got a 67, and your new grade ends up
being a 79, how does that impact the Standard Deviation?
New Data Set: 78, 95, 100, 79, 82, 90, and 89
New Mean: 87.6
New Standard Deviation: 7.9
So, the Standard Deviation decreased from 10.3 to 7.9. A
smaller Standard Deviation means tighter distribution (less
variation) in the data.
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