Download Statistics_basics_2007

Basic Statistics Concepts
Marketing Logistics
Basic Statistics Concepts
Including: histograms, means,
normal distributions,
standard deviations.
Basic Statistics Concepts
Developing a histogram.
Developing a Histogram
• Let’s say we are looking at the test scores
of 42 students.
• For the sake of our discussion, we will call
each test score an “observation.”
• Therefore, we have 42 observations.
• The next slide shows the 42 test scores or
observations.
70
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75
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80
85
75
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85
80
85
95
180
85
80
75
170
70
1
90
65
90
75
75
85
95
70
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75
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95
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85Observations 60
55
60
80
1
75
80
90
85
1
65
1
70
80
x
Plotting Scores on a Histogram
• We decide to start figuring out how many
times students made a specific score.
• In other words, how many students got a
score of 85? How many got a 95? And so
on…
• We list all the scores, then begin recording
how many times a student got that score.
• The next slide shows a list of all the
scores.
55
60
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70
75
80
85
90
95
Scores made by students
55
60
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70
75
80
85
90
95
Back to Our Observations.
70
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75
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85
75
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85
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85
95
180
85
80
75
170
70
1
90
65
90
75
75
85
95
70
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75
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80
95
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85Observations 60
55
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80
1
75
80
90
85
1
65
1
70
80
x
Back to Our Observations.
• How many times did someone get a 70?
• Look on the next slide and count the
number of scores of 70.
70
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70
75
60
75
80
85
75
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85
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85
95
180
85
80
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170
70
1
90
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90
75
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85
95
70
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75
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80
95
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85Observations 60
55
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80
1
75
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90
85
1
65
1
70
80
x
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95
180
85
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170
70
1
90
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90
75
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85
95
70
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75
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60
80
95
55
85Observations 60
55
60
80
1
75
80
90
85
1
65
1
70
80
There are six
scores of 70
x
Back to Our List of Scores
Back to Our List of Scores
55
60
65
70
75
80
85
90
95
Back to Our List of Scores
For each of the
scores of 70 we
make one mark.
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1
1
1
1
1
55
60
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70
75
80
85
90
95
We continue to count
the number of specific
observations having a
specific score.
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1
55
60
65
70
1
75
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1
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1
80
85
90
95
We continue to count
the number of specific
observations having a
specific score.
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1
1
1
1
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1
55
60
65
1
1
1
1
1
1
70
We are
making what
is called a
“histogram.”
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75
80
85
90
95
Histogram
Many times our
histogram will
end up looking
much like this:
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
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60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
When things occur
at what we would
call random, they
frequently fall into a
normal distribution.
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60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
In a normal distribution
the highest number of
observations occurs at
the mean.
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60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
In a normal distribution
the highest number of
observations occurs at
the mean. There were
seven scores of 75.
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Then they tend to taper
off as you go to the
higher scores…
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Then they tend to taper
off as you go to the
higher scores. Only six
scores of 80…
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Then they tend to taper
off as you go to the
higher scores. Only four
scores of 85…
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Then they tend to taper
off as you go to the
higher scores. Three
scores of 90.
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Then they tend to taper
off as you go to the
higher scores. Two
scores of 95.
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
Below the mean, scores
tend to taper off, usually
at about an identical rate
as the scores we just
looked at that were
above the mean.
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
This phenomenon
often occurs in
events that we
consider to be at
random…
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60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
…the scores tend
to be distributed in
a predictable way…
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60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
…so we say it’s a…
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
…so we say it’s a…
1
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55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
It is usually graphed
somewhat like this:
1
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1
55
60
65
70
75
80
85
90
95
Histogram
This is what is
called a
“normal
distribution.”
It is usually graphed
somewhat like this:
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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60
65
70
75
80
85
90
95
Histogram
It is usually graphed
somewhat like this:
While this is a rather crude graphing,
the next slide shows several
examples of normal distributions.
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55
60
65
70
75
80
85
90
95
Total Order Cycle with Variability
1. Order preparation
and transmittal
2. Order entry
and processing
Frequency:
1
3
1
5. Transportation
6. Customer
receiving
Frequency:
Frequency:
1
3
Frequency:
Frequency:
2
5
.5
1
3. Order picking
or production
2
1
3
9
TOTAL
Frequency:
1.5
3.5 days
8
20 days
Total Order Cycle with Variability
1. Order preparation
and transmittal
2. Order entry
and processing
Frequency:
1
3
1
5. Transportation
6. Customer
receiving
Frequency:
Frequency:
1
3
Frequency:
Frequency:
2
5
.5
1
3. Order picking
or production
2
1
3
9
TOTAL
Frequency:
1.5
3.5 days
8
20 days
The line in the middle of each normal distribution indicates the average or,
more correctly, the “mean.” It’s the place where we have the highest number
of observations.
TotalMeanOrder
Cycle
with
Variability
or average of about 2.
Mean of just under 1
1. Order preparation
and transmittal
2. Order entry
and processing
Frequency:
2
3
1
2
Mean of about 3
1
3
6. Customer
receiving
5. Transportation
Frequency:
Frequency:
Frequency:
1
3. Order picking
or production
TOTAL
9
Mean of about 10.
Frequency:
Frequency:
Mean of 1
1
3
5
.5
1
1.5
3.5 days
8
20 days
The line in the middle of each normal distribution indicates the average or,
more correctly, the “mean.” It’s the place where we have the highest number
of observations.
Histogram
If the highest part of our histrogram is on 75,
it stands to reason that 75 is our mean of our test scores.
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60
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70
75
80
85
90
95
85
95
70
180
85
65
80
75
70
1
70
70
1
Sure enough, if you
75 to average90
65
were
out
all
60of our
90
75
observations…
75
75
85
80
95
70
85
65
70
75
75
65
55
60
80
85
95
55
80
85Observations 60
55
60
80
1
75
80
90
85
1
65
1
70
80
x
55
85
95
70
180
85
60
65
80
75
80
70
1
70
70
1
1
75
Sure enough, if you
75 to average90
65
80
were
out
all
60of our
90
75
90
observations…
75
75
85
85
80
95
70
1
65
85
65
70
1
70
mean of 75. 65
75 You get a75
80
55
60
80
x
85
95
55 Mean = 75
80
85Observations 60
A mean can be a good predictor…
• A mean or average can often help me
predict what will happen in the future.
• For instance, if students usually get a
mean of 75 on tests, by giving basically
the same kinds of tests, an instructor can
usually predict that in the future students
will usually score an average of 75 on the
test.
A mean can be a good predictor,
but…
• Sometimes a mean is not enough for a
prediction or determination.
A mean can be a good predictor,
but…
• Sometimes a mean is not enough for a
prediction or determination.
• For instance, if I tell you that the average, or
mean, depth of the Mississippi River is about 18
feet, I’m not giving you a clear picture of the
nature of the river.
A mean can be a good predictor,
but…
• Sometimes a mean is not enough for a
prediction or determination.
• For instance, if I tell you that the average, or
mean, depth of the Mississippi River is about 18
feet, I’m not giving you a clear picture of the
nature of the river.
• That’s because at its headwaters, the river
averages around 3 feet deep. But in certain
places around New Orleans, it is 200 feet deep.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
There is a big difference between 3 feet and 200 feet.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
There is a big difference between 3 feet and 200 feet.
That means my description of the Mississippi River
as having an 18-foot mean depth really doesn’t tell me much.
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
There is a big difference between 3 feet and 200 feet.
That means my description of the Mississippi River
as having an 18-foot mean depth really doesn’t tell me much.
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
3 feet
New Orleans
Depth 5 feet, 13 feet less than mean of 18 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
3 feet
New Orleans
Depth 100 feet, 82 feet more than mean of 18 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
3 feet
New Orleans
Depth 190 feet, 172 feet more than mean of 18 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
And so on…
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
If I can now average all of these measurements
-- how far away each depth is from the mean of
18 feet – I can get a clearer picture of how deep
the Mississippi River actually is.
To get an accurate picture or prediction of Mississippi River depth, I need to find
out how much variation there is around the mean. In other words, at different places
along the river, how much different is the depth at that place, compared to the mean.
Mississippi River Surface
Headwaters
New Orleans
3 feet
Mean depth = 18 feet
200 feet
Mississippi River Bottom
In other words, what I want to know is the
“standard deviation” – what is the average
all of the depth measurements are away
from the mean of 18 feet.
Let’s go back to our test scores.
In other words, what I want to know is the
“standard deviation” – what is the average
all of the depth measurements are away
from the mean of 18 feet.
70
65
70
75
60
75
80
85
75
55
85
80
55
85
95
180
85
60
80
75
80
170
70
1
1
75
90
65
80
90
75
90
75
85
85
95
70
1
65
65
70
1
70
75
65
80
60
80
x
95
55 Mean = 75
85Observations 60
Standard Deviation
• How far away from the mean do the
observations generally fall?
Standard Deviation
• How far away from the mean do the
observations generally fall?
• There is a formula to show us…
Standard Deviation
• How far away from the mean do the
observations generally fall?
(Observation – mean)2
SD =
N-1
Standard Deviation
• How far away from the mean do the
observations generally fall?
(Observation – mean)2
SD =
To find standard deviation…
N-1
Standard Deviation
• How far away from the mean do the
observations generally fall?
• We take the square root of…
(Observation – mean)2
SD =
N-1
Standard Deviation
• How far away from the mean do the
observations generally fall?
(Observation – mean)2
SD =
N-1
The sum of…
Standard Deviation
• How far away from the mean do the
observations generally fall?
The difference between the
observation minus the mean…
(Observation – mean)2
SD =
N-1
Standard Deviation
• How far away from the mean do the
observations generally fall?
The difference between the
observation minus the mean…
(Observation – mean)2
SD =
N-1
…Squared
…
Standard Deviation
• How far away from the mean do the
observations generally fall?
(Observation – mean)2
SD =
N-1
…divided by one less than
the number of observations
Standard Deviation
• Note for the Statistics Police, but something you
don’t have to worry about…
N-1 may not always be
technically correct. In
some cases it should be
just N, the number of
(Observation – mean)2
observations.
SD =
N-1
However, in this class
we will always use N-1.*
…divided by one less than
the number of observations
*For population use N; for sample use N-1.
Standard Deviation
• Let’s find the standard deviation for our
test score observations…
(Observation – mean)2
SD =
N-1
Standard Deviation
• Let’s begin with just this part of the
formula…
(Observation – mean)2
SD =
N-1
Standard Deviation
• Let’s begin with just this part of the
formula…
(Observation – mean)2
Standard Deviation
• Let’s begin with just this part of the formula
and look at the test score of 70, one of our
observations.
(Observation – mean)2
Standard Deviation
• Let’s begin with just this part of the formula
and look at the test score of 70, one of our
observations.
(Observation – mean)2
70
Standard Deviation
• And let’s include our mean, which we
determined to be 75.
(Observation – mean)2
70
75
Standard Deviation
(Observation – mean)2
70
- 75
Subtract the mean from the
observation, or
take 75 away from 70.
Standard Deviation
(Observation – mean)2
70
- 75 = -5
Subtract the mean from the
observation, or
take 75 away from 70.
It equals minus 5.
Standard Deviation
We cannot have negative numbers, so to
make -5 positive, we square it, since a
negative times a
2
(Observation
–
mean)
negative equals
a positive.
- 75 = -5
70
Standard Deviation
We cannot have negative numbers, so to
make -5 positive, we square it, since a
negative times a
2
(Observation
–
mean)
negative equals
a positive.
- 75 = -5 X - 5
70
= 25
Standard Deviation
We cannot have negative numbers, so to
make -5 positive, we square it, since a
negative times a
2
(Observation
–
mean)
negative equals
a positive.
- 75 = -5 X - 5
70
= 25
This is called a “square.”
Expressed another way:
Observation
Mean
70
65
75
75
Observation
minus mean
squared known
as a“square.”
-5
-10
Observation
Minus mean
25
100
S
Observation
70
65
70
75
60
75
80
First six
85
observations…
Mean
75
75
75
75
75
75
75
75
Observation
minus mean
-5
-10
-5
0
-15
0
5
10
We go through all our observations,
subtracting the mean from them and
squaring the results.
Observation
minus mean
squared
25
100
25
0
225
0
25
100
Squares
Observation
Mean
80
85
75
55
85
80
80
70
90
75
75
75
75
75
75
75
75
75
…next seven
observations. Rather than
have slides showing all
the observations…
Observation
minus mean
5
10
0
-20
10
5
5
-5
15
Observation
minus mean
squared
25
100
0
400
100
25
25
25
225
Squares
Observation
Mean
90
85
65
70
80
…I will skip to the final
four observations.
75
75
75
75
75
Observation
minus mean
15
10
-10
-5
5
Observation
minus mean
squared
225
100
100
25
25
4300
102.381 Po
Squares
Standard Deviation
• And come to the next part of our formula…
(Observation – mean)2
SD =
N-1
Standard Deviation
• And come to the next part of our formula…
• …adding up all of the squares.
(Observation – mean)2
SD =
N-1
Standard Deviation
• And come to the next part of our formula…
• …adding up all of the squares.
(Observation – mean)2
SD =
Observation
Mean
70
65
70
75
60
75
80
85
75
75
75
75
75
75
75
75
Observation
minus mean
-5
-10
-5
0
-15
0
5
10
Observation
minus mean
squared
25
100
25
0
225
0
25
100
Add up all squares
First six
observations…
A
D
D
Observation
Mean
80
85
75
55
85
80
80
70
90
75
75
75
75
75
75
75
75
75
…next seven
observations. Rather than
have slides showing all
the observations…
Observation
minus mean
5
10
0
-20
10
5
5
-5
15
Observation
minus mean
squared
25
100
0
400
100
25
25
25
225
Add up all squares
A
D
D
Observation
90
85
65
70
80
Mean
75
75
75
75
75
…I will skip to the final
four observations.
Observation
minus mean
15
10
-10
-5
5
Observation
minus mean
squared
225
100
A
A
100
D
D
D
25
D
25
4300
102.381 Popul
Observation
90
85
65
70
80
…total of all
observations.
Mean
75
75
75
75
75
Sum of squares
Observation
minus mean
15
10
-10
-5
5
Observation
minus mean
squared
225
100
100
25
25
4300
102.381 Popul
Standard Deviation
• We now have some data for our formula…
(Observation – mean)2
SD =
N-1
Standard Deviation
• We now have some data for our formula…
Sum of squares
(Observation – mean)2
SD =
N-1
Standard Deviation
• We now have some data for our formula…
Sum of squares
4300
SD =
N-1
Standard Deviation
• Now for the next part of our formula…
4300
SD =
N-1
Standard Deviation
• Now to the next part of our formula…
• …divide the sum of squares by the
number of observations minus 1.
4300
SD =
N-1
Count the number of observations…
Count the number of observations…
Observation
1
2
3
4
5
6
Mean
70
65
70
75
60
75
80
85
First six
observations…
75
75
75
75
75
75
75
75
Observation
minus mean
-5
-10
-5
0
-15
0
5
10
Observation
minus mean
squared
25
100
25
0
225
0
25
100
Count the number of observations…
Observation
7
8
9
10
11
12
13
Mean
80
85
75
55
85
80
80
70
90
75
75
75
75
75
75
75
75
75
…next seven
observations. Rather than
have slides showing all
the observations…
Observation
minus mean
5
10
0
-20
10
5
5
-5
15
Observation
minus mean
squared
25
100
0
400
100
25
25
25
225
Count the number of observations…
Observation
39
40
41
42
Mean
90
85
65
70
80
Observation
minus mean
75
75
75
75
75
15
10
-10
-5
5
Observation
minus mean
squared
225
100
100
25
25
4300
102.381 P
Sum of squares
…I will skip to the final
four observations.
Count the number of observations…
Observation
39
40
41
42
Mean
90
85
65
70
80
Observation
minus mean
75
75
75
75
75
15
10
-10
-5
5
Observation
minus mean
squared
225
100
100
25
25
4300
102.381 P
Sum of squares
There are 42
observations
Standard Deviation
• More data for the formula…
4300
SD =
N-1
Standard Deviation
•More data for the formula…
4300
SD =
N-1
We have 42 observations so n is 42.
Standard Deviation
•More data for the formula…
4300
SD =
42-1
We have 42 observations so n is 42.
Standard Deviation
•More data for the formula…
4300
SD =
42-1
42 minus 1 is 41.
Standard Deviation
•More data for the formula…
4300
SD =
41
42 minus 1 is 41.
Standard Deviation
•Process part of the formula…
4300
SD =
41
4300 divided by 41…
Standard Deviation
•Process part of the formula…
4300
SD =
41
4300 divided by 41…
= 104.87
…equals
104.87
Standard Deviation
•Finish the formula…
SD =
104.87
Find the square root of 104.87
Standard Deviation
•Finish the formula…
Which is 10.24
SD =
= 104.87
Find the square root of 104.87
= 10.24
Standard Deviation
•Finish the formula…
SD =
10.24
Therefore, our standard deviation is 10.24
Standard Deviation
•This means that our observations
average 10.24 away from the mean.
SD =
10.24
Therefore, our standard deviation is 10.24
To Review…
To Review…
• We developed a histogram…
Histogram
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
55
60
65
70
75
80
85
90
95
To Review…
• We examined a normal distribution...
Normal Distribution
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
55
60
65
70
75
80
85
90
95
Total Order Cycle with Variability
Normal Distribution
1. Order preparation
and transmittal
2. Order entry
and processing
Frequency:
1
3
1
5. Transportation
6. Customer
receiving
Frequency:
Frequency:
1
3
Frequency:
Frequency:
2
5
.5
1
3. Order picking
or production
2
1
3
9
TOTAL
Frequency:
1.5
3.5 days
8
20 days
To Review…
• We learned how to determine standard
deviation, or the average of how far
observations are different from the mean.
Standard Deviation
• How far away from the mean do the
observations generally fall?
(Observation – mean)2
SD =
N-1
End of Program.