Basic Statistics Concepts Marketing Logistics. Basic Statistics Concepts Including: histograms,...

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.

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858080709090759565756095

958575706575857070658055

55608075809085657080

Observations

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 65 70 75 80 85 90 95

55 60 65 70 75 80 85 90 95

Scores made by students

Back to Our Observations.

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858080709090759565756095

958575706575857070658055

55608075809085657080

Observations

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.

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858080709090759565756095

958575706575857070658055

55608075809085657080

Observations

7065707560758085755585

858080709090759565756095

958575706575857070658055

55608075809085657080

There are six

scores of 70Observations

Back to Our List of Scores

55 60 65 70 75 80 85 90 95

Back to Our List of Scores

55 60 65 70 75 80 85 90 95

1 1 1 1 1 1

Back to Our List of Scores

For each of thescores of 70 wemake one mark.

55 60 65 70 75 80 85 90 95

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

We continue to count the number of specific observations having a specific score.

55 60 65 70 75 80 85 90 95

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

We continue to count the number of specific observations having a specific score.

We are making

what is called a

“histogram.”

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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

HistogramMany times our histogram will end up looking much like this:

55 60 65 70 75 80 85 90 95

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

HistogramThis is what is called a “normal distribution.”

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

When things occur at what we would call random, they frequently fall into a normal distribution.

Histogram

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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

This is what is called a “normal distribution.”

In a normal distribution the highest number of observations occurs at the mean.

Histogram

55 60 65 70 75 80 85 90 95

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

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.

Histogram

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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

This is what is called a “normal distribution.”

Then they tend to taper off as you go to the higher scores…

Histogram

55 60 65 70 75 80 85 90 95

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

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…

Histogram

55 60 65 70 75 80 85 90 95

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

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…

Histogram

55 60 65 70 75 80 85 90 95

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

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.

Histogram

55 60 65 70 75 80 85 90 95

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

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.

Histogram

55 60 65 70 75 80 85 90 95

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

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.

Histogram

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

This phenomenon often occurs in events that we consider to be at random…

Histogram

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

…the scores tend to be distributed in a predictable way…

Histogram

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

…so we say it’s a…Histogram

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

…so we say it’s a…Histogram

55 60 65 70 75 80 85 90 95

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

This is what is called a “normal distribution.”

HistogramIt is usually graphed somewhat like this:

55 60 65 70 75 80 85 90 95

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

HistogramIt is usually graphed somewhat like this:

This is what is called a “normal distribution.”

55 60 65 70 75 80 85 90 95

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

HistogramIt is usually graphed somewhat like this:

While this is a rather crude graphing, the next slide shows several examples of normal distributions.

Total Order Cycle with Variability

2. Order entry and processing

Frequency:

1 2 3

1. Order preparation and transmittal

Frequency:

1 2 3

3. Order picking or production

Frequency:

1 9

Frequency:

TOTAL

3.5 days 8 20 days

5. Transportation

Frequency:

1 3 5

6. Customer receiving

Frequency:

.5 1 1.5

From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

Total Order Cycle with Variability

2. Order entry and processing

Frequency:

1 2 3

1. Order preparation and transmittal

Frequency:

1 2 3

3. Order picking or production

Frequency:

1 9

Frequency:

TOTAL

3.5 days 8 20 days

5. Transportation

Frequency:

1 3 5

6. Customer receiving

Frequency:

.5 1 1.5

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.

From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

Total Order Cycle with Variability

2. Order entry and processing

Frequency:

1 2 3

1. Order preparation and transmittal

Frequency:

1 2 3

3. Order picking or production

Frequency:

1 9

Frequency:

TOTAL

3.5 days 8 20 days

5. Transportation

Frequency:

1 3 5

6. Customer receiving

Frequency:

.5 1 1.5

Mean or average of about 2.Mean of just under 1

Mean of about 10.

Mean of 1

Mean of about 3

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.

From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

55 60 65 70 75 80 85 90 95

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

HistogramIf 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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858080709090759565756095

958575706575857070658055

55608075809085657080

Observations

Sure enough, if you were to average out all of ourobservations…

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858080709090759565756095

958575706575857070658055

55608075809085657080

Mean = 75

Observations

Sure enough, if you were to average out all of our observations…

You get a mean of 75.

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

Mississippi River Bottom

3 feet

200 feet

Headwaters New Orleans

Mean depth = 18 feet

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

There is a big difference between 3 feet and 200 feet.

Headwaters New Orleans

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

There is a big difference between 3 feet and 200 feet.

Headwaters New Orleans

That means my description of the Mississippi River as having an 18-foot mean depth really doesn’t tell me much.

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

There is a big difference between 3 feet and 200 feet.

Headwaters New Orleans

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?

That means my description of the Mississippi River as having an 18-foot mean depth really doesn’t tell me much.

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

Depth 5 feet, 13 feet less than mean of 18 feet

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

Depth 100 feet, 82 feet more than mean of 18 feet

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

Depth 190 feet, 172 feet more than mean of 18 feet

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

And so on…

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

If I can now average all of these measurements-- how far away each depth is from the mean of18 feet – I can get a clearer picture of how deepthe Mississippi River actually is.

Mississippi River Surface

Mississippi River Bottom

3 feet

200 feet

Mean depth = 18 feet

Headwaters New Orleans

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.

In other words, what I want to know is the“standard deviation” – what is the averageall of the depth measurements are away from the mean of 18 feet.

In other words, what I want to know is the“standard deviation” – what is the averageall of the depth measurements are away from the mean of 18 feet.

Let’s go back to our test scores.

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858080709090759565756095

958575706575857070658055

55608075809085657080

Mean = 75

Observations

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?

SD =(Observation – mean)2

N-1

Standard Deviation

• How far away from the mean do the observations generally fall?

SD =(Observation – mean)2

N-1

To find standard deviation…

Standard Deviation

• How far away from the mean do the observations generally fall?

• We take the square root of…

SD =(Observation – mean)2

N-1

Standard Deviation

• How far away from the mean do the observations generally fall?

SD =(Observation – mean)2

N-1

The sum of…

Standard Deviation

• How far away from the mean do the observations generally fall?

SD =(Observation – mean)2

N-1

The difference between the

observation minus the mean…

Standard Deviation

• How far away from the mean do the observations generally fall?

SD =(Observation – mean)2

N-1

The difference between the

observation minus the mean…

…Squared…

Standard Deviation

• How far away from the mean do the observations generally fall?

SD =(Observation – mean)2

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…

SD =(Observation – mean)2

N-1

…divided by one less than the number of observations

N-1 may not always be technically correct. In some cases it should be just N, the number of observations.However, in this classwe will always use N-1.*

*For population use N; for sample use N-1.

Standard Deviation

• Let’s find the standard deviation for our test score observations…

SD =(Observation – mean)2

N-1

Standard Deviation

• Let’s begin with just this part of the formula…

SD =(Observation – mean)2

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 75Subtract the mean from the observation, ortake 75 away from 70.

-

Standard Deviation

(Observation – mean)2

70 75Subtract the mean from the observation, ortake 75 away from 70.

It equals minus 5.

- = -5

Standard Deviation

(Observation – mean)2

70 75- = -5

We cannot have negative numbers, so to

make -5 positive, we square it, since a

negative times a

negative equals

a positive.

Standard Deviation

(Observation – mean)2

70 75- = -5 X - 5

= 25

We cannot have negative numbers, so to

make -5 positive, we square it, since a

negative times a

negative equals

a positive.

Standard Deviation

(Observation – mean)2

70 75- = -5 X - 5

= 25

We cannot have negative numbers, so to

make -5 positive, we square it, since a

negative times a

negative equals

a positive.This is called a “square.”

25-57570

Observation Mean

ObservationMinus mean

Observationminus meansquared knownas a“square.”

Expressed another way:

S

25-57570100-10756525-57570007575

225-157560007575

2557580

Observation Mean Observation minus mean

Observationminus meansquared

We go through all our observations,

subtracting the mean from them and squaring

the results.

Squares

First six observations…

Observation Mean Observation minus mean

Observationminus meansquared

2557580100107585

007575400-2075551001075852557580255758025-57570

Squares

…next seven observations. Rather than have slides showing all the observations…

Observation Mean Observation minus mean

Observationminus meansquared

225157590100107585100-10756525-575702557580

4300

…I will skip to the final four

observations.

Squares

Standard Deviation

• And come to the next part of our formula…

SD =(Observation – mean)2

N-1

Standard Deviation

• And come to the next part of our formula…

• …adding up all of the squares.

SD =(Observation – mean)2

N-1

Standard Deviation

• And come to the next part of our formula…

• …adding up all of the squares.

SD =(Observation – mean)2

25-57570100-10756525-57570007575

225-157560007575

2557580

Observation Mean Observation minus mean

Observationminus meansquared

Add up all squares

First six observations…

A

D

D

Observation Mean Observation minus mean

Observationminus meansquared

2557580100107585

007575400-2075551001075852557580255758025-57570

Add up all squares

A

D

D

…next seven observations. Rather than have slides showing all the observations…

Observation Mean Observation minus mean

Observationminus meansquared

225157590100107585100-10756525-575702557580

4300

A

D

D

A

D

D

…I will skip to the final four

observations.

Observation Mean Observation minus mean

Observationminus meansquared

225157590100107585100-10756525-575702557580

4300

Sum of squares…total of all

observations.

Standard Deviation

• We now have some data for our formula…

SD =(Observation – mean)2

N-1

Standard Deviation

• We now have some data for our formula…

SD =(Observation – mean)2

N-1

Sum of squares

Standard Deviation

• We now have some data for our formula…

SD = 4300

N-1

Sum of squares

Standard Deviation

• Now for the next part of our formula…

SD = 4300

N-1

Standard Deviation

• Now to the next part of our formula…

• …divide the sum of squares by the number of observations minus 1.

SD = N-1

4300

Count the number of observations…

25-57570100-10756525-57570007575

225-157560007575

2557580

Observation Mean Observation minus mean

Observationminus meansquared

First six

observations…

1

2

3

4

5

6

Count the number of observations…

Observation Mean Observation minus mean

Observationminus meansquared

2557580100107585

007575400-2075551001075852557580255758025-57570

…next seven observations. Rather than have slides showing all the observations…

7

8

9

10

11

12

13

Count the number of observations…

Observation Mean Observation minus mean

Observationminus meansquared

225157590100107585100-10756525-575702557580

4300

Sum of squares

39

40

41

42

Count the number of observations…

…I will skip to the final four

observations.

Observation Mean Observation minus mean

Observationminus meansquared

225157590100107585100-10756525-575702557580

4300

39

40

41

42

Count the number of observations…

There are 42 observations

Sum of squares

Standard Deviation

SD = N-1

4300

• More data for the formula…

Standard Deviation

SD = N-1

4300

•More data for the formula…

We have 42 observations so n is 42.

Standard Deviation

SD = 42-1

4300

•More data for the formula…

We have 42 observations so n is 42.

Standard Deviation

SD = 42-1

4300

•More data for the formula…

42 minus 1 is 41.

Standard Deviation

SD = 41

4300

•More data for the formula…

42 minus 1 is 41.

Standard Deviation

SD = 41

4300

•Process part of the formula…

4300 divided by 41…

Standard Deviation

SD = 41

4300

•Process part of the formula…

4300 divided by 41…

= 104.87

…equals

104.87

Standard Deviation

SD =

•Finish the formula…

Find the square root of 104.87

104.87

Standard Deviation

SD =

•Finish the formula…

Find the square root of 104.87

= 104.87 = 10.24

Which is 10.24

Standard Deviation

SD =

•Finish the formula…

Therefore, our standard deviation is 10.24

10.24

Standard Deviation

SD =

•This means that our observations average 10.24 away from the mean.

Therefore, our standard deviation is 10.24

10.24

To Review…

To Review…

• We developed a histogram…

55 60 65 70 75 80 85 90 95

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

Histogram

To Review…

• We examined a normal distribution...

55 60 65 70 75 80 85 90 95

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

Normal Distribution

Total Order Cycle with Variability

2. Order entry and processing

Frequency:

1 2 3

1. Order preparation and transmittal

Frequency:

1 2 3

3. Order picking or production

Frequency:

1 9

Frequency:

TOTAL

3.5 days 8 20 days

5. Transportation

Frequency:

1 3 5

6. Customer receiving

Frequency:

.5 1 1.5

Normal Distribution

From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

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?

SD =(Observation – mean)2

N-1

End of Program.