measurement scales (1)

Scales of Measurement

Learning Module

Different scales of measurement use the same numerals (i.e., 1, 2, 3, 4 . . .)

But, the numerals carry different information and carry different information and symbolize different phenomena across scales (i.e., 1 = Catholic, 2 = Mormon . . . or 1 = Agree, 2 = Disagree, or 1 = correct, 0 = incorrect)

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But, the numerals carry different information and carry different information and symbolize different phenomena across scales (i.e., 1 = Catholic, 2 = Mormon . . . or 1 = Agree, 2 = Disagree, or 1 = correct, 0 = incorrect)

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But, the numerals carry different information and symbolize different phenomena across scales (i.e., 1 = Catholic, 2 = Mormon . . . or 1 = Agree, 2 = Disagree, or 1 = correct, 0 = incorrect)

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But, the numerals carry different information and symbolize different phenomena across scales (i.e.,

• 1 = Catholic, 2 = Mormon . . .

• 1 = Agree, 2 = Disagree

• 1 = correct, 0 = incorrect

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The four common scales of measurement are:

Nominal (1 = Male, 2 = Female)

Ordinal (1 = Private, 2 = Sergeant, 3 = Lieutenant . . .)

Interval (30OF, 40OF, 50O . . .)

Ratio (0 meters, 10 meters, 100 meters . . .)

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Interval (30OF, 40OF, 50OF. . .)


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Nominal, Ordinal, Interval, Ratio

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Nominal scales use numbers as replacements for names.

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1 = American

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1 = American

2 = Canadian

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1 = American

2 = Canadian

3 = Mexican

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1 = American

2 = Canadian

3 = Mexican

Data Set

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1 = American

2 = Canadian

3 = Mexican

Student Nationality Test Scores

1 3 32

2 1 28

3 3 33

4 2 27

5 1 34

6 2 31

Data Set

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1 = American

2 = Canadian

3 = Mexican


1 3 32

2 1 28

3 3 33

4 2 27

5 1 34

6 2 31

Data Set

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Nominal


1 = American

2 = Canadian

3 = Mexican


1 3 32

2 1 28

3 3 33

4 2 27

5 1 34

6 2 31

Data Set

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1 = American

2 = Canadian

3 = Mexican


1 3 32

2 1 28

3 3 33

4 2 27

5 1 34

6 2 31

Data Set

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The root of the term “nominal” is “nom” meaning “name”.

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Nominal scales

• assume no quantity of the attribute.

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Nominal scales


1 = American

2 = Canadian

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Nominal scales


1 is not more than 2 and2 is not less than 1 in this context

1 = American

2 = Canadian

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Nominal scales


• has no particular interval

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Nominal scales


• has no particular interval

1 and 2 and 3 are not equal intervals because there is no quantity involved.

1 = American2 = Canadian3 = Mexican

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Nominal scales


• has no particular interval.

• has no zero or starting point.

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Nominal scales



• has no zero or starting point.1 = American2 = Canadian3 = Mexican

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Nominal scales



• has no zero or starting point.

Because there is no quantity involved there is no such thing as a zero point (ie., complete absence of nationality).

1 = American2 = Canadian3 = Mexican

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Ordinal scales use numbers to represent relative amounts of an attribute.

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Private1

Corporal2

Sargent3

Lieutenant4

Major5

Colonel6

General7

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Private1

Corporal2

Sargent3

Lieutenant4

Major5

Colonel6

General7

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3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

Relative in terms of PLACEMENT (1st, 2nd, & 3rd)Slide 42 of 85



3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

Relative in terms of PLACEMENT (1st, 2nd, & 3rd)Slide 43 of 85


Ordinal scales

• assume quantity of the attribute.

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Ordinal scales


Lieutenant4

Colonel6

A colonel has more authority than a Lieutenant

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Ordinal scales


1st place is higher than 3rd place

3rd

Place1st

Place

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Ordinal scales


• do not have equal intervals.

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Ordinal scales



3rd

Place15’ 2”

2nd

Place16’ 1”

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Ordinal scales



The distance between 3rd and 2nd place (11”) is not the same interval as the distance between 2nd and 1st place (1”)

3rd

Place15’ 2”

2nd

Place16’ 1”

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Ordinal scales



3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

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Ordinal scales




3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

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Ordinal scales




3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

A higher number only

represents more of the attribute

than a lower number,

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Ordinal scales




3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

. . . but how much more is

undefined.

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Ordinal scales




3rd

Place15’ 2”

2nd

Place16’ 1”

1st

Place16’ 3”

The difference between points

on the scale varies from

point to point

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Ordinal scales



• may have an arbitrary zero or starting point.

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Ordinal scales




O Completely Disagree

O Mostly Disagree

O Mostly Agree

O Completely Agree

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Ordinal scales




O Completely Disagree

O Mostly Disagree

O Mostly Agree

O Completely Agree

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Ordinal scales




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Ordinal scales




O Not at All

O Very Little

O Somewhat

O Quite a Bit

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Ordinal scales




O Not at All

O Very Little

O Somewhat

O Quite a Bit

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Ordinal scales




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Ordinal scales




O Not Important

O Slightly Important

O Somewhat Important

O Very Important

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Ordinal scales




O Not Important

O Slightly Important

O Somewhat Important

O Very Important

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Important Point

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Important Point

Numbers on an ordinal scale are limited in the information they carry (i.e., no equal intervals,

no zero point)

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Interesting Note

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Interesting Note

Technically, numbers on an ordinal scale cannot be added or subtracted.

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Interesting Note

Technically, numbers on an ordinal scale cannot be added or subtracted.

(but we frequently do it anyway !)

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Ordinal Numbers in a Data Set

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Student Nationality Place Test Scores

1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

Nominal

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

OrdinalNominal

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Interval scales


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Interval scales


Temperature

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Interval scales


• have equal intervals.

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Interval scales



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Interval scales



40o - 41o

100o - 101o

70o - 71o

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Interval scales



40o - 41o

100o - 101o

70o - 71o

Each set of readings are the same distance apart: 1o

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Interval scales




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Interval scales



• may have an arbitrary zero or starting point.Daniel Gabriel Fahrenheit (1686–1736) determined that equal amounts of ice, water, and salt mixed together reached a

stable temperature at 0o F

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Interval scales



• may have an arbitrary zero or starting point.Daniel Gabriel Fahrenheit (1686–1736) determined that equal amounts of ice, water, and salt mixed together reached a

stable temperature at 0o F

That has an arbitrary feel to it.

Doesn’t it? Slide 83 of 85

Technically, numbers on an interval scale can be added and subtracted

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70o

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100o

70o

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100o

70o

100o is 30o more (+) than 70o

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100o

70o

100o is 30o more (+) than 70o

70o is 30o less (-) than 100o

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Technically, numbers on an interval scale can be added and subtracted but not divided and multiplied.

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100o

50o

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100o

50oAnd 50o is NOT half (/) as hot as 100o

100o is NOT twice (x) as hot as 50o

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100o

50oAnd 50o is NOT half (/) as hot as 100o

But 100o is NOT twice (x) as hot as 50o

But many do so anyways

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Interval Numbers in a Data Set

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set

OrdinalNominal Interval

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1 3 3 32

2 1 5 28

3 3 2 33

4 2 6 27

5 1 1 34

6 2 4 31

Data Set


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Ratio scales


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Ratio scales


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Ratio scales


6’5” 5’4”5’3” 6’4” 5’11”5’10”

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Ratio scales



6’5” 5’4”5’3” 6’4” 5’11”5’10”

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Ratio scales



6’5” 5’4”5’3” 6’4” 5’11”5’10”

Every inch represents a unit of measure that is the same across all inches Slide 102 of 85


Ratio scales



6’5” 5’4”5’3” 6’4” 5’11”5’10”

With the interval nature of the data, you can say that player 4 (blue team) is 6 inches taller than Player 19 (yellow team)Slide 103 of 85


Ratio scales



• has a zero or starting point.

6’5” 5’4”5’3” 6’4” 5’11”5’10”

With a zero starting point (0’0”) you can say that player 6 (blue team) is 4/5 the size of player 4 (blue team) Slide 104 of 85

Ratio Numbers in a Data Set

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Student Nationality Place Test Scores Height

1 3 3 32 5’2”

2 1 5 28 6’3”

3 3 2 33 6’0”

4 2 6 27 5’8”

5 1 1 34 6’1”

6 2 4 31 5’5”

Data Set


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Student Nationality Place Test Scores Height

1 3 3 32 5’2”

2 1 5 28 6’3”

3 3 2 33 6’0”

4 2 6 27 5’8”

5 1 1 34 6’1”

6 2 4 31 5’5”

Data Set

OrdinalNominal Interval Ratio

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Important Point

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Important Point

Numbers on a ratio scale

• carry more information than the same numbers on an interval or ordinal scale.

• can be

– added,

– subtracted,

– multiplied, or

– divided.

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Important Point



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Important Point



• can be

– added,

– subtracted,

– multiplied, or

– divided.

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Two more Important Points

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1. More adequate scales can be easily converted to less adequate scales.

2. Most statistical programs will treat interval and ratio data the same.

Ratio - - - > Interval - - - > Ordinal - - - > Nominal

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1. More adequate scales can be easily converted to less adequate scales.

2. Most statistical programs will treat interval and ratio data the same.

Ratio - - - > Interval - - - > Ordinal - - - > Nominal

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Let’s Review

1. Which scale does not measure quantity or amount?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

1. Which scale does not measure quantity or amount?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

2. Which scale has a zero or starting point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

2. Which scale has a zero or starting point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

3. Which scale captures amount but does not have equal distances between units of measure?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

3. Which scale captures amount but does not have equal distances between units of measure?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

4. Which scale has equal distance between adjacent points but no zero point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

4. Which scale has equal distance between adjacent points but no zero point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

5. Which scale expresses more of an attribute across the scale, but does not express the distance between each point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

5. Which scale expresses more of an attribute across the scale, but does not express the distance between each point?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review

6. Which scale is represented by the highlighted column in the data set?

A. Nominal

B. Ordinal

C. Interval

D. RatioStudent Test Scores Place Nationality Height

1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 125 of 85

Let’s Review


A. Nominal

B. Ordinal

C. Interval


1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 126 of 85

Let’s Review


A. Nominal

B. Ordinal

C. Interval


1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 127 of 85

Let’s Review


A. Nominal

B. Ordinal

C. Interval


1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 128 of 85

Let’s Review


A. Nominal

B. Ordinal

C. Interval


1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 129 of 85

Let’s Review


A. Nominal

B. Ordinal

C. Interval


1 32 3 3 5’2”

2 28 5 1 6’3”

3 33 2 3 6’0”

4 27 6 2 5’8”

5 34 1 1 6’1”

6 31 4 2 5’5”Slide 130 of 85

Let’s Review

9. Under which scale would you classify the Kelvin scale?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

What is the Kelvin Scale?The Kelvin scale assumes quantity of heat and has equal intervals along the scale with an absolute zero

Absolute zero heat represents zero molecular motion and is a good starting point for measurement.

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Let’s Review

9. Under which scale would you classify the Kelvin scale?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

What is the Kelvin Scale?The Kelvin scale assumes quantity of heat and has equal intervals along the scale with an absolute zero

Absolute zero heat represents zero molecular motion and is a good starting point for measurement.

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Let’s Review

10. Under which scale would you classify a Likertscale?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

O Strongly DisagreeO DisagreeO Slightly DisagreeO Slightly AgreeO Strongly Disagree.

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Let’s Review

10. Under which scale would you classify a Likertscale?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

O Strongly DisagreeO DisagreeO Slightly DisagreeO Slightly AgreeO Strongly Disagree.

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Let’s Review

11. Under which scale would you classify social security numbers?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

987-65-4321

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Let’s Review

11. Under which scale would you classify social security numbers?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

987-65-4321

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Let’s Review

12. Under which scale would you classify

the College Football Top 25 ranking?

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review



A. Nominal

B. Ordinal

C. Interval

D. Ratio

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Let’s Review



A. Nominal

B. Ordinal

C. Interval

D. Ratio

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13. What scale is represented in each row?

Scale Quantity Assumed

Equal Intervals Zero Point Calculations

? Yes Yes Absolute Add, subtract, multiply, divide

Yes Yes Arbitrary Add, subtract

Yes No Arbitrary None

No No Irrelevant None

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Ratio Yes Yes Absolute Add, subtract, multiply, divide

Yes Yes Arbitrary Add, subtract



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? Yes Yes Arbitrary Add, subtract



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Interval Yes Yes Arbitrary Add, subtract

? Yes No Arbitrary None


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Ordinal Yes No Arbitrary None


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? No No Irrelevant None

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Nominal No No Irrelevant None

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More Practice Problems

What type of data is represented in this problem?

A. Nominal? If yes, what is it? ___________________

B. Ordinal? If yes, what is it? ___________________

C. Interval? If yes, what is it? ___________________

D. Ratio? If yes, what is it? ___________________

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14. Suppose a researcher wants to analyze whether different ethnic groups vary in terms of their level of public religious devotion. She also thinks that there might be a relationship between public religious devotion and the length of hair for both men and women.

What type of data is represented in this problem?A. Nominal? If yes, what is it? ___________________B. Ordinal? If yes, what is it? ___________________C. Interval? If yes, what is it? ___________________D. Ratio? If yes, what is it? ___________________

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Ethnic Group

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Ethnic Group

Level of public religious devotion

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Ethnic Group


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None




Ethnic Group


None

Length of hair

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Ethnic Group


None

Length of hair

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Gender


16. Which data type is represented in the scenario below:

A researcher created an assessment of depression that included ten T/F questions. Subjects were given 1 point for every question that they answered correctly. Scores could range from 0 to 10.

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

1. higher scores represent more depression2. the difference between 10 and 9 is the same as the

difference between 9 and 8. 3. a score of 0 is an arbitrary starting point based on the

limited number of questions selected by the researcher. It is probable that there is some degree of depression in subjects that score 0.

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

1. higher scores represent more depression2. the difference between 10 and 9 is the same as the

difference between 9 and 8. 3. a score of 0 is an arbitrary starting point based on the

limited number of questions selected by the researcher. It is probable that there is some degree of depression in subjects that score 0.

In many cases, this can be a subjective determination. In this case, it can be argued that this is actually an ordinal scale because different questions might carry different predictive weight. For example “I feel suicidal” might be more indicative of depression than “I feel blue more days than not”.

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A researcher believes that the number of broken bones that someone suffers can be counted as discrete trauma. She includes an item in her survey that reads “How many bones have you broken in your life time?”

A. Nominal

B. Ordinal

C. Interval

D. Ratio

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

1. higher scores represent more trauma to the body2. the difference between 3 bones and 2 bones is the same

as the difference between 2 bones and 1 bone. 3. the starting point, zero broken bones, is an absolute zero.

Zero broken bones is really zero broken bones.

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

1. higher scores represent more trauma to the body2. the difference between 3 bones and 2 bones is the same

as the difference between 2 bones and 1 bone. 3. the starting point, zero broken bones, is an absolute zero.

Zero broken bones is really zero broken bones.

Once again, technically this can be categorized as an ordinal scale, because if most of your broken bones occurred when you were two years old, that might be less traumatic to the body than if they occurred at age 90.

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A. Nominal

B. Ordinal

C. Interval

D. Ratio

Obviously higher scores represent more trauma to the body. The difference between 3 bones and 2 bones is the same as the difference between 2 bones and 1 bone. The starting point, zero broken bones, is an absolute zero. Zero broken bones is really zero broken bones.

Once again, technically this can be categorized as an ordinal scale, because if most of my broken bones occurred when I was two years old, that might be less traumatic to the body than if they occurred at age 90.

While we categorize scales as interval and ratio there could always be some technical reason or rationale for reclassify them as ordinal.

The degree of technicality depends on your audience and the purposes of your research.

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In Summary

Here is a basic decision tree that may be useful in determining the type of data you are working with:

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In Summary


Is there an assumption of quantity?

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In Summary


NOMINAL

no


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In Summary


NOMINAL

yes no


Are there equal

intervals?

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In Summary

NOMINAL

ORDINAL

yes no

no


Are there equal

intervals?

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In Summary


NOMINAL

ORDINAL

yes

yes

no

no


Are there equal

intervals?

Is there an absolute

zero?

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In Summary


NOMINAL

ORDINAL

INTERVAL

yes

yes

no

no

no


Are there equal

intervals?


zero?

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In Summary


NOMINAL

ORDINAL

INTERVALRATIO

yes

yes

yes

no

no

no


Are there equal

intervals?


zero?

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measurement scales (1)

Business