Distributions of Variables

Anthony J Greene 1

Distributions of Variables

I. Properties of Variables

II. Nominal Data & Bar Charts

III. Ordinal Data

IV. Interval & Ratio Data, Histograms & Frequency Distributions

V. Cumulative Frequency Distributions & Percentile Ranks

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Variables

Variable: A characteristic that takes on multiple values. I.e.,varies from one person or thing to another.

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Variables

Cause and Effect

• The Independent Variable

• The Dependent Variable

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Distributions

• The distribution of population data is called the population distribution or the distribution of the variable.

• The distribution of sample data is called a sample distribution.

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Variables

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VariablesKinds of Variables (any of which can be an independent

or dependent variable)

• Qualitative variable: A nonnumerically valued variable.

• Quantitative variable: A numerically valued variable.

• Discrete Variable: A quantitative variable whose possible values form a finite (or countably infinite) set of numbers.

• Continuous variable: A quantitative variable whose possible values form some interval of numbers.

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Quantitative Variables

• Discrete data: Data obtained by observing values of a discrete variable.

• Continuous data: Data obtained by observing values of a continuous variable.

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The Four Scales

• Nominal: Categories• Ordinal: Sequence• Interval: Mathematical Scale w/o a true zero• Ratio: Mathematical Scale with a true zero

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The Four Scales

• Nominal: Classes or Categories. Also called a Categorical scale. E.g., Catholic, Methodist, Jewish, Hindu, Buddhist, …

Qualitative Data

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The Four Scales

• Ordinal: Sequential Categories. e.g., 1st, 2nd, 3rd, … with no indication of the distance between classes

Discrete Data

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The Four Scales

• Interval: Data where equal spacing in the variable corresponds to equal spacing in the scale. E.g., 1940s, 1950s, 1960s… : or SAT Scores.

Discrete or Continuous

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The Four Scales

• Ratio: An interval scale with a mathematically meaningful zero. e.g., latencies of 1252 ms, 1856 ms, ….: mg of Prozac

Discrete or Continuous

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The Four Scales

Nominal: No mathematical operations

Ordinal: <, >, =

Interval: +, -, and ordinal operations

Ratio: , , and interval operations

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Nominal Variables

• Classes: Categories for grouping data. • Frequency: The number of observations that

fall in a class.• Frequency distribution: A listing of all

classes along with their frequencies.• Relative frequency: The ratio of the

frequency of a class to the total number of observations.

• Relative-frequency distribution: A listing of all classes along with their relative frequencies.

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Frequencies of Nominal Variables

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Sample Pie Charts and Bar Charts of Nominal Data

44%

23%

33%

DemocraticRepublicanOther

0

5

10

15

20

25

30

35

40

45

50

Republican Democratic Other

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Frequency Bar Charts

• Frequency bar chart: A graph that displays the independent variable on the horizontal axis -- categories -- and the frequencies -- dependent variable -- on the vertical axis. The frequency is represented by a vertical bar whose height is equal to the frequency of cases that fall within a given class of the I.V.

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Frequency Charts of Nominal Data

90

180

130

Democratic

RepublicanOther

0

20

40

60

80

100

120

140

160

180

200

Republican Democratic Other

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Relative Frequency Bar Charts• Relative-frequency bar chart: A graph that displays

the I.V. on the horizontal axis -- categories -- and the relative frequencies -- D.V. -- on the vertical axis. The relative frequency of each class is represented by a vertical bar whose height is equal to the relative frequency of the class.

• The difference between this and a frequency bar chart is that the proportion or percentage (always between zero and one) is listed instead of the numbers that fall into a given class.

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Relative Frequency Charts of Nominal Data

44%

23%

33%


05

101520253035404550

%

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Probability Distribution and Probability Bar Chart

Frequency Distributions and Charts for a whole population

Probability distribution: A listing of the possible values and corresponding probabilities of a discrete random variable; or a formula for the probabilities.

Probability bar chart: A graph of the probability distribution that displays the possible values of a discrete random variable on the horizontal axis and the probabilities of those values on the vertical axis. The probability of each value is represented by a vertical bar whose height is equal to the probability.

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Probability Charts of Nominal Data

44%

23%

33%


05

101520253035404550

Repub

lican

Democ

ratic

Other

%

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Bar Chart

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The Bar Graph: Nominal Data

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Sum of the Probabilities of a Discrete Random Variable

For any discrete random variable, X, the sum of the probabilities of its possible values equals 1; in symbols, we have P(X = x) = 1.

For example Republicans: 32.5%, Democrats 45.0%, Other 22.5%

0.325 + 0.450 + 0.225 = 1.00 or 100%

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Ordinal Variables

Note that “Rank” is the ordinal variable. “Mortality” is a ratio variable but can easily be downgraded to an ordinal variable with a loss of information

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Distributions and Charts for Ordinal Data

• Frequency distributions, relative frequency distribution, and probability distributions are done exactly as they were for Nominal Data

• Bar charts are used.

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Distribution of Education Level

Level P(x)

Elementary 0.03

High School 0.45

Associates 0.12

Bachelors 0.28

Masters 0.10

Doctorate 0.020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Elemen

tary

High

Schoo

l

Assoc

iate

s

Bachelo

rs

Mas

ters

Docto

rate

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Interval and Ratio Data

• Frequency: The number of observations that fall in a class.

• Frequency distribution: A listing of all classes along with their frequencies.

• Relative frequency: The ratio of the frequency of a class to the total number of observations.

• Relative-frequency distribution: A listing of all classes along with their relative frequencies.

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Histograms

• Frequency histogram: A graph that displays the independent variable on the horizontal axis and the frequencies -- dependent variable -- on the vertical axis. The frequency is represented by a vertical bar whose height is equal to the frequency of cases that fall within a given range of the I.V.

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Interval and Ratio Variables

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Years of Education

Avg.Income (in thousands)

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Enrollment in Milwaukee Public Elementary Schools

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Relative Frequency distribution of Enrollments in MPS

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Probability distribution of a randomly selected elementary-school student

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Probability distribution of the age of a randomly selected student

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Probability Histogram

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

19 20 21 22 23 24 25 26 27

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Another Example

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Frequency vs. Relative Frequency

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Frequency vs. Relative Frequency

This is also the Probability Distribution

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More Examples:Frequency Histogram

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More Examples: Grouped Frequency Histogram

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Grouped Frequency Histogram

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Proportions and Frequency

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Frequency Groupings

9 intervals with each interval 5 points wide. The frequency column (f) lists the number of individuals with scores in each of the class intervals.

Groupings: There had

to be a catch

• What to do with the in-betweens?

• Only a concern for continuous variables

• Real Limits -- those in the “14” bar are really from 13.5 to 14.5

• Upper Real Limits & Lower Real Limits: For the case of whole numbers, simply add 0.5 to the high score and subtract 0.5 from the lowest observed score (these observed scores are the “apparent limits”)

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Understanding Real Limits

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Real Limits & Apparent Limits

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Frequency & Cumulative FrequencyI.Q. Range Real Limits Frequency Cuml. Freq.

< 52 0 – 52.5 1 1

52-67 52.5-67.5 4 5

68-84 67.5-84.5 11 16

85-100 84.5-100.5 34 50

101-116 100.5-116.5 34 84

117-132 116.5-132.5 11 95

133-148 132.5-148.5 4 99

>148 148.5 + 1 100

Frequency (Normal Distribution)

0

5

10

15

20

25

30

35

40

>52 52-68 68-84 84-100 100-116 116-132 132-148 >148

Cumulative Frequency (Ogive)

0

10

20

30

40

50

60

70

80

90

100

>52 52-68 68-84 84-100 100-116 116-132 132-148 >148

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Computing Percentile RanksPounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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Computing Percentile Ranks

• Remember that each value has real limits

• What is the 90th %ile?

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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• What is the 90th %ile? 2.5 because at or below “2” are 90% of the scores, but “2” includes all from 1.5 to 2.5

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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• What is the 20th %ile? 0.5

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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• What about the in-betweens?


• What %ile corresponds to 2 lbs?

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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Linear Interpolation

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And Percentiles


Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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And Percentiles


Where’s the 80th %ile? 17.5/27.5 = 0.63. The interval is 1.0 lb, so 1.5 + 1(0.63) = 2.13

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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And Percentiles




Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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And Percentiles




2 lbs. Is halfway into the interval (0.5). So its halfway between 62.5 - 90.0. Since 27.5% of the scores are in this interval we need to go up 0.5(27.5%) = 13.75%. 62.5% + 13.75% = 76.25%

Pounds x

Real Limits

Freq

f

RelativeFreq.

Cuml.Freq.

%ile

0 0-0.5 8 0.200 0.200 20.0

1 0.5-1.5 17 0.425 0.625 62.5

2 1.5-2.5 11 0.275 0.900 90.0

3 2.5-3.5 3 0.075 0.975 97.5

4 3.5-4.5 1 0.025 1.000 100

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The Stem & Leaf Diagram

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Stem & Leaf Plots

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Comparison of Frequency Histogram vs.

Stem & Leaf Diagram

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The Blocked Frequency Histogram

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The Frequency Distribution Polygon –or– Line Graph

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Grouped Frequency Polygon

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The Normal Distribution

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Variants on the Normal Distribution

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Comparing Two Distributions

Number of Sentences recalled from each category

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Comparing Distributions

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Distributions

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Variables and Distributions

In Class Exercise

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The Math You’ll Need To Know

Calculate:

ΣX =

ΣX2 =

(ΣX)2 =

X

1

2

0

4

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Calculate:

ΣX = 7

ΣX2 =

(ΣX)2 =

X

1

2

0

4

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Calculate:

ΣX = 7

ΣX2 = 21

(ΣX)2 =

X

1

2

0

4

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Calculate:

ΣX = 7

ΣX2 = 21

(ΣX)2 = 49

X

1

2

0

4

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Calculate:ΣX = ΣY = ΣX ΣY = ΣXY =ΣX2 = (ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = ΣX ΣY = ΣXY =ΣX2 = (ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = ΣXY =ΣX2 = (ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY =ΣX2 = (ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY = -2ΣX2 = (ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY = -2ΣX2 = 14(ΣX)2 = ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY = -2ΣX2 = 14(ΣX)2 = 36ΣY2 = (ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY = -2ΣX2 = 14(ΣX)2 = 36ΣY2 = 30(ΣY)2 =

X Y

1 3

3 1

0 -2

2 -4

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Calculate:ΣX = 6ΣY = -2ΣX ΣY = -12ΣXY = -2ΣX2 = 14(ΣX)2 = 36ΣY2 = 30(ΣY)2 = 4

X Y

1 3

3 1

0 -2

2 -4

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The Mean

Σx/n = M

where n = sample size

X

1

4

8

3

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Calculate:

Σ(x-M) =

Σ(x-M)2 =

Σ(x2 –M2) =

X M = 4

1

4

8

3

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Calculate:

Σ(x-M) = 0

Σ(x-M)2 =

Σ(x2 –M2) =

X M = 4

1

4

8

3

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Calculate:

Σ(x-M) = 0

Σ(x-M)2 = 26

Σ(x2 –M2) =

X M = 4

1

4

8

3

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Calculate:

Σ(x-M) = 0

Σ(x-M)2 = 26

Σ(x2 –M2) = 26

X M = 4

1

4

8

3

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Calculate:

sp = 13

n1= 8

n2= 102

2

1

2

n

s

n

s pp

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Calculate:

sp = 13

n1= 8

n2= 10

17.6025.38

9.16125.21

10

169

8

169

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Calculate:

sp = 13

n1= 8

n2= 1017.62

2

1

2

n

s

n

s pp

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What Type of Data?

Years Spent in the Military

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What Type of Data?

Military Rank:

Lieutenant

Captain

Major

Lt. Colonel

Colonel

General

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What Type of Data?

Branch of Service:

Army

Air Force

Navy

Marine Corps

Coast Guard

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What Type of Data?

Time taken to complete a 30 mile bicycle race

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What Type of Data?

Finishing place in a 30 mile bicycle race


Frequency Dist. & Percentile

Raw Scores:

15, 18, 21, 23, 27, 33, 33, 35, 36, 36, 39, 41

44, 47, 49, 50



X f

10-19 2

20-29 3

30-39 6

40-49 4

50-59 1



Compute the 52%ile X f

10-19 2

20-29 3

30-39 6

40-49 4

50-59 1



Compute the 52%ile X f Cum f

10-19 2 2

20-29 3 5

30-39 6 11

40-49 4 15

50-59 1 16



Compute the 52%ile• The 52%ile is

somewhere between 30-39.

X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0



Compute the 52%ile• The 52%ile is


• That interval is from 0.3125 – 0.6875

X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0


Frequency Dist. & PercentileCompute the 52%ile• The 52%ile is



• That interval is 0.375 wide

X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0


Frequency Dist. & PercentileCompute the 52%ile• The 52%ile is




• To get from 0.3125 to 0.52 we go 0.2075 into the interval

X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0


Frequency Dist. & Percentile• That interval is from 0.3125 – 0.6875


• To get from 0.3125 to 0.52 we go 0.2075 into the interval

• That’s 0.553 of the way into the interval (0.2075/0.375)

X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0


Frequency Dist. & Percentile• That’s 0.553 of the

way into the interval (0.2075/0.375)

• The real limits are from 19.5 to 29.5 (a range of 10)

• 52%ile is 29.5 + 5.53 = 35.03

This Process is called


X f Cum f

10-19 2 2 0.125

20-29 3 5 0.3125

30-39 6 11 0.6875

40-49 4 15 0.9375

50-59 1 16 1.0

Documents

Distributions of Variables