Variable

An item of data Examples:

– gender– test scores– weight

Value varies from one observation to another

Types/Classifications of Variables

QualitativeQuantitative

– Discrete– Continuous

Qualitative Data

Describes the quality Non-numerical format

Counts Cannot order or measure Examples

– gender– marital status– geographical region – job title….

Categorical data

Non-overlapping categories or characteristics

Examples:– Completes/Incompletes

– Professions

– Gender

Quantitative Data

Frequencies

Measurements

Discrete

Measurements are integers

Examples:

– number of employees of a company

– number of incorrect answers on a test

– number of participants in a program…

Continuous

Measurements can take on any value - usually within some range

Examples:

– Age

– Income

Arithmetic operations such as differences and averages make sense.

Qualitatiave or Quantitative?Discrete or Continuous?

Score on a placement examPreferred restaurantDollar amount of a loanHeight SalaryLength of time to complete a taskNumber of applicantsEthnic origin

Treatment as Ranks

Natural order Not strictly measuredExamples:

– Age group – Likert Scale data

Distinction between adjacent points on the scale is not necessarily the same

AnalysisQualitative Data

Frequency tables

Modes - most frequently occurring

Graphs: Bar Charts and Pie Charts

AnalysisQuantitative Data

Any form

Create groups or categories and generate

frequency tables

All descriptive statistics

Effective Graphs: Quantitative Data

Histograms

Stem-and-Leaf plots

Dot Plots

Box plots

XY Scatter Plots (2 variables).

Examples of Graphs

Pie Chart

Performance Appraisals

10%

14%

33%

38% More Difficult

Difficult

Same

Much Easier

Easier

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

EastWestNorth

Histogram

Histogram

0

2

4

6

8

10

12

49 59 69 79 89 99

Score

Fre

qu

ency

Boxplot

20 30 40 50 60 70 80 90 100 110

C1

Boxplot of C1

Stem and Leaf PlotStem and Leaf PlotWeight of Meat

7 58 38 79999 239 66789

1010 68811 224411 78812 412 81313 814 1

Analyze Ranked Data

Frequency tables Mode, Median, Quartiles Graphs:

– Bar Charts– Dot Plots, Pie Charts– Line Charts (2 variables)

Data Example

Suggest some ways you could analyze these items.

Score on a placement exam Preferred restaurant Dollar amount of a loan Height Salary Length of time to complete a task Number of applicants Ethnic origin

Tables and Graphs

Note Excel will create any graph that you specify

Consider the type of data before selecting your graph.

Frequency Table/Frequency Distribution

Summarize data:categoricalnominal Continuous data - the data set has been

divided into meaningful groups

Frequency Distribution

Count the number of observations that fall into each category.

Frequency: the number associated with each category

Relative Frequency Distribution

Proportion of observations falling in a given category

Report relative frequencies or percentages

ExampleFrequency Distribution

No. of Defective Parts n

0 & < 2 02 4 44 6 56 8 128 10 14

10 12 912 14 1014 16 6

GraphsCategorical/Qualitative Data

Pie Charts

Circle - divided

proportionately

Segment - percentage of

the whole that falls into

each category Native Language

English55%

Viet Namese

15%

Swedish5%

Spanish25%

Bar Charts

Bar charts - % in various

categories Vertical scale -

frequencies, relative

frequencies Horizontal scale -

categories Allows comparisons

Ave ra ge Units Sold (pe r pe rs on) by Produc t

0

5

10

15

20

B41 BA42 B41F C21 Other

Product

Ave

rag

e S

old

/Pe

rso

nBefore Training

A f ter Training

Constructing Bar Charts

All boxes should have the same width Gaps between the boxes - no connection

between Any order. Use to represent two categorical variables

simultaneously

Graphs: MeasuredContinues Quantitative Data

Histograms Stem and LeafBox plotsLine GraphsXY Scatter Charts (2 variables)

Histograms

Frequency distributions of continuous variables

Drawn without gaps between the bars

Grade Distribution

02468

1012

59 69 79 89 99Grade

Fre

qu

ency

Constructing Histograms

Non-overlapping intervals

Intervals - generally the same length

Number of values in each interval -class frequency

Relative frequencies o

Grade Distribution

02468

1012

59 69 79 89 99Grade

Fre

qu

ency

XY Scatter Chart

Two variables Variables: quantitative

and continuous. Plot pairs - rectangular

coordinate system Examine the relationship

between two variables

Abscent by Age

0

5

10

15

20

0 10 20 30 40 50 60 70

Age

Days

Abs

ent

Line Chart

Similar to the scatter chart

Values of the independent variable (shown on the horizontal axis) can be ranked values (i.e.. they do not have to be continuous variables).

1997 Monthly Sales

125130135140145150155160165170

Jan

Feb

Mar

Apr

May

June

Month

Sale

s (x

$10,

000)

Basic Principles for Constructing All Plots

Data should stand out clearly from background

The information should be clearly labeled – title– axes, bars, pie segments, etc. - include units

that are needed to interpret data– scale including starting points.

Principles cont.

Source No clutter Minimize information or data on one graph.Try several approaches

Describing Data

Shape of the Distribution– Symmetry– Skewness– Modality: most frequently occurring value – Unimodal or bimodal or uniform

Right Skewed Left Skewed

Histogram

0

2

4

6

8

10

12

59 69 79 89 99

Grade

Fre

qu

ency

Histogram

0

2

4

6

8

10

12

59 69 79 89 99

Grade

Fre

qu

ency

Histogram

0

2

4

6

8

10

12

59 69 79 89 99

Grade

Fre

qu

ency

Symmetrical

Describing Data

CentralitySpreadExtreme values

Measures of Centrality

MeanMedianMode

Mean

Most common measure

Extremely large values in a data set will

increase the value of the mean

Extremely low values will decrease it.

Calculating the Mean

T1 T2 T3

85 85 85

90 90 90

75 35 75

90 90 110

340 300 360 Sum

85 75 90 Mean

Median

Central point .Half of the data has a value than the medianHalf of the data has a higher value than the

medianNot affected by extremely large or small

values

Find the Median

85 90 75 92 95 Data

75 85 90 92 95 Sorted Data

Median is 90.

Find the Median

95 90 92 85 Data

85 90 92 95 Sorted Data

Median:

(90 + 92)/2 = 91

Measures of Spread

Range

Subtract the smallest value from the largest

Report the smallest and largest values.

85 90 75 92 95 Scores

Range: 75 to 95

or 20

Variance/Standard Deviation

Average variation of the data values from

the mean of the valuesVariance.

The Empirical Rule

Symmetrical DataAt least:

68% of the data values are within one standard deviation of the mean

90% of the data values are within two standard deviation of the mean

99% of the data values are within three standard deviations of the mean

Tchybychef’s Inequality

Skewed DataAt least:

75% of the data values are within two standard deviation of the mean.

90% of the data values are within one standard deviation of the mean.

Measures of Relative Standing

Percentiles

Quartiles

Quartiles

The lower quartile is the same as the 25th percentile.– 25% of the scores are lower and– 75% of the scores are higher than the lower

quartile.

The upper quartile is the same as the 75th percentile.– 75% of the scores are lower and

Correlation

Describes the strength of the relationship

between two (or more) variables

Pearson Product-moment Correlation

Coefficient - assumes continuous

quantitative data

Relationship between Variables

Positive Negative No relationship.

Interpreting Correlation Coefficients.

0.20 to 0.35- show a slight relationship(little value in practical prediction situations)

0.50 - crude group prediction

(Correlations this low do not suggest a good relationship)

0.65 to 0.85 - group predictions that are good

Over 0.85 - a close relationship between the two

variables.

Even a high correlation coefficient does not establish a

cause and effect relationship!!!!!

Coefficient of Determination

Square root of the correlation coefficient

Gives the percent of variation in the

dependent variable that is ‘explained’ by

the independent variable.

Look at an XY scatter plot

Least Square Line

Describe the relationship between the two

variables

Make predictions of the dependent variable

from the independent variable

Positive Relationship

0

1

2

3

4

5

6

7

8

0 2 4 6

X

Y

r will be a positive number.

Negative Relationship

0

1

2

3

4

5

6

7

8

0 2 4 6

X

Y

r will be a negative number.