Sample DistributionsSample Distributions

ReviewReview

Parameter vs. StatisticParameter vs. Statistic Population and SamplePopulation and Sample ConstructConstruct Variables (IV and DV)Variables (IV and DV) OperationalizationOperationalization Levels of measurementLevels of measurement Summation notationSummation notation

• What is the value of What is the value of (X+2)(X+2)22 where X{1, 3, 4, 5} where X{1, 3, 4, 5}

By the end of today we will have three ways to By the end of today we will have three ways to describe a distributiondescribe a distribution ShapeShape

• SkewSkew• KurtosisKurtosis

Central TendencyCentral Tendency• MeanMean• MedianMedian• ModeMode

VariabilityVariability• Standard DeviationStandard Deviation• VarianceVariance

What is the job of descriptive What is the job of descriptive statistics?statistics?

To simplify and organize dataTo simplify and organize data

How Best to Accomplish This?How Best to Accomplish This?

Among other techniques, a frequency Among other techniques, a frequency distribution is helpfuldistribution is helpful

A A frequency distributionfrequency distribution is an organized is an organized tabulation of the number of individuals tabulation of the number of individuals located in each category (from a sample) located in each category (from a sample) on the scale of measurement.on the scale of measurement.

Example of a Frequency Example of a Frequency Distribution TableDistribution Table

X f

10

9

8

7

6

5

4

2

5

7

3

2

0

1

What other information can this What other information can this type of table give us?type of table give us?

It can help us find It can help us find XX It gives an easy way to see proportions and It gives an easy way to see proportions and

percentagespercentages Definition: A Definition: A proportion proportion measures the fraction of measures the fraction of

the total group that is associated with each score.the total group that is associated with each score.• proportion = p = f/Nproportion = p = f/N

A percentage gives essentially the same information A percentage gives essentially the same information just in a different formjust in a different form

• Percentage = proportion(100) = p(100) = (f/N)(100) then Percentage = proportion(100) = p(100) = (f/N)(100) then just add the percent signjust add the percent sign

ExampleExample

X

10

9

8

7

6

5

4

2

5

7

3

2

0

1

f

20

p = f/N

2/20 = 0.10

5/20 = 0.25

7/20 = 0.35

3/20 = 0.15

2/20 = 0.10

0/20 = 0.00

1/20 = 0.05

% = p(100)

10%

25%

35%

15%

10%

0%

5%

1.00 100%

What might be another way we would What might be another way we would want to look at this information?want to look at this information?

A A frequency distribution graphfrequency distribution graph is a is a picture of the information available in a picture of the information available in a frequency distribution tablefrequency distribution table

What are the basic parts of any graph?What are the basic parts of any graph? X-axisX-axis Y-axisY-axis ScaleScale OriginOrigin

The Basic Types of Graphs (Data The Basic Types of Graphs (Data From a Sample)From a Sample)

HistogramsHistograms A A histogramhistogram lists the numerical scores lists the numerical scores

(categories of measurement) along the x-(categories of measurement) along the x-axis and creates bars that extend up to the axis and creates bars that extend up to the height on the y-axis which represents the height on the y-axis which represents the frequency for that categoryfrequency for that category

Example of a HistogramExample of a Histogram

Basic Graphs ContinuedBasic Graphs Continued

Bar GraphsBar Graphs A A bar graphbar graph is essentially the same as a is essentially the same as a

histogram, except that spaces are left histogram, except that spaces are left between adjacent bars. This space between adjacent bars. This space emphasizes that the scale is either nominal, emphasizes that the scale is either nominal, or ordinal not interval or ratio. or ordinal not interval or ratio.

Example of a Bar GraphExample of a Bar Graph

Describing Individual Points or Sub-Describing Individual Points or Sub-Sets of ScoresSets of Scores

We have used the distributions to describe We have used the distributions to describe entire sets of scores. How might we use entire sets of scores. How might we use them to describe individual scores?them to describe individual scores? A A rankrank or or percentile rankpercentile rank is defined as the is defined as the

percentage of individuals in the distribution percentage of individuals in the distribution with scores at or below a particular valuewith scores at or below a particular value

A A percentilepercentile is used to identify an individual is used to identify an individual score’s percentile rankscore’s percentile rank

Example of Frequency and Cumulative Example of Frequency and Cumulative PercentagePercentage

X f cf c%

54321

15842

20191462

100%95%70%30%10%

Highest Year School Completed, Father

17 1.1 1.6 1.6

7 .5 .7 2.2

31 2.0 2.9 5.1

22 1.5 2.1 7.2

22 1.5 2.1 9.3

61 4.0 5.7 15.0

27 1.8 2.5 17.5

165 10.9 15.4 32.9

39 2.6 3.6 36.6

49 3.2 4.6 41.2

38 2.5 3.6 44.7

300 19.8 28.1 72.8

28 1.8 2.6 75.4

77 5.1 7.2 82.6

12 .8 1.1 83.7

103 6.8 9.6 93.4

12 .8 1.1 94.5

24 1.6 2.2 96.7

13 .9 1.2 97.9

22 1.5 2.1 100.0

1069 70.5 100.0

205 13.5

211 13.9

32 2.1

448 29.5

1517 100.0

0

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Total

Valid

NAP

DK

NA

Total

Missing

Total

Frequency Percent Valid PercentCumulative

Percent