Introduction to Descriptive StatisticsIntroduction to Descriptive Statistics

Objectives:Objectives:1.1. Explain the general role of statistics in Explain the general role of statistics in

assessment & evaluationassessment & evaluation

2.2. Explain three methods for describing a data set: Explain three methods for describing a data set: shape, center, and spreadshape, center, and spread

3.3. Explain the relationship between the standard Explain the relationship between the standard deviation and the normal curvedeviation and the normal curve

““Data have a story to tell. Statistical analysis is Data have a story to tell. Statistical analysis is detective work in which we apply our intelligence detective work in which we apply our intelligence and our tools to discover parts of that story.”and our tools to discover parts of that story.”

-Hamilton (1990)-Hamilton (1990)

Levels of MeasurementLevels of Measurement

NominalNominalOrdinalOrdinal IntervalIntervalRatioRatioDetermining what statistics are Determining what statistics are

appropriateappropriate

NominalNominal

Naming things.Naming things.

Creating groups that are Creating groups that are qualitatively different or qualitatively different or unique…unique…

But not necessarily But not necessarily quantitatively different.quantitatively different.

NominalNominal

Placing individuals or Placing individuals or objects into categories.objects into categories.

Making mutually Making mutually excusive categories.excusive categories.

Numbers assigned to Numbers assigned to categories are arbitrary.categories are arbitrary.

NominalNominal

Sample variables:Sample variables:– GenderGender– RaceRace– EthnicityEthnicity– Geographic locationGeographic location– Hair or eye colorHair or eye color

OrdinalOrdinal

Rank ordering things.Rank ordering things.

Creating groups or Creating groups or categories when only rank categories when only rank order is known.order is known.

Numbers imply order but not Numbers imply order but not exact quantity of anything.exact quantity of anything.

OrdinalOrdinal

The difference between The difference between individuals with adjacent individuals with adjacent ranks, on relevant ranks, on relevant quantitative variables, is quantitative variables, is not necessarily the same not necessarily the same across the distribution.across the distribution.

OrdinalOrdinal

Sample variables:Sample variables:– Class RankClass Rank– Place of finish in a race (1Place of finish in a race (1stst, 2, 2ndnd, ,

etc.)etc.)– Judges ratingsJudges ratings– Responses to Likert scale items Responses to Likert scale items

(for example – SD, D, N, A, SA) (for example – SD, D, N, A, SA)

IntervalInterval

Orders observations Orders observations according to the quantity of according to the quantity of some attribute.some attribute.

Arbitrary origin.Arbitrary origin. Equal intervals.Equal intervals. Equal differences expressed Equal differences expressed

as equal distances.as equal distances.

IntervalInterval

Sample variables:Sample variables:– Test ScoresTest Scores

•SATSAT•GREGRE• IQ testsIQ tests

– Temperature Temperature •CelsiusCelsius•FahrenheitFahrenheit

RatioRatio

Quantitative measurement.Quantitative measurement. Equal intervals.Equal intervals. True zero point.True zero point. Ratios between values are Ratios between values are

useful.useful.

RatioRatio

Sample variables:Sample variables:– Financial variablesFinancial variables– Finish times in a raceFinish times in a race– Number of units soldNumber of units sold– Test scores scaled as percent Test scores scaled as percent

correct or number correctcorrect or number correct

Levels of Measurement Levels of Measurement ReviewReview What level of measurement?What level of measurement?

– Today is a fall day.Today is a fall day.– Today is the third hottest day of Today is the third hottest day of

the month.the month.– The high today was 70The high today was 70o o

Fahrenheit.Fahrenheit.– The high today was 20The high today was 20oo Celsius. Celsius.– The high today was 294The high today was 294oo Kelvin. Kelvin.

Levels of Measurement Levels of Measurement ReviewReview What level of measurement?What level of measurement?

– Student #1256 is:Student #1256 is:– a malea male– from Lawrenceville, GA.from Lawrenceville, GA.– He came in third place in the race He came in third place in the race

today.today.– He scored 550 on the SAT verbal He scored 550 on the SAT verbal

section.section.– He has turned in 8 out of the 10 He has turned in 8 out of the 10

homework assignments.homework assignments.

Levels of Measurement Levels of Measurement ReviewReview What level of measurement?What level of measurement?

– Student #3654 is:Student #3654 is:– in the third reading group.in the third reading group.– Nominal?Nominal?– Ordinal?Ordinal?– Interval?Interval?– Ratio?Ratio?

Descriptive StatisticsDescriptive Statistics

Used to describe the basic features of a Used to describe the basic features of a batch of data. Uses graphical displays batch of data. Uses graphical displays and descriptive quantitative indicators. and descriptive quantitative indicators.

The purpose of descriptive statistics is to The purpose of descriptive statistics is to organize and summarize data so that organize and summarize data so that the data is more readily comprehended. the data is more readily comprehended. That is, descriptive statistics describes That is, descriptive statistics describes distributions with numbers.distributions with numbers.

Five Descriptive QuestionsFive Descriptive Questions

What is the middle of the set of What is the middle of the set of scores?scores?

How spread out are the scores?How spread out are the scores? Where do specific scores fall in the Where do specific scores fall in the

distribution of scores?distribution of scores? What is the shape of the distribution?What is the shape of the distribution? How do different variables relate to How do different variables relate to

each other?each other?

Five Descriptive QuestionsFive Descriptive Questions

MiddleMiddle SpreadSpread Rank or Relative PositionRank or Relative Position ShapeShape CorrelationCorrelation

MiddleMiddle

MeanMeanMedianMedianModeMode

Examples of these measuresExamples of these measures

Mean of: 2, 3, 6, 7, 3, 5, 10Mean of: 2, 3, 6, 7, 3, 5, 10(2 + 3 + 6 + 7 + 3 + 5 + 10)/ 7 = 36/ 7 = 5.14(2 + 3 + 6 + 7 + 3 + 5 + 10)/ 7 = 36/ 7 = 5.14

Mode of: 2, 3, 6, 7, 3, 5, 10 is 3Mode of: 2, 3, 6, 7, 3, 5, 10 is 3

Median of: 2, 3, 6, 7, 3, 5, 10Median of: 2, 3, 6, 7, 3, 5, 10

First data is ordered: 2, 3, 3, 5, 6, 7, 10. Middle First data is ordered: 2, 3, 3, 5, 6, 7, 10. Middle value is 5 therefore that is the median. value is 5 therefore that is the median.

Some Important PointsSome Important Points

Mode is the only descriptive measure Mode is the only descriptive measure used for nominal dataused for nominal data

Median is unaffected by extreme values, Median is unaffected by extreme values, it is resistant to extreme observations.it is resistant to extreme observations.

Mean or Average is affected by extremely Mean or Average is affected by extremely small or large values. We say that it is small or large values. We say that it is sensitive or sensitive or nonresistantnonresistant to the influence to the influence of extreme observations. The mean is the of extreme observations. The mean is the balance point of the distribution. balance point of the distribution.

In symmetric distributions the mean and In symmetric distributions the mean and median are close together.median are close together.

More important pointsMore important points

In skewed data the mean is pulled to the In skewed data the mean is pulled to the tail of the distribution.tail of the distribution.

Median is not necessarily preferred over Median is not necessarily preferred over the mean even if it is resistant. However if the mean even if it is resistant. However if data is known to be strongly skewed then data is known to be strongly skewed then the median is preferable. the median is preferable.

Finally, the average is usually the Finally, the average is usually the measurement of central tendency of measurement of central tendency of choice because it is stable during choice because it is stable during sampling. sampling.

SpreadSpread

Standard DeviationStandard Deviation VarianceVariance RangeRange IQRIQR

Large S

Average or Normal S Small S

X = Mean

S = Standard Deviation

X = 50 (S = 20)

X = 50 (S = 10) X = 50 (S = 5)

How do measures of variability differ How do measures of variability differ when distributions are spread out?when distributions are spread out?

Describing Data: Center & Describing Data: Center & SpreadSpread

Rank or Relative PositionRank or Relative Position

Five number summaryFive number summary Min, 25Min, 25thth, 50, 50thth, 75, 75thth, Max, Max Identifying specific values that Identifying specific values that

have interpretive meaninghave interpretive meaning Identifying where they fall in Identifying where they fall in

the set of scoresthe set of scoresBox plotsBox plotsOutliersOutliers

ShapeShape

Positive SkewnessPositive Skewness Negative SkewnessNegative Skewness NormalityNormality HistogramsHistograms

Shape - NormalityShape - Normality

Scanning

50.0

47.5

45.0

42.5

40.0

37.5

35.0

32.5

30.0

27.5

25.0

100

80

60

40

20

0

Std. Dev = 4.84

Mean = 38.0

N = 344.00

344N =

Scanning

60

50

40

30

20

184719125

23312240

Shape- Positive SkewnessShape- Positive Skewness

Total for IIP

50

40

30

20

10

0

Std. Dev = .56

Mean = 2.10

N = 344.00

344N =

Total for IIP

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

.5

1102733625610710429

Shape – Negative Shape – Negative SkewnessSkewness

PREACT

40

30

20

10

0

Std. Dev = .42

Mean = 3.32

N = 154.00

154N =

PREACT

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

13079

91119

11164118

When a distribution of When a distribution of data resembles a normal data resembles a normal distribution (or normal distribution (or normal curve): curve):

68% of the data lies 68% of the data lies within + or – 1 standard within + or – 1 standard deviationdeviation

95% of the data lie 95% of the data lie within + or – 2 standard within + or – 2 standard deviationsdeviations

99.7% of the data lie 99.7% of the data lie within + or – 3 standard within + or – 3 standard deviations from the deviations from the meanmean

99.7%

95%

68%

Relating the Relating the Standard DeviationStandard Deviation (S) to the (S) to the normal distribution.normal distribution.““68-95-99.7% Rule”68-95-99.7% Rule”

Describing Data: Center & Describing Data: Center & SpreadSpread

OutliersOutliers

344N =

BDI Total

50

40

30

20

10

0

-10

107321196125276851132930018336

22061

71

82120

OutliersOutliers

BDI Total

40.0

35.0

30.0

25.0

20.0

15.0

10.0

5.0

0.0

BDI TotalF

requ

ency

140

120

100

80

60

40

20

0

Std. Dev = 7.10

Mean = 7.1

N = 344.00

OutliersOutliers

Statistics

BDI Total344

0

7.12

5.00

0

7.101

50.426

0

40

2.00

5.00

10.00

Valid

Missing

N

Mean

Median

Mode

Std. Deviation

Variance

Minimum

Maximum

25

50

75

Percentiles