Statistics An Introduction

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StatisticsStatisticsAn IntroductionAn Introduction

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Learning ObjectivesLearning Objectives

1.1. Define StatisticsDefine Statistics2.2. Describe the Uses of StatisticsDescribe the Uses of Statistics3.3. Distinguish Descriptive & Inferential Distinguish Descriptive & Inferential

StatisticsStatistics4.4. Define Population, Sample, Parameter, Define Population, Sample, Parameter,

& Statistic& Statistic5.5. Identify data typesIdentify data types

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What is Statistics?What is Statistics?

The practice (science?) of data analysisThe practice (science?) of data analysisSummarizing data and drawing inferences Summarizing data and drawing inferences

about the larger population from which about the larger population from which it was drawnit was drawn

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Statistical MethodsStatistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

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Descriptive Descriptive StatisticsStatistics

1.1. InvolvesInvolves Collecting DataCollecting Data Presenting DataPresenting Data Characterizing Characterizing

DataData

2.2. PurposePurpose Describe DataDescribe Data X = 30.5 SX = 30.5 S22 = 113 = 113

00

2525

5050

Q1Q1 Q2Q2 Q3Q3 Q4Q4

$$

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Inferential StatisticsInferential Statistics

1.1. InvolvesInvolves EstimationEstimation Hypothesis Hypothesis

TestingTesting

2.2. PurposePurpose Make Decisions About Make Decisions About

Population Based on Population Based on Sample CharacteristicsSample Characteristics

Population?Population?

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Key TermsKey Terms

1.1. Population (Universe)Population (Universe) All Items of InterestAll Items of Interest

2.2. SampleSample Portion of PopulationPortion of Population

3.3. ParameterParameter Summary Measure about PopulationSummary Measure about Population

4.4. StatisticStatistic Summary Measure about SampleSummary Measure about Sample

• PP in in PPopulation opulation

& & PParameterarameter• SS in in SSample ample

& & SStatistictatistic

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Data TypesData Types

QuantitativeQuantitative DiscreteDiscrete ContinuousContinuous

QualitativeQualitative Nominal (categorical)Nominal (categorical) Ordinal (rank ordered categories)Ordinal (rank ordered categories)

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SamplingSampling

Representative sampleRepresentative sample Same characteristics as the populationSame characteristics as the population

Random sampleRandom sample Every subset of the population has an Every subset of the population has an

equal chance of being selectedequal chance of being selected

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ReviewReview

Descriptive vs. Inferential StatisticsDescriptive vs. Inferential StatisticsVocabularyVocabulary

PopulationPopulation (Random, representative) sample(Random, representative) sample ParameterParameter StatisticStatistic

Data typesData types

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Methods for Describing DataMethods for Describing Data

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Learning ObjectivesLearning Objectives

1.1. Describe Qualitative Data GraphicallyDescribe Qualitative Data Graphically2.2. Describe Numerical Data GraphicallyDescribe Numerical Data Graphically3.3. Create & Interpret Graphical DisplaysCreate & Interpret Graphical Displays4.4. Explain Numerical Data PropertiesExplain Numerical Data Properties5.5. Describe Summary MeasuresDescribe Summary Measures

6.6. Analyze Numerical Data Using Summary Analyze Numerical Data Using Summary Measures Measures

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Data PresentationData Presentation

SummaryTable

DotChart

PieChart

NumericalData

DataPresentation

BarChart

QualitativeData

Stem-&-LeafDisplay

FrequencyDistribution

Histogram

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Presenting Presenting Qualitative DataQualitative Data

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SummaryTable

DotChart

PieChart

NumericalData

DataPresentation

BarChart

QualitativeData

Stem-&-LeafDisplay


Histogram

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Student Student SpecializationsSpecializations

HCIIEMP

LISUndecided

0123456789

10

HCI IEMP LIS Undecided

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Calculus KnowledgeCalculus Knowledge

integrals | Freq. Percent Cum.integrals | Freq. Percent Cum.------------+-----------------------------------------------+----------------------------------- 1 | 3 13.04 13.041 | 3 13.04 13.04 2 | 1 4.35 17.392 | 1 4.35 17.39 3 | 11 47.83 65.223 | 11 47.83 65.22 4 | 6 26.09 91.304 | 6 26.09 91.30 5 | 2 8.70 100.005 | 2 8.70 100.00------------+-----------------------------------------------+----------------------------------- Total | 23 100.00Total | 23 100.00

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Presenting Presenting Numerical DataNumerical Data

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SummaryTable

DotChart

PieChart

NumericalData

DataPresentation

BarChart

QualitativeData

Stem-&-LeafDisplay


Histogram

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Student Age Student Age (Reported) Data(Reported) Data

Stem-and-leaf plot for ageStem-and-leaf plot for age

2* | 222334445557778992* | 22233444555777899 3* | 012573* | 01257 4* | 4* | 5* | 5* | 6* | 6* | 7* | 67* | 6

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HistogramHistogram

02

46

810

Freq

uenc

y

20 30 40 50 60 70age

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Starting Salaries (in Starting Salaries (in $K)$K)

3* | 83* | 8 4* | 0000254* | 000025 5* | 00005* | 0000 6* | 00000056* | 0000005 7* | 57* | 5 8* | 08* | 0

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Numerical Data Numerical Data PropertiesProperties

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Thinking ChallengeThinking Challenge

... employees cite low ... employees cite low pay -- most workers pay -- most workers earn only $20,000.earn only $20,000.... President claims ... President claims average pay is $70,000!average pay is $70,000!

$400,000$400,000

$70,000$70,000

$50,000$50,000

$30,000$30,000

$20,000$20,000

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Standard NotationStandard Notation

MeasureMeasure SampleSample PopulationPopulation

MeanMean xx

Stand. Dev.Stand. Dev. ss

VarianceVariance ss 22 22

SizeSize nn NN

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Numerical Data Numerical Data PropertiesProperties

Central Tendency Central Tendency (Location)(Location)

Variation Variation (Dispersion)(Dispersion)

ShapeShape

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Numerical DataNumerical DataProperties & Properties &

MeasuresMeasuresNumerical Data

Properties

MeanMean

MedianMedian

ModeMode

CentralTendency

RangeRange

VarianceVariance

Standard DeviationStandard Deviation

Variation

SkewSkew

Shape

Interquartile RangeInterquartile Range

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Central TendencyCentral Tendency

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Properties

MeanMean

MedianMedian

ModeMode

CentralTendency

RangeRange

VarianceVariance


Variation

SkewSkew

Shape


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What’s wrong with What’s wrong with this?this?

Measurements 1 4 2 9 8Measurements 1 4 2 9 8Middle measurement is 2, so that’s the Middle measurement is 2, so that’s the

medianmedian

XXXX

nnXX XX XX

nn

iiii nn

11 11 22

8.45/24

589241

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AgesAges

Mean = 29Mean = 29Median = 27Median = 27

2* | 222334445557778992* | 22233444555777899 3* | 012573* | 01257 4* | 4* | 5* | 5* | 6* | 6* | 7* | 67* | 6

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Summary of Summary of Central Tendency Central Tendency

Measures Measures MeasureMeasure EquationEquation DescriptionDescription

MeanMean XXii / / nn Balance PointBalance PointMedianMedian ((nn+1)+1) PositionPosition

22Middle ValueMiddle ValueWhen OrderedWhen Ordered

ModeMode nonenone Most FrequentMost Frequent

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ShapeShape

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Properties

MeanMean

MedianMedian

ModeMode

CentralTendency

RangeRange


VarianceVariance


Variation

SkewSkew

Shape

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ShapeShape

1.1. Describes How Data Are DistributedDescribes How Data Are Distributed

2.2. Measures of ShapeMeasures of Shape Skew = SymmetrySkew = Symmetry

Right-SkewedRight-SkewedLeft-SkewedLeft-Skewed SymmetricSymmetricMeanMean = = MedianMedian = = ModeModeMeanMean MedianMedian ModeMode ModeMode MedianMedian MeanMean

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VariationVariation

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Properties

MeanMean

MedianMedian

ModeMode

CentralTendency

RangeRange

VarianceVariance


Variation

SkewSkew

Shape


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QuartilesQuartiles

1.1. Measure of Measure of NoncentralNoncentral Tendency Tendency2.2. Split Ordered Data into 4 QuartersSplit Ordered Data into 4 Quarters

3.3. Position of i-th QuartilePosition of i-th Quartile

25%25% 25%25% 25%25% 25%25%

QQ11 QQ22 QQ33

PositioninPositioning Point ofg Point of QQ ii (n(nii

1)1)44

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AgesAges

RangeRangeQuartilesQuartiles

2* | 222334445557778992* | 22233444555777899 3* | 012573* | 01257 4* | 4* | 5* | 5* | 6* | 6* | 7* | 67* | 6

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Box Plots - Age and Box Plots - Age and SalarySalary

Quartiles: 24, 27, 30Quartiles: 24, 27, 30Inner fences: (15,39)Inner fences: (15,39)Outer fences: (6, 48)Outer fences: (6, 48)

Quartiles: 41K, 50K, 60KQuartiles: 41K, 50K, 60KInner fences: ??Inner fences: ??Outer fences: ??Outer fences: ??

2040

6080

40,000

50,000

60,000

70,000

80,000

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Variance & Variance & Standard DeviationStandard Deviation

1.1. Measures of DispersionMeasures of Dispersion

2.2. Most Common MeasuresMost Common Measures

3.3. Consider How Data Are DistributedConsider How Data Are Distributed

4.4. Show Variation About Mean (Show Variation About Mean (XX or or ))

44 66 88 1010 1212

XX = 8.3= 8.3

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Sample Variance Sample Variance FormulaFormula

nn - 1 in denominator! - 1 in denominator! (Use (Use NN if if PopulationPopulation Variance)Variance)SS

(X(X X)X)

nn

(X(X X)X) (X(X X)X) (X(X X)X)nn

iiii

nn

nn

22

22

11

1122

2222 22

11

11

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Equivalent FormulaEquivalent Formula

1

1

2

12

12

1

2

1

2222

2222

1

22

1

2

2

nxnx

nxnxnxx

nxnxxx

nxxxx

n

xxxx

n

xxs

ii

iiii

n

iii

n

ii

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Another Equivalent Another Equivalent FormulaFormula

1

1

1

22

22

222

nnxx

nnxnx

nxnxs

ii

ii

i

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Empirical RuleEmpirical Rule

If x has a “symmetric, mound-shaped” If x has a “symmetric, mound-shaped” distributiondistribution

Justification: Known properties of the “normal” Justification: Known properties of the “normal” distribution, to be studied later in the coursedistribution, to be studied later in the course

%3.03Pr

%52Pr%32Pr

i

i

i

xxx

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Preview of Preview of Statistical InferenceStatistical Inference

You observe one data pointYou observe one data pointMake hypothesis about mean and standard Make hypothesis about mean and standard

deviation from which it was drawndeviation from which it was drawnEmpirical Rule tells you how (un)likely the data Empirical Rule tells you how (un)likely the data

point ispoint is If very unlikely, you are suspicious of the If very unlikely, you are suspicious of the

hypothesis about mean and standard deviation, hypothesis about mean and standard deviation, and reject it and reject it

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Summary of Summary of Variation Measures Variation Measures

MeasureMeasure EquationEquation DescriptionDescriptionRangeRange XXlargest largest - - XXsmallestsmallest Total SpreadTotal Spread

Interquartile RangeInterquartile Range QQ3 3 - - QQ11 Spread of Middle 50%Spread of Middle 50%

Standard DeviationStandard Deviation(Sample)(Sample)

XX XX

nnii

22

11

Dispersion aboutDispersion aboutSample MeanSample Mean

Standard DeviationStandard Deviation(Population)(Population)

XX

NNii XX

22 Dispersion aboutDispersion aboutPopulation MeanPopulation Mean

VarianceVariance(Sample)(Sample)

((XXii --XX ))22

nn - 1- 1Squared DispersionSquared Dispersionabout Sample Meanabout Sample Mean

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Z-scoresZ-scores

Number of standard deviations from the Number of standard deviations from the meanmean

i

ixz

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ConclusionConclusion

1.1. Described Qualitative Data GraphicallyDescribed Qualitative Data Graphically2.2. Described Numerical Data GraphicallyDescribed Numerical Data Graphically3.3. Created & Interpreted Graphical DisplaysCreated & Interpreted Graphical Displays4.4. Explained Numerical Data PropertiesExplained Numerical Data Properties5.5. Described Summary MeasuresDescribed Summary Measures

6.6. Analyzed Numerical Data Using Summary Analyzed Numerical Data Using Summary Measures Measures

Statistics An Introduction

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