Edpsy 511Edpsy 511

Homework 1: Due 2/6Homework 1: Due 2/6

Descriptive StatisticsDescriptive Statistics

Statistics vs. ParametersStatistics vs. Parameters

► A parameter is a characteristic of a A parameter is a characteristic of a population.population. It is a numerical or graphic way to summarize It is a numerical or graphic way to summarize

data obtained from the populationdata obtained from the population

► A statistic is a characteristic of a sample.A statistic is a characteristic of a sample. It is a numerical or graphic way to summarize It is a numerical or graphic way to summarize

data obtained from a sampledata obtained from a sample

Types of Numerical DataTypes of Numerical Data

► There are two fundamental types of There are two fundamental types of numerical data:numerical data:

1)1) Categorical data: obtained by determining Categorical data: obtained by determining the frequency of occurrences in each of the frequency of occurrences in each of several categoriesseveral categories

2)2) Quantitative data: obtained by determining Quantitative data: obtained by determining placement on a scale that indicates amount placement on a scale that indicates amount or degreeor degree

Techniques for Summarizing Techniques for Summarizing Quantitative DataQuantitative Data

► Frequency DistributionsFrequency Distributions► HistogramsHistograms► Stem and Leaf PlotsStem and Leaf Plots► Distribution curvesDistribution curves► AveragesAverages► VariabilityVariability

Summary MeasuresSummary Measures

Central Tendency

Arithmetic Mean

Median Mode

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Range

Measures of Central Measures of Central TendencyTendency

Central Tendency

Average (Mean) Median Mode

1

1

n

ii

N

ii

XX

n

X

N

Mean (Arithmetic Mean)Mean (Arithmetic Mean)

►Mean (arithmetic mean) of data Mean (arithmetic mean) of data valuesvalues SampleSample mean mean

PopulationPopulation mean mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

MeanMean

►The most common measure of central The most common measure of central tendencytendency

►Affected by extreme values (outliers)Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Mean of Grouped FrequencyMean of Grouped Frequency

XX ff fXfX

1010 11

99 33

88 22

77 44

66 66

55 55

TotalTotal NN

2121 fX NfXX /

Weighted MeanWeighted Mean

A form of mean A form of mean obtained from obtained from groups of data in groups of data in which the different which the different sizes of the groups sizes of the groups are accounted for or are accounted for or weighted.weighted.

total

wN

xfx

)(

GroupGroup xbarxbar NN f(xbar)f(xbar)

11 3030 1010

22 2525 1515

33 4040 2525

total

wN

xfx

)(

MedianMedian

►Robust measure of central tendencyRobust measure of central tendency►NotNot affected by extreme values affected by extreme values

►In an In an Ordered arrayOrdered array, median is the , median is the “middle” number“middle” number If n or N is odd, median is the middle numberIf n or N is odd, median is the middle number If n or N is even, median is the average of If n or N is even, median is the average of

the two middle numbersthe two middle numbers

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

ModeMode► A measure of central tendencyA measure of central tendency► Value that occurs most oftenValue that occurs most often► Not affected by extreme valuesNot affected by extreme values► Used for either numerical or categorical Used for either numerical or categorical

datadata► There may may be no modeThere may may be no mode► There may be several modesThere may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode

VariabilityVariability

► Refers to the extent to which the scores on a Refers to the extent to which the scores on a quantitative variable in a distribution are spread quantitative variable in a distribution are spread out.out.

► The The rangerange represents the difference between the represents the difference between the highest and lowest scores in a distribution.highest and lowest scores in a distribution.

► A A five number summaryfive number summary reports the lowest, the first reports the lowest, the first quartile, the median, the third quartile, and highest quartile, the median, the third quartile, and highest score.score. Five number summaries are often portrayed graphically by Five number summaries are often portrayed graphically by

the use of the use of box plots.box plots.

VarianceVariance► The Variance, sThe Variance, s22, represents the amount of variability of , represents the amount of variability of

the data relative to their meanthe data relative to their mean► As shown below, the variance is the “average” of the As shown below, the variance is the “average” of the

squared deviations of the observations about their meansquared deviations of the observations about their mean

1

)( 22

n

xxs i

► The Variance, sThe Variance, s22, is the , is the samplesample variance, and is variance, and is used to estimate the actual used to estimate the actual populationpopulation variance, variance, 22

N

xi

22 )(

Standard DeviationStandard Deviation

► Considered the most useful index of variability.Considered the most useful index of variability.► It is a single number that represents the spread of a It is a single number that represents the spread of a

distribution.distribution.► If a distribution is normal, then the mean plus or If a distribution is normal, then the mean plus or

minus 3 SD will encompass about 99% of all scores minus 3 SD will encompass about 99% of all scores in the distribution.in the distribution.

Calculation of the Variance and Standard Calculation of the Variance and Standard Deviation of a Distribution (Definitional Deviation of a Distribution (Definitional

formula)formula)

√

RawScore Mean X – X (X – X)

2

85 54 31 96180 54 26 67670 54 16 25660 54 6 3655 54 1 150 54 -4 1645 54 -9 8140 54 -14 19630 54 -24 57625 54 -29 841

Variance (SD2) =

Σ(X – X)2

N-1

= 3640

9 =404.44

Standard deviation (SD) = Σ(X – X)2

N-1

Definitional vs. Definitional vs. ComputationalComputational

► DefinitionalDefinitional An equation that An equation that

defines a measuredefines a measure

► ComputationalComputational An equation that An equation that

simplifies the simplifies the calculation of the calculation of the measuremeasure

1

)( 22

2

nn

XX

s

1

)( 22

n

xxs i

Calculate the variance using Calculate the variance using the computational and the computational and definitional formulas.definitional formulas.

►10, 12, 12, 12, 13, 13, 1410, 12, 12, 12, 13, 13, 14

1

)( 22

n

xxs i

1

)( 22

2

nn

XX

s

In small groups: calculate the In small groups: calculate the variance using the variance using the

computational and definitional computational and definitional formulas.formulas.

►2, 8, 9, 11, 15, 17, 202, 8, 9, 11, 15, 17, 20

1

)( 22

n

xxs i

1

)( 22

2

nn

XX

s

Comparing Standard Comparing Standard DeviationsDeviations

Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 S = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 S = 4.57

Data C

Facts about the Normal Facts about the Normal DistributionDistribution

► 50% of all the observations fall on each side of 50% of all the observations fall on each side of the mean. the mean.

► 68% of scores fall within 1 SD of the mean in a 68% of scores fall within 1 SD of the mean in a normal distribution.normal distribution.

► 27% of the observations fall between 1 and 2 27% of the observations fall between 1 and 2 SD from the mean.SD from the mean.

► 99.7% of all scores fall within 3 SD of the mean. 99.7% of all scores fall within 3 SD of the mean. ► This is often referred to as the This is often referred to as the 68-95-99.7 rule68-95-99.7 rule

The Normal CurveThe Normal Curve

Different Distributions ComparedDifferent Distributions Compared

Fifty Percent of All Scores in a Normal Fifty Percent of All Scores in a Normal Curve Fall on Each Side of the MeanCurve Fall on Each Side of the Mean

Probabilities Under the Normal Probabilities Under the Normal CurveCurve

Standard ScoresStandard Scores

► Standard scores use a common scale to indicate how Standard scores use a common scale to indicate how an individual compares to other individuals in a an individual compares to other individuals in a group.group.

► The simplest form of a standard score is a The simplest form of a standard score is a Z scoreZ score..► A A Z score Z score expresses how far a raw score is from the expresses how far a raw score is from the

mean in standard deviation units. mean in standard deviation units. ► Standard scores provide a better basis for comparing Standard scores provide a better basis for comparing

performance on different measures than do raw performance on different measures than do raw scores.scores.

► A A Probability Probability is a percent stated in decimal form and is a percent stated in decimal form and refers to the likelihood of an event occurring.refers to the likelihood of an event occurring.

► T scores T scores are z scores expressed in a different form (z are z scores expressed in a different form (z score x 10 + 50).score x 10 + 50).

Probability Areas Between the Mean Probability Areas Between the Mean and Different Z Scoresand Different Z Scores

Examples of Standard Scores Examples of Standard Scores