CHAPTER 3CHAPTER 3

Descriptive Descriptive StatisticsStatisticsMeasures of Central Measures of Central TendencyTendency

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Descriptive StatisticsDescriptive StatisticsMeasures of Central TendencyMeasures of Central Tendency

Mean-Mean---------------IntervalInterval or or Ratio scaleRatio scale Polygon Polygon – The sum of the values divided by the number of The sum of the values divided by the number of

values--often called the values--often called the "average." "average." μ=ΣX/Nμ=ΣX/N– Add all of the values together. Divide by the number Add all of the values together. Divide by the number

of values to obtain the mean. of values to obtain the mean. – Example: Example: XX 771212242420201919

????????22

DescrDescriptive Statisticsiptive Statistics

The Mean is:The Mean is:

μ=ΣX/N= 82/5=16.4μ=ΣX/N= 82/5=16.4

(7 + 12 + 24 + 20 + 19) / 5 = (7 + 12 + 24 + 20 + 19) / 5 = 16.4.16.4.

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The Characteristics of MeanThe Characteristics of Mean 1. 1. Changing a score Changing a score in a distribution in a distribution will will

change the meanchange the mean 2. 2. Introducing or removing a score Introducing or removing a score from from

the distribution the distribution willwill change the mean change the mean 3. 3. Adding or subtracting Adding or subtracting a constant from a constant from

each score each score willwill change the mean change the mean 4. 4. Multiplying or dividing Multiplying or dividing each score by a each score by a

constant constant willwill change the mean change the mean 5. Adding a score which is5. Adding a score which is same as the same as the

mean mean will not will not change the meanchange the mean44

Descriptive StatisticsDescriptive Statistics Measures of Central TendencyMeasures of Central Tendency Median/Median/MiddleMiddleOrdinal ScaleOrdinal ScaleBar/HistogramBar/Histogram

– Divides the values into two equal Divides the values into two equal halves, halves, with with half of the values being lower than the median half of the values being lower than the median and half higher than the median. and half higher than the median. Sort the values into Sort the values into ascending order. ascending order. If you have an If you have an odd number odd number of values, the of values, the

median is the median is the middle value. middle value. If you have an If you have an even number even number of values, the of values, the

median is the arithmetic median is the arithmetic mean mean (see above) of (see above) of the the two middle two middle values. values.

– Example: The median of the same five numbers Example: The median of the same five numbers (7, 12, 24, 20, 19) is ???. (7, 12, 24, 20, 19) is ???.

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StatisticsStatistics The median is 19.The median is 19. MModeode--Nominal ScaleNominal Scale Bar/Histogram Bar/Histogram

– The most The most frequentlyfrequently-occurring value (or -occurring value (or values). values). Calculate the frequencies for all of the Calculate the frequencies for all of the

values in the data. values in the data. The mode is the value (or values) with The mode is the value (or values) with

the highest frequency. the highest frequency. – Example: For individuals having the Example: For individuals having the

following ages -- 18, 18, 19, 20, 20, 20, 21, following ages -- 18, 18, 19, 20, 20, 20, 21, and 23, the mode is ???? and 23, the mode is ???? The Mode is 20The Mode is 20

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CHARACTERISTICS OF MODECHARACTERISTICS OF MODE

Nominal Scale Nominal Scale

Discrete VariableDiscrete Variable

Describing ShapeDescribing Shape

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The RangeThe Range The Range:The Range: The Range is the difference between The Range is the difference between

the highest number –lowest number +1the highest number –lowest number +1 2, 4, 7, 8, and 10 -> 2, 4, 7, 8, and 10 -> Discrete NumbersDiscrete Numbers 2, 4.6, 7.3, 8.4, and 10 -> 2, 4.6, 7.3, 8.4, and 10 -> Continues Continues

NumbersNumbers The difference between the The difference between the upper real upper real

limit limit of the of the highest number highest number and the and the lower real limit lower real limit of the of the lowest number.lowest number.

CHAPTER 4CHAPTER 4

VariabilityVariability

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VariabilityVariabilityVariability is a measure of Variability is a measure of

dispersiondispersion or spreading of or spreading of scores around the mean, and scores around the mean, and has 2 purposes:has 2 purposes:

1. Describes the distribution1. Describes the distribution Next slideNext slide

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Range, Interquartile Range, Semi-Interquartile Range, Interquartile Range, Semi-Interquartile Range, Range, Standard Deviation, and Variance are the Standard Deviation, and Variance are the

Measures of VariabilityMeasures of Variability

The Range:The Range: The Range is the difference between the The Range is the difference between the

highest number –lowest number +1highest number –lowest number +1 2, 4, 7, 8, and 10 -> 2, 4, 7, 8, and 10 -> Discrete NumbersDiscrete Numbers 2, 4.6, 7.3, 8.4, and 10 -> 2, 4.6, 7.3, 8.4, and 10 -> Continues Continues

NumbersNumbers The difference between the The difference between the upper real upper real

limit limit of the of the highest number highest number and the and the lower lower real limit real limit of the of the lowest number.lowest number.

VariabilityVariability 2. How well an individual score (or 2. How well an individual score (or

group of scores) represents the group of scores) represents the entire distribution. i.e. Z Scoreentire distribution. i.e. Z Score

Ex. In Ex. In inferential statistics inferential statistics we we collect information from a small collect information from a small samplesample then, then, generalizegeneralize the results the results obtained from the sample to the obtained from the sample to the entire entire population.population.

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Interquartile Range (IQR) Interquartile Range (IQR) In descriptive statistics, the In descriptive statistics, the

Interquartile Range Interquartile Range (IQR), (IQR), also called the also called the midspread or midspread or middle fifty, middle fifty, is a measure of is a measure of statistical dispersion, being statistical dispersion, being equal to the difference equal to the difference between the upper and lower between the upper and lower quartiles. (quartiles. (QQ33 − − QQ11)=IQR)=IQR 1414

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Interquartile Range (IQR)Interquartile Range (IQR) IQR IQR is the range covered is the range covered by the by the middle 50% middle 50% of the of the distribution.distribution.

IQR IQR is the distance is the distance between the 3between the 3rdrd Quartile Quartile and 1and 1stst Quartile. Quartile.

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Semi-Interquartile Range (SIQR)Semi-Interquartile Range (SIQR)

SIQR SIQR is ½ or half of is ½ or half of the Interquartile the Interquartile Range.Range.

SIQR SIQR = = (Q3-Q1)/2(Q3-Q1)/2

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VariabilityVariability

2020

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Variability Variability Range, Range, SS,SS, Standard Deviations Standard Deviations and and VariancesVariances

X X σ² = ss/N σ² = ss/N PopPop 1 1 σ = √σ = √ss/Nss/N 22 4 4 s² = ss/n-1 or ss/df s² = ss/n-1 or ss/df Standard deviationStandard deviation

5 5 s = √ss/df s = √ss/df Sample Sample

SS=SS=ΣxΣx²-(Σx)²/N ²-(Σx)²/N Computation Computation

SS=SS=ΣΣ(( x-x-μμ))² ² Definition Definition

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMeanVarianceVariance (σ²)(σ²) is the is the MeanMean of of SquaredSquared Deviations= Deviations=MMSS

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Practical Implication for Test Practical Implication for Test ConstructionConstruction

Variance and Covariance measure the quality of Variance and Covariance measure the quality of each each item in a test.item in a test.

Reliability and validity measure the quality of Reliability and validity measure the quality of the the entire test.entire test.

σ²=SS/N σ²=SS/N used for one set of data used for one set of data

VarianceVariance is the degree of variability is the degree of variability

of scores from meanof scores from mean..Correlation is based on a statistic called Covariance (Cov xy Correlation is based on a statistic called Covariance (Cov xy

or S xy) ….. r=sp/√ssx.ssyor S xy) ….. r=sp/√ssx.ssy

COVxy=SP/N-1COVxy=SP/N-1 used for 2 sets of data used for 2 sets of data

CovarianceCovariance is a number that reflects the degree to is a number that reflects the degree to which 2 variables vary togetherwhich 2 variables vary together. .

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VarianceVariance

X X σ² = ss/N Popσ² = ss/N Pop 1 s² = ss/n-1 or ss/df Sample1 s² = ss/n-1 or ss/df Sample 2 2 4 4 55

SS=SS=ΣxΣx²-(Σx)²/N²-(Σx)²/N

SS=SS=ΣΣ(( x-x-μμ))²²

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean

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CovarianceCovariance CorrelationCorrelation is based on a statistic called is based on a statistic called

CovarianceCovariance (Cov (Cov xy xy or Sor S xy xy) ….. ) ….. COVCOVxyxy=SP/N-1=SP/N-1

Correlation--Correlation-- r=sp/√ssx.ssy r=sp/√ssx.ssy CovarianceCovariance is a number that reflects the is a number that reflects the

degree to which degree to which 2 variables 2 variables varyvary together. together.

Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

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CovarianceCovariance CorrelationCorrelation is based on a statistic called is based on a statistic called

CovarianceCovariance (Cov (Cov xy xy or Sor S xy xy) ….. ) ….. COVCOVxyxy=SP/N-1=SP/N-1

Correlation--Correlation-- r=sp/√ssx.ssy r=sp/√ssx.ssy CovarianceCovariance is a number that reflects the is a number that reflects the

degree to which degree to which 2 variables 2 variables varyvary together. together.

Original DataOriginal Data X YX Y 8 18 1 1 01 0 3 63 6 0 10 1

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CovarianceCovariance

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Descriptive Statistics for Descriptive Statistics for Nondichotomous VariablesNondichotomous Variables

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Descriptive Statistics for Descriptive Statistics for Dichotomous DataDichotomous Data

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Descriptive Statistics for Descriptive Statistics for Dichotomous DataDichotomous Data

Item Variance & Covariance Item Variance & Covariance

FACTORS THAT AFFECT FACTORS THAT AFFECT VARIABILITYVARIABILITY

1. Extreme Scores 1. Extreme Scores i.e. 1, 3, 8, 11, 1,000,000.00 . We can’t use i.e. 1, 3, 8, 11, 1,000,000.00 . We can’t use the Range in this situation but we can use the other measures of the Range in this situation but we can use the other measures of variability.variability.

2. Sample Size 2. Sample Size If we increase the sample size will change the If we increase the sample size will change the Range therefore we can’t use the Range in this situation but we can Range therefore we can’t use the Range in this situation but we can use the other measures of variability.use the other measures of variability.

3. Stability 3. Stability Under SamplingUnder Sampling (see next slide) p.130 The (see next slide) p.130 The S and S² for all samples should be the same because they come from S and S² for all samples should be the same because they come from same population (all slices of a pizza should taste the same).same population (all slices of a pizza should taste the same).

4. Open-Ended Distribution 4. Open-Ended Distribution When we don’t have When we don’t have highest score and lowest score in a distributionhighest score and lowest score in a distribution

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