chapter 4 variability. variability in statistics, our goal is to measure the amount of variability...

Chapter 4Chapter 4

VariabilityVariability


In statistics, our goal is to measure the In statistics, our goal is to measure the amount of variability for a particular set of amount of variability for a particular set of scores, a distribution.scores, a distribution. If all the scores are the same no variabilityIf all the scores are the same no variability If small difference, variability is smallIf small difference, variability is small If large difference, variability is largeIf large difference, variability is large


Variability provides a quantitative measure Variability provides a quantitative measure of the degree to which scores in a of the degree to which scores in a distribution are spread out or clustered distribution are spread out or clustered together.together.

Goal: to describe how spread out the Goal: to describe how spread out the scores are in a distributionscores are in a distribution

Figure 4.1Figure 4.1

Population distributions of heights and weightsPopulation distributions of heights and weights

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Variability (cont.)Variability (cont.)

Variability will serve two purposesVariability will serve two purposesDescribe the distributionDescribe the distribution

Close togetherClose togetherSpread out over a large distanceSpread out over a large distance

Measure how well an individual score (or Measure how well an individual score (or group of scores) represents the entire group of scores) represents the entire distributiondistribution

Variability (cont.)Variability (cont.)

Variability provides information about how Variability provides information about how much error to expect when you are using a much error to expect when you are using a sample to represent a population.sample to represent a population.

Three measures of variabilityThree measures of variabilityRangeRange Interquartile rangeInterquartile rangeStandard deviationStandard deviation

RangeRange

The range is the difference between the The range is the difference between the upper real limit of the largest (maximum) X upper real limit of the largest (maximum) X value and the lower real limit of the value and the lower real limit of the smallest (minimum) X value.smallest (minimum) X value.

Range is the most obvious way to Range is the most obvious way to describe how spread out the scores are.describe how spread out the scores are.

Range (cont.)Range (cont.)

Problem: Completely determined by the Problem: Completely determined by the two extreme values and ignores the other two extreme values and ignores the other scores in the distribution.scores in the distribution.

It often does not give an accurate It often does not give an accurate description of the variability for the entire description of the variability for the entire distribution.distribution.

Considered a crude and unreliable Considered a crude and unreliable measure of variabilitymeasure of variability

Interquartile Range and Interquartile Range and Semi-Interquartile RangeSemi-Interquartile Range

Divide the distribution into four equal partsDivide the distribution into four equal partsQ1, Q2, Q3Q1, Q2, Q3The interquartile range is defined as the The interquartile range is defined as the

distance between the first quartile and the distance between the first quartile and the third quartilethird quartile

Interquartile RangeInterquartile Range

Q1 Q2 Q3

25% 25% 25% 25%

Semi-interquartile RangeSemi-interquartile Range


The interquartile rangeThe interquartile range

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Interquartile Range (cont.)Interquartile Range (cont.)

When the interquartile range is used to When the interquartile range is used to describe variability, it commonly is describe variability, it commonly is transformed into the semi-interquartile transformed into the semi-interquartile range.range.

Semi-interquartile range is one-half of the Semi-interquartile range is one-half of the interquartile rangeinterquartile range


Because the semi-interquartile range is Because the semi-interquartile range is derived from the middle 50% of a derived from the middle 50% of a distribution, it is less likely to be influenced distribution, it is less likely to be influenced by extreme scores and therefore gives a by extreme scores and therefore gives a better and more stable measure of better and more stable measure of variability than the range.variability than the range.


Does not take into account distances Does not take into account distances between individual scoresbetween individual scores

Does not give a complete picture of how Does not give a complete picture of how scattered or clustered the scores are.scattered or clustered the scores are.

Standard DeviationStandard Deviation

Most commonly usedMost commonly usedMost important measure of variabilityMost important measure of variabilityStandard deviation uses the mean of the Standard deviation uses the mean of the

distribution as a reference point and distribution as a reference point and measures variability by considering the measures variability by considering the distance between each score and the distance between each score and the mean.mean.

Standard Deviation (cont.)Standard Deviation (cont.)

Are the scores clustered or scattered?Are the scores clustered or scattered?Deviation is the average distance and Deviation is the average distance and

direction from the mean.direction from the mean.


Goal of standard deviation is to measure Goal of standard deviation is to measure the standard, or typical, distance from the the standard, or typical, distance from the mean.mean.

Deviation is the distance and direction Deviation is the distance and direction from the meanfrom the mean

deviation score = X - deviation score = X -


Step 1Step 1Determine the deviation or distance from the Determine the deviation or distance from the

mean for each individual score.mean for each individual score.

IfIf X = 53X = 53

deviation score = X – deviation score = X –

= +3= +3


IfIf X = 45X = 45

deviation score = X – deviation score = X –

= -5= -5


Step 2: Calculate the Step 2: Calculate the meanmean of the of the deviation scoresdeviation scoresAdd the derivation scoresAdd the derivation scoresDivide by NDivide by N


XX X – X –

88 +5+5

11 -2-2

33 00

00 -3-3

X – X – Deviation scores must add up to zero


Step 3: Square each deviation score.Step 3: Square each deviation score.Why? The average of the deviation Why? The average of the deviation

scores will not work as a measure of scores will not work as a measure of variability.variability.

Why? They always add up to zeroWhy? They always add up to zero


Step 3 cont.: Step 3 cont.: Using the squared values, you can now Using the squared values, you can now

compute the mean squared deviationcompute the mean squared deviationThis is called This is called variancevariance

Variance = mean squared deviationVariance = mean squared deviation


By squaring the deviation scores:By squaring the deviation scores:You get rid of the + and –You get rid of the + and –You get a measure of variability based on You get a measure of variability based on

squared distancessquared distancesThis is useful for some inferential statisticsThis is useful for some inferential statisticsNote: This distance is not the best descriptive Note: This distance is not the best descriptive

measure for variabilitymeasure for variability


Step 4: Make a correction for squaring the Step 4: Make a correction for squaring the distances by getting the square root.distances by getting the square root.

Standard deviation = varianceStandard deviation = variance

Sum of Squared Deviations (SS)Sum of Squared Deviations (SS)

Variance = mean squared deviation = SSVariance = mean squared deviation = SS

NN

Definitional FormulaDefinitional Formula

SS = SS = X – X –

Sum of Squared Deviations (SS) Sum of Squared Deviations (SS) Definitional FormulaDefinitional Formula

XX X – X – X – X – = 8= 8

11 -1-1 11

00 -2-2 44

66 +4+4 1616

11 -1-1 11

2222 = = X X – – ))22

Computational FormulaComputational Formula

SS = SS = XX22 – ( – (XX

NN

Computational Formula for SSComputational Formula for SS

XX XX22

11 11

00 00

66 3636

11 11

X = 8X = 8 XX22 = 38 = 38

SS = XX2 2 – (– (X)X)22

NN

= 38 – (8)= 38 – (8)22

44

= 38 – 64= 38 – 64

44

= 38 – 16= 38 – 16

= 22= 22

Definitional vs. Computational?Definitional vs. Computational?

Definitional is most direct way of Definitional is most direct way of calculating the sum of squarescalculating the sum of squares

However if you have numbers with However if you have numbers with decimals, it can become cumbersomedecimals, it can become cumbersome

Computation is most commonly usedComputation is most commonly used

FormulasFormulas

Variance = SSVariance = SS

NNStandard deviation = variance = SSStandard deviation = variance = SS

NN

Formulas (cont.)Formulas (cont.)

Variance and standard deviation are Variance and standard deviation are parameters of a population and will be parameters of a population and will be identified with a Greek letter – identified with a Greek letter – ororsigmasigma

Population standard deviation = Population standard deviation = SSSS

NN

Population variance = Population variance = SSSS

NN


Graphic presentation of the mean and standard deviationGraphic presentation of the mean and standard deviation

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Variability of a sample selected from a populationVariability of a sample selected from a population

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Largest and smallest distance from the meanLargest and smallest distance from the mean

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Example (pg. 94)Example (pg. 94)

XX X – X – X – X –

X = 35X = 35

11 -4-4 1616

66 11 11 n=7n=7

44 -1-1 11

33 -2-2 44

88 33 99

77 22 44

66 11 11

3636

= = X X – – ))2 2 = SS= SS

Degrees of FreedomDegrees of Freedom

Degrees of freedom, useDegrees of freedom, use for for samplesample variance variance

where n is the number of scores in the where n is the number of scores in the samplesample..

With a sample of n scores, the first n-1 With a sample of n scores, the first n-1 scores are free to varyscores are free to vary

but the final score is restricted.but the final score is restricted.As a result, the sample is said to have As a result, the sample is said to have

n-1 degrees of freedomn-1 degrees of freedom

Degrees of FreedomDegrees of Freedom

Degrees of freedom, or Degrees of freedom, or df, df, for sample for sample variance are defined as variance are defined as

df = n – 1df = n – 1

where n is the number of scores in the where n is the number of scores in the sample.sample.

Table 4.2Table 4.2

Reporting the mean and standard deviation in APA formatReporting the mean and standard deviation in APA format

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chapter 4 variability. variability in statistics, our goal is to measure the amount of variability...

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