VariabilityIbrahim Altubasi, PT, PhDThe University of Jordan

Variability • Look at the following two population distributions

of student scores on a 5-point scale3, 3, 3, 3, 3, 3, 3, 3, 3, 31, 1, 2, 2, 3, 3, 4, 4, 5, 5

• Your task: Calculate the means of the two distributions

• Think: Does the mean tell the difference• between the two distributions?

Descriptive Statistics Descriptive Statistics: statistical procedures used tosummarize, organize, and simplify data.

Shape of Distribution

Central Tendency

Variability

RangeInterquartile RangeVarianceStandard DeviationVariability provides a quantitative

measure of the degree to which scores in a distribution are spread-out or clustered together

Variability Variability

RangeInterquartile RangeVariance Standard Deviation

The distance between the largest score andthe smallest score

Variability Variability

RangeInterquartile RangeVariance Standard Deviation

Interquartile range (IQR) is the distancebetween Q1 and Q3.Semi-Interquartile range is one half ofthe IQR.

The 1st quartile (Q1) is the score at the 25th percentile (the scoreseparates the lowest 25% of the distribution from the rest.The 2nd quartile (Q2) is…The 3rd quartile (Q3) is the score that divides the bottom threequarters of the distribution from the top quarter (the score at the ??percentile)

Variability• Find the range and the semi-interquartile range of

the following data set:

3, 5, 3, 2, 8, 4, 6, 7, 1, 4, 3, 2

Variability Variability

RangeInterquartile RangeVariance Standard Deviation

Variance and Standard deviation (SD) arethe most commonly used and mostimportant measure of variability. They usethe mean as a reference and measurevariability by considering the distancebetween each score and the mean.

The SD approximates the average distancefrom the mean.

Variability in the Population

Population Mean

μ = Σ X N

Deviation is the distance from the mean:Deviation Score = X- μ

Population Sample Size

X μ Deviation

35 38 -3

30 38 -8

38 38 0

60 38 22

35 38 -3

30 38 -8

Σ 228

0

μ 38Questions:•Can we use the average of the deviation scores as a measure of variability?•How can we represent the average distance from the mean?

Variability in the Population

Sum of squared deviations, also called Sum of squares (SS):

SS = Σ (X – μ)2 = Σ(X2) – (ΣX)2/N

X μ Deviation

Squared Deviation

35 38 -3 9

30 38 -8 64

38 38 0 0

60 38 22 484

35 38 -3 9

30 38 -8 64

Σ 228

0 630

μ 38

N 6

Squared Deviations (X – μ) 2

SS: Σ(X – μ) 2

Question: Is SS a goodmeasure for variability?Why or why not?

Variability in the Population

Variance is the average of the squared deviations. Populationvariance is denoted as σ2.

σ2= SS/N = Σ (X – μ) 2 /N.

X μ Deviation

Squared Deviation

35 38 -3 9

30 38 -8 64

38 38 0 0

60 38 22 484

35 38 -3 9

30 38 -8 64

Σ 228

0 630

μ 38 105

N 6

Squared Deviations (X – μ) 2

SS: Σ(X – μ) 2

σ2= Σ (X – μ) 2 /N

Variability in the Population

Standard deviation (SD) is the square root of the variance. It has the same unit as the raw score. Population SD is denoted as σ.

σ = √(Σ (X – μ) 2 /N).

X μ Deviation

Squared Deviation

35 38 -3 9

30 38 -8 64

38 38 0 0

60 38 22 484

35 38 -3 9

30 38 -8 64

Σ 228

0 630

μ 38 105

N 6

SS: Σ(X – μ) 2

σ2= Σ (X – μ) 2 /N

Variability in the Sample

The population of adultheights forms a normaldistribution.

If you select a samplefrom this population, youare most likely to obtainindividuals who are nearaverage in height.

As a result, the scores inthe sample will be lessvariable (spread out) thanthe scores in thepopulation

Variability of the Sample

Standard deviation (SD) and Variance for the samples

Sample Mean: = ΣX n

Deviation Score = X –

Sum of Squares from the sample

SS = Σ (X- ) 2 = ΣX2 – (ΣX) 2

n

Sample Variance (denoted as s2): s2 = SS n-1

Sample SD (denoted as s): s= √ SS n-1

Variability The sample variance s2 = SS / (n-1) is an unbiased estimate of the population variance.

A sample statistic is unbiased if the average value of the sample statistics, obtained over many different samples, is equal to the population parameter.

A sample statistic is biased if the average value of the sample statistics, obtained over many different samples, consistently underestimates or overestimates the population parameter.

Sample mean ( is an unbiased estimate for population mean(μ).

Sample variance (s2) calculated through SS/(n-1) is an unbiasedestimate for population variance (σ2)

Variability

Adding a constant to each score will change the mean but will notchange the SD or variance.

Multiplying each score by a constant will cause the mean and theSD to be multiplied by the same constant, and will cause thevariance to be multiplied by the square of the constant.

Variability Add or subtracta constant fromeach score inthe data …;

s2 = 1.47

s = 1.21

s2 = ?

s = ?

VariabilityMultiplying ordividing eachscore by aconstant in thedata…

s2 = 2.25

s = 1.58

s2 = ?

s = ?

Descriptive Statistics

Mean is the most common measure of central tendency; SD (and related variance) is the most common measure of variability.

When median is used to report central tendency, interquartile range (IQR) or semi-interquartile range is commonly used to report variability.

SIQR provides the best measure of variability for distributionsthat are very skewed or that have a few extreme scores.