VariabilityVariability

What Do We Mean by Variability?What Do We Mean by Variability?

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

What Does Variability Tell Us?What Does Variability Tell Us?

It describes the distributionIt describes the distribution It represents how well an individual score It represents how well an individual score

is likely to represent the entire populationis likely to represent the entire population The more variability, the less alike scores are The more variability, the less alike scores are

from individual to individualfrom individual to individual

What Does the Range Tell Us?What Does the Range Tell Us?

The range tells us how spread the scores The range tells us how spread the scores areare

Problems with using the rangeProblems with using the range Ignores the middle scoresIgnores the middle scores

• Outliers can throw off the rangeOutliers can throw off the range Gives us no idea of the scaleGives us no idea of the scale

What Is the Interquartile Range?What Is the Interquartile Range?

The interquartile range ignores extreme scores The interquartile range ignores extreme scores and measures the range covered by the middle and measures the range covered by the middle 50% of the distribution50% of the distribution

How does SPSS find quartilesHow does SPSS find quartiles (1/4)(n+1) – Gives us the number of the lower bound (1/4)(n+1) – Gives us the number of the lower bound

elementelement (3/4)(n+1) – Gives us the number of the upper bound (3/4)(n+1) – Gives us the number of the upper bound

elementelement How do we find the interquartile range?How do we find the interquartile range?

Q3 – Q1Q3 – Q1 Semi-Interquartile range (divided in half)Semi-Interquartile range (divided in half)

What Are the Weaknesses of These What Are the Weaknesses of These Methods?Methods?

Don’t take into account the spread of the Don’t take into account the spread of the other scoresother scores

Don’t take into account scaling issuesDon’t take into account scaling issues

What Is a Standard Deviation?What Is a Standard Deviation?

This is the way we generally measure This is the way we generally measure variabilityvariability

This measure is used to determine what This measure is used to determine what the typical distance from the mean is for the typical distance from the mean is for any given score in our dataany given score in our data

How Do We Calculate a Standard How Do We Calculate a Standard Deviation?Deviation?

First you must find the deviation of each score, First you must find the deviation of each score, or the deviationor the deviation X - X - μμ

Next we attempt to find the mean of the Next we attempt to find the mean of the deviationsdeviations What do we notice about the mean of the deviations?What do we notice about the mean of the deviations? Why is this happening?Why is this happening?

How else might we find the mean of the How else might we find the mean of the deviations?deviations?

How Else Might We Find the Mean How Else Might We Find the Mean of the Deviations?of the Deviations?

Absolute values?Absolute values? Some use these, though they are harder to Some use these, though they are harder to

use mathematically for inferential statistics use mathematically for inferential statistics since the normal curve is based on S.D.since the normal curve is based on S.D.

Squaring the numbers gives us a positive Squaring the numbers gives us a positive value.value. It also weights deviants differentially.It also weights deviants differentially.

Taking the Mean of Squared Taking the Mean of Squared Deviations Gives Us VarianceDeviations Gives Us Variance

What is variance?What is variance? This is also a measure of variationThis is also a measure of variation Helpful, but not quite as valuable in describing Helpful, but not quite as valuable in describing

the average deviation from the meanthe average deviation from the mean How do we solve this?How do we solve this?

Take the square rootTake the square root This yields the Standard DeviationThis yields the Standard Deviation

Samples Versus PopulationsSamples Versus Populations s vs. s vs. σσ ss22 vs. vs. σσ22

As always M vs. As always M vs. μμ n-1 versus Nn-1 versus N

This increases the size of the average deviant and This increases the size of the average deviant and makes it a more accurate, unbiased estimator of the makes it a more accurate, unbiased estimator of the population scorepopulation score

This is in essence a penalty for samplingThis is in essence a penalty for sampling Another way to think about it is because of the Another way to think about it is because of the

degrees of freedomdegrees of freedom

Degrees of FreedomDegrees of Freedom

For a sample of n scores, the For a sample of n scores, the degrees of degrees of freedomfreedom or or dfdf for the sample variance are for the sample variance are defined as defined as df = n – 1df = n – 1. The degrees of . The degrees of freedom determine the number of scores freedom determine the number of scores in the sample that are independent and in the sample that are independent and free to vary.free to vary.

Formulae? Formulas?Formulae? Formulas?

Sum of Squared ErrorSum of Squared Error SS = SS = Σ(X-μ)Σ(X-μ)22

VarianceVariance σσ2 2 = SS/N= SS/N ss22 = SS/(n-1) = SS/df = SS/(n-1) = SS/df

Standard DeviationStandard Deviation σ = √(SS/n)σ = √(SS/n) s = √(SS/n-1) = √(SS/df)s = √(SS/n-1) = √(SS/df)