Variability

What Do We Mean by Variability?  Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.

What Does Variability Tell Us?  It describes the distribution  It represents how well an individual score is likely to represent the entire population The more variability, the less alike scores are from individual to individual The more variability, the less alike scores are from individual to individual

What Does the Range Tell Us? TTTThe range tells us how spread the scores are PPPProblems with using the range Ignores the middle scores Outliers can throw off the range Gives us no idea of the scale

What Is the Interquartile Range?  The interquartile range ignores extreme scores and measures the range covered by the middle 50% of the distribution  How does SPSS find quartiles (1/4)(n+1) – Gives us the number of the lower bound element (1/4)(n+1) – Gives us the number of the lower bound element (3/4)(n+1) – Gives us the number of the upper bound element (3/4)(n+1) – Gives us the number of the upper bound element  How do we find the interquartile range? Q3 – Q1 Q3 – Q1  Semi-Interquartile range (divided in half)

What Are the Weaknesses of These Methods? DDDDon’t take into account the spread of the other scores DDDDon’t take into account scaling issues

What Is a Standard Deviation?  This is the way we generally measure variability  This measure is used to determine what the typical distance from the mean is for any given score in our data

How Do We Calculate a Standard Deviation?  First you must find the deviation of each score, or the deviation X - μ X - μ  Next we attempt to find the mean of the deviations 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 deviations?

How Else Might We Find the Mean of the Deviations?  Absolute values? Some use these, though they are harder to use mathematically for inferential statistics since the normal curve is based on S.D. Some use these, though they are harder to use mathematically for inferential statistics since the normal curve is based on S.D.  Squaring the numbers gives us a positive value. It also weights deviants differentially. It also weights deviants differentially.

Taking the Mean of Squared Deviations Gives Us Variance  What is variance? This is also a measure of variation This is also a measure of variation Helpful, but not quite as valuable in describing the average deviation from the mean Helpful, but not quite as valuable in describing the average deviation from the mean  How do we solve this? Take the square root Take the square root This yields the Standard Deviation This yields the Standard Deviation

Samples Versus Populations  s vs. σ  s 2 vs. σ 2  As always M vs. μ  n-1 versus N This increases the size of the average deviant and makes it a more accurate, unbiased estimator of the population score This increases the size of the average deviant and makes it a more accurate, unbiased estimator of the population score This is in essence a penalty for sampling This is in essence a penalty for sampling Another way to think about it is because of the degrees of freedom Another way to think about it is because of the degrees of freedom

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

Formulae? Formulas?  Sum of Squared Error SS = Σ(X-μ) 2 SS = Σ(X-μ) 2  Variance σ 2 = SS/N σ 2 = SS/N s 2 = SS/(n-1) = SS/df s 2 = SS/(n-1) = SS/df  Standard Deviation σ = √(SS/n) σ = √(SS/n) s = √(SS/n-1) = √(SS/df) s = √(SS/n-1) = √(SS/df)