PsychStats-Chapter 4: Variabilty

1. Introduction 2. Defining Standard Deviation and Variance 3. Measuring Variance and Standard Deviation for a Population 4. Measuring Standard Deviation and Variance for a Sample 5. Sample Variance as an Unbiased Statistic

Variability

provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered

The Range

The first step in defining and measuring variability is the range. Which is the distance covered by the scores in a distribution.

range formula

= 𝑥𝑚𝑎𝑥 − 𝑥𝑚𝑖n

Standard Deviation

the most commonly used and most important measure of variability.

Deviation

the distance from the mean.

deviation score

= X - u

Variance

the square of the standard deviation

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = √variance

Remember, that Standard Deviation, is to get the standard distance of the scores from the mean, since we used squares of the deviations in taking the variance, we now take the square root of the variance to finally solve for the Standard Deviation.

Variance

= mean squared deviation = Σ (X - 𝜇) 2 𝑁

Sum of squared deviation

SS =Σ (X - 𝜇) ^2

Variance

=SS/N

Standard Deviation

=√SS/N

Population Standard Deviation

***σ=√***σ² =√SS/N

Bias

To correct the _________ in sample variability, it is necessary to make an adjustment in the formulas for sample variance and standard deviation

Sample Variance

s² = SS/n-1

Sample Standard Deviation

s= √s² = √ SS/ (n-1)

The mean

(average) for every sample may vary, and this places a restriction in the sample variability.

A sample statistic is unbiased

if the average value of the statistic is equal to the population parameter. (The average value of the statistic is obtained from all the possible samples for a specific sample size, n.)

A sample statistic is biased

if the average value of the statistic either underestimates or overestimates the corresponding population parameter

TRANSFORMATION OF SCALE

Adding a constant to each score does not change the standard deviation. • Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.