Chap 4

A sample variance is divided by (n - 1) instead of n to make it an

unbiased estimate of the population variance.

Population standard deviation

σ=√σ^2=√SS/N Standard distance from μ √(SS/N)

population variance

σ²=SS/N Average squared distance from the population mean ∑(X - μ)² / N

No

Is it possible to obtain a negative value for SS (sum of squared deviations), variance, and standard deviation?

Explain why the two formulas use different values in the denominator.

The two formulas use different values because the sample mean statistic is an unbiased estimator of the population mean, and sample variance uses n - 1 to be an unbiased estimator of σ².

Explain why the formula for sample variance is different from the formula for population variance. Why is it inappropriate to use the formula for population variance in calculating the variance of a sample?

Variance is defined as the mean squared deviation, and, for a population, is computed as the sum of squared deviations divided by N. Without some adjustment, the sample variance will be biased and will consistently underestimate the corresponding population value.

What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample.

When standard deviation is zero, the scores have no variability.

Under what circumstances is the definitional formula easy to use?

When the mean is a whole number and there are relatively few scores

IQR is a better measurement of variability than the simple range

because most of the scores are clustered together within a range of two points.

A sample statistic is biased

consistently underestimates or overestimates the corresponding population parameter.

A sample mean (that is, M = ∑X/n) is an unbiased

estimator of the population mean.

Sample Statistic is unbiased

if the average value of the sample statistic, obtained over all possible samples, is equal to the population parameter.

median

midpoint in a list of scores listed in order from smallest to largest

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 together.

sample standard deviation

s=√s^2=√SS/n-1 Square root of the sample variance

sample variance

s^2 = SS/n-1 Average squared distance from the sample mean SS / (n - 1)

Standard deviation

square root of the variance

central tendency

statistical measure to determine a single score that defines the midpoint of a distribution

Under what circumstances is the computational formula preferred?

the mean is not a whole number or when there are many scores.

Variance

the mean of the squared deviations