Ch 4 Social Statistics: Variability

definitional formula =

SS = E (X-M)^2 Find the deviation from the mean for each score: Deviation = X-M Square each deviation: squared deviation = (X-M)^2 Add the squared deviations: SS= E(X-M)^2

equation for population variance

SS/N

What is the range for the following set of scores? Scores: 5, 7, 9, 15

10 or 11 points

Describe the scores in a sample that has a standard deviation of zero.

A standard deviation of zero indicates there is no variability. In this case, all of the scores in the sample have exactly the same value.

computational formula

SS= EX^2 - (EX^2)/n)

Computational formula SS

SS= Ex^2 - (EX)^2/N

range =

URL for Xmax - LRL for Xmin

In words, explain what is measured by variance and standard deviation.

Variance is the mean of the squared deviations. Standard deviation is the square root of the variance and provides a measure of the standard distance from the mean.

population varience is

Varience = SS/N

scores are all whole numbers, the range can be obtained by

Xmax - Xmin + 1

A sample statistic is _________ if the average value of the statistic either underestimates or overestimates the corresponding population parameter.

biased

For the following scores, which of the following actions will increase the range? Scores: 3, 7, 10, 15 a. Add 4 points to the score b. Add 4 points to the score c. Add 4 points to the score d. Add 4 points to the score

d. add 4 points to the score X=15

SS formula

definitional SS= E(X-u)^2 find each deviation score (x-μ) Square each deviation score (x-μ)^2 Add the squared deviations.

sample varience

s^2 = SS/n-1

population standard deviation is

sd = square root of SS/N

Standard deviation is the square root of the variance and provides a measure of the standard, or average distance from the mean.

sd = square root of variance

lower real limit (LRL) for the

smallest score (xmin)

SS =

sum of squares

A sample statistic is _____________ 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.)

unbiased

sum of squares

is the sum of the squared deviation scores

standard deviation

provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered.

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.

A researcher takes all of the possible samples of n=4 from a population. Next, the researcher computes a statistic for each sample and calculates the average of all the statistics. Which of the following statements is the most accurate? a.If the average statistic overestimates the corresponding population parameter, then the statistic is biased. b. If the average statistic underestimates the corresponding population parameter, then the statistic is biased. c. If the average statistic is equal to the corresponding population parameter, then the statistic is unbiased. d. All of the above.

d. All of the above.

degrees of freedom

degrees of freedom to be used in calculations would be n - 1. To calculate the degrees of freedom for a sample size of N=9. subtract 1 from 9 (df=9-1=8).

sample varience

is represented by the symbol and equals the mean squared distance from the mean. Sample variance is obtained by dividing the sum of squares (SS) by n − 1.

sample standard deviation

is represented by the symbol s and equals the square root of the sample variance.

Population varience

is represented by the symbol σ and equals the mean squared distance from the mean. Population variance is obtained by dividing the sum of squares (SS) by N.

population standard deviation

is represented by the symbol σ and equals the square root of the population variance.

upper real limit (URL) for the

largest scores (xmax)

equation for sample varience

E(X-M)^2 / (n-1)

equation for population standard deviation

sq r (E(X-u)^2 /N)

equation for sample standard deviation

sq r of SS / (n-1)

sample standard deviation is the square root of the varience

ssd = s= square root of s^2 = square root of SS/n-1

how to find a deviation or deviation score

the difference between a score and the mean calculated: D = X - u (mean) ex. u=50, if score is X= 53 then deviation score= 3 points

+ or - indicated

the direction from the mean (below or above it)

Sample Standard deviation

The square root of the variance and provides a measure of the standard, or average distance from the SAMPLE mean. S Square root of the average squared distance from M

Calculate SS, variance, and standard deviation for the following sample of n=8 scores: 0, 4, 1, 3, 2, 1, 1, 0.

SS= 14,s^2 = 2, and s= sq r 2 = 1.41

Which of the following is a consequence of increasing variability?

The distance from one score to another tends to increase and a single score tends to provide a less accurate representation of the entire distribution.

population variance

The mean squared deviation. POPULATION Variance is the average squared distance from the POPULATION mean. σ² Mean squared deviation from μ

sample variance

The mean squared deviation. SAMPLE Variance is the average squared distance from the SAMPLE mean. s² Mean squared deviation from the sample mean

population standard deviation

The square root of the variance and provides a measure of the standard, or average distance from the POPULATION mean. σ Square root of the population variance