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