Variability
it simply means that things aren’t always the same; they vary.
- 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 good measure of variability serves two purposes:
- Variability describes the distribution. Specifically, it tells whether the scores are clustered close together or are spread out over a large distance - It tells how much distance to expect between one score and another
- Variability measures how well an individual score (or group of scores) represents the entire distribution.
range
the distance covered by the scores in a distribution, from the smallest score to the largest score.
range = X (max) - X (min)
When the scores are measurements of a continuous variable:
range = upper real limit for X max - lower real limit for X min
When the scores are whole numbers, the range can also be defined as the number of measurement categories:
range = X max - X min + 1
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.
Deviation
distance from the mean:
deviation score = X - mu (mean)
Deviation score
he distance (and direction) from the mean to a specific score.
deviation score = X - mu (mean)
how to find standard deviation
- calculate deviation score
- calculate the mean of the deviation scores (first add up deviation scores than divide by N)
- get rid of the signs (+ and -). The standard procedure for accomplishing this is to square each deviation score. Using the squared values, you then compute the mean squared deviation, which is called variance.
- The final step simply takes the square root of the variance to obtain the standard deviation, which measures the standard distance from the mean.
Variance
Variance equals the mean of the squared deviations. Variance is the average squared distance from the mean.
Standard deviation
Standard deviation is the square root of the variance and provides a measure of the standard, or average distance from the mean.
Standard deviation = Square root of Variance
Variance
Variance = mean squared deviation = sum of squared deviations/number of scores
SS, or sum of squares
SS, or sum of squares, is the sum of the squared deviation scores.
Definitional formula (to compute SS)
SS = sigma(X-mu)^2
(use when mean is a whole number)
Computational formula (to compute SS)
SS = sigmaX^2 - (sigmaX)^2 / N
(use when mean is not a whole number)
Variance equation using SS
Variance = SS/N
Standard deviation equation using SS
standard deviation = square root of SS/N
Population variance
The average squared distance from the mean; the mean of the squared deviations.
o^2 = SS/N
Population standard deviation
Population standard deviation is represented by the symbol o and equals the square root of the population variance.
o = square root of SS/N
Definitional formula for sum of squared deviation for a sample
SS = sigma(X-M)^2
Computational formula for sum of squared deviation for a sample
SS = sigmaX^2 - (sigmaX)^2 / n
sample variance
The sum of the squared deviations divided by df = n-1. An unbiased estimate of the population variance.
Sample variance is represented by the symbol s^2 and equals the mean squared distance from the mean. Sample variance is obtained by dividing the sum of squares by n − 1.
s^2 = SS/n-1
Sample standard deviation
Sample standard deviation is represented by the symbol s and equal the square root of the sample variance.
s = square root of s^2 = square root of SS/n-1
degrees of freedom, or df
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.
unbiased
A statistic that, on average, provides an accurate estimate of the corresponding population parameter. The sample mean and sample variance are unbiased statistics.
- 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.)
biased
A sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter.
