Variability• Measures of spread of scores• range: highest - lowest• standard deviation: average difference
from mean• variance: average squared difference from
mean
Range• Subtract lowest score from highest score• For continuous variables, add a point for
real limits
EXAMPLE: Find the range of this set of scores:3,7,8,10,15,17
Range = 17- 3 = 14 (for discrete variable)Range = 17 - 3 + 1 = 15 (for continuous variable)
Population or Sample Standard Deviation
x (x )2
N
EXAMPLE: Find the population standard deviation for this set of scores: 3,7,8,10,15,17
Sx (x x )2
N
STEP 1: Calculate the mean. = (3+7+8+10+15+17) / 6 = 10.00 STEP 2: Subtract the mean from each score.
x x - 3 -77 -38 -210 015 517 7
• STEP 3: Square each (x- ).
x x - (x - )2
3 -7 497 -3 98 -2 410 0 015 5 2517 7 49
• STEP 4: Sum the (x - )2
x x - (x - )2
3 -7 497 -3 98 -2 410 0 015 5 2517 7 49
136 = (x - )2
STEP 5: Divide by N and take the square root.
x 136
6 22.67 4.76
Population or Sample Variance
• Same as x or Sx, but don’t take the square root.
EXAMPLE: Calculate the population variance of this set of scores: 3,7,8,10,15,17
x2 136
622.67
Estimated Population Standard Deviation or Variance
• Same as x and x2, but divide by N-1 instead of
N.
EXAMPLE: Calculate the estimated population standard deviation. 3,7,8,10,15,17
sx 136
5 27.20 5.22
More about deviating from standards...
Why are the formulae different for estimating? - sample variability is usually less than the
population variability -dividing by N-1 compensates for that - unbiased estimate
Comparing Measures of Variability• range:
easy to compute highly unstable
• standard deviation: very commonly used takes all scores into account
• variance: used in inferential statistics hard to interpret