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