2. Variability Measures of spread of scores range: highest - lowest standard deviation: average difference from mean variance: average squared difference from mean

3. 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)

4. Population or Sample Standard Deviation EXAMPLE: Find the population standard deviation for this set of scores: 3,7,8,10,15,17

5. STEP 1: Calculate the mean.  = (3+7+8+10+15+17) / 6 = 10.00 STEP 2: Subtract the mean from each score. x x -  3-7 7-3 8-2 10 0 15 5 17 7

6. STEP 3: Square each (x-  ). x x -  (x -  ) 2 3-749 7-39 8-24 10 00 15 525 17 749

7. STEP 4: Sum the (x -  ) 2 x x -  (x -  ) 2 3-749 7-39 8-24 10 00 15 525 17 749 136 =  (x -  ) 2

8. STEP 5: Divide by N and take the square root.

9. Population or Sample Variance Same as  x or S x, but don’t take the square root. EXAMPLE: Calculate the population variance of this set of scores: 3,7,8,10,15,17

10. Estimated Population Standard Deviation or Variance Same as  x and  x 2, but divide by N-1 instead of N. EXAMPLE: Calculate the estimated population standard deviation. 3,7,8,10,15,17

11. 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

12. 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