Lecture Notes

Chapter 4 Combination Statistics
(Modified: 2003-07-16)

• z-Scores
  • used to describe a single score
  • provides a standard way to compare scores from any distribution(s)
  • also known as standard scores
  • when used descriptively, the range of z scores is approximately -3 to +3. What do you think the mean is? Why?
  • formula:

• (note the similarity to the deviation, and note you are dividing the deviation by the standard deviation)
• The following table is similar to the one on p. 69.

Student	Raw Score T1	Calculate	z score T1	Calculate	Raw Score T2	z score T2
Kris	76	76-54/10	2.20	76-86/6	76	-.1.67
Robin	54	54-54/10	.00	86-86/6	86	.00
Marty	58	58-54/10	.40	72-86/6	72	-.67
Terry	58	58-54/10	.40	90-86/6	90	.67
	Mean T1 = 54				Mean T2 = 86
	S = 10				S = 6

Another familiar example...

Raw Score	Calculation	z-score
40	40-30.12/7.22	1.369
38	38-30.12/7.22	1.092
37	37-30.12/7.22	.953
34	34-30.12/7.22	.538
34	34-30.12/7.22	.538
33	33-30.12/7.22	.399
33	33-30.12/7.22	.399
33	33-30.12/7.22	.399
33	33-30.12/7.22	.399
29	29-30.12/7.22	-.155
29	29-30.12/7.22	-.155
29	29-30.12/7.22	-.155
28	28-30.12/7.22	-.293
27	27-30.12/7.22	-.432
26	26-30.12/7.22	-.570
19	19-30.12/7.22	-1.540
10	10-30.12/7.22	-2.787
Mean = 30.12		Mean = 0
SD = 7.22		SD = 1.00

Boxplot of Scores Above

• Boxplots
  • Formerly called box and whisker plots
  • Show:
    • Interquartile Range
    • Range
    • Median
    • Mean
    • Variability
    • Skew
    • Outliers
  • Boxplot Interpretation

  

• Effect Size Index (d)
  • Answer questions about SIZE of differences between distributions
  • d = effect size
  • Magnitude of d
    • Small effect, d = .20
    • Medium effect, d = .50
    • Large effect, d = .80
  • APA and d
    • APA now recommends that researchers report d, the effect size index in all papers involving numberical data
  • Calculating d, the effect size index

• Descriptive Statistics Report
  • Construct Boxplots
  • Find Effect Size Index
  • Tell the story
    • Form of distributions
      • symmetrical or skewed?
    • Central Tendency
      • give mean and median (What about mode?)
    • Overlap of distributions
      • For two or more distributions, how close are they to each other?
    • Interpretation of Effect Size Index
      • Do you have a small, medium, or large effect.
      • What other adjectives can you use (i.e., very, moderately, extremely)
  • See p. 76 for an example

• z-Score Problems--problems and answers for z-scores