1. Descriptive Statistics I REVIEW
Measurement scales
Nominal, Ordinal, Continuous (interval, ratio)
Summation Notation:
3, 4, 5, 5, 8 Determine: ∑ X, (∑ X)2, ∑X2
Percentiles: so what?
Chapter 3

2. Measures of central tendency
Mean, median mode
3, 4, 5, 5, 8
Distribution shapes
Chapter 3

3. Variability
Range Hi – Low scores only (least reliable measure; 2 scores only)
Variance (S2) Spread of scores based on the squared
deviation of each score from mean
Most stable measure
Standard Deviation The square root of the variance
Most commonly used measure of variability
Error
Total
variance
True Variance
These are measures of variability. The range is the most unreliable measure because it depends only two scores. The standard deviation is the square root of the variance.
Chapter 3

4. Variance (Table 3.2) The didactic formula 4+1+0+1+4=10 10 = 2.5
=10 10 = 2.5
5-1=4 4
The calculating formula
Introduce the variance from the didactic equation (i.e., the average squared deviation from the mean) and then the calculating formula. Use Table 3-2 on page 40 to illustrate that you get the same answer with both formulae. Then illustrate that you obtain the standard deviation by simply taking the square root of the variance.
= 55-45=10 = 2.5 4
Chapter 3

5. Standard Deviation The square root of the variance
Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations
Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.
Chapter 3

6. The Normal Distribution
Indicate the Mean plus and minus 3 standard deviations captures nearly 100% of the scores in a normal distribution.
M + 1s = 68.26% of observations
M + 2s = 95.44% of observations
M + 3s = 99.74% of observations
Chapter 3

7. Calculating Standard Deviation
Raw scores 3 7 4 5 1 ∑
Mean: 4
(X-M) -1 3 1 -3
(X-M)2 1 9 20
S= √20 5
S= √4
S=2
Chapter 3

8. Coefficient of Variation (V)
Relative variability around the mean OR
Determines homogeneity of scores S M
Helps more fully describe different data sets that have a common std deviation (S) but unique means (M)
Lower V=mean accounts for most variability in scores
=homogeneous >.5=heterogeneous
Chapter 3

9. Descriptive Statistics II
What is the “muddiest” thing you learned today?
Chapter 3

10. Descriptive Statistics II REVIEW
Variability
Range
Variance: Spread of scores based on the squared deviation of each score from mean Most stable measure
Standard deviation Most commonly used measure
Coefficient of variation
Relative variability around the mean (homogeneity of scores)
Helps more fully describe different data sets that have a common std deviation (S) but unique means (M)
50+10
What does this tell you?
Chapter 3

11. Standard Scores
Set of observations standardized around a given M and standard deviation
Score transformed based on its magnitude relative to other scores in the group
Converting scores to Z scores expresses a score’s distance from its own mean in sd units
Use of standard scores: determine composite scores from different measures (bball: shoot, dribble); weight?
Chapter 3

12. Standard Scores Z-score M=0, s=1 T-score T = 50 + 10 * (Z) M=50, s=10
Provide an example of the use of standard scores For example, determining a composite score for basketball from shooting (high is better) and dribbling (low is better). You might want to weight shooting 2 or 3 times that of dribbling and show how scores change.
Chapter 3

13. Conversion to Standard Scores
Raw scores 3 7 4 5 1
Mean: 4
St. Dev: 2
X-M -1 3 1 -3 Z
-.5
1.5 .5
-1.5
SO WHAT?
You have a Z score but what do you do with it? What does it tell you?
Provide an example of the use of standard scores For example, determining a composite score for basketball from shooting (high is better) and dribbling (low is better). You might want to weight shooting 2 or 3 times that of dribbling and show how scores change.
Allows the comparison of scores using different scales to compare “apples to apples”
Chapter 3

14. Normal distribution of scores Figure 3.7
Provide illustrations of use interpretation of the normal distribution.
Point out that the only numbers that ever change on the figure are for the specific test that one is using. Here it is VO2 but it could be body fat or written test score, or a scale from the affective domain
99.9
Chapter 3

15. Normal-curve Areas Table 3-3
Z scores are on the left and across the top
Z=1.64: 1.6 on left , .04 on top=44.95
Values in the body of the table are percentage between the mean and a given standard deviation distance
The "reference point" is the mean
Students need to clearly see that the reference point is the mean and that the values in the BODY of the table are percentages of observations between the MEAN and the given standard deviation units away from the mean.
Use many of the homework problems for students to get practice working with the Z-table
Chapter 3

17. Area of normal curve between 1 and 1.5 std dev above the mean Figure 3.9

18. Normal curve practice Z score Z = (X-M)/S T score T = 50 + 10 * (Z)
Percentile P = 50 + Z percentile (+: add to 50, -: subtract from 50)
Raw scores
Hints
Draw a picture
What is the z score?
Can the z table help?
There a numerous homework problems on the WWW about converting between z, T, percentile and raw scores
The animation shows that when z = 1, T = 60, Percentile = 84, and VO2 = 65
Chapter 3

19. Assume M=700, S=100
Percentile
T score
z score
Raw score 64
53.7
.37
737 43
–1.23
618 17 68
835
.57
There a numerous homework problems on the WWW about converting between z, T, percentile and raw scores
Chapter 3

20. Descriptive Statistics III
Explain one thing that you learned today to a classmate
What is the “muddiest” thing you learned today?
Chapter 3