1. Since When is it Standard to Be Deviant?

2. Passing the “So What?” Test
Mean Age = 32.4 (So What?)
Ever meet the average person?
Measures of dispersion provide an idea of what the mean may mean for the rest of us.

3. Calculating the Range
Age 12 17 20 30 31 34 36 42 60
60 – 12 = 48

4. At Home with a Range One of the easiest calculations in statistics.
Maximum – Minimum
Very close to useless
No very helpful in getting a good sense of dispersion
Far too general to set the groundwork for more advanced statistics

5. Variance: Please Disperse!
To properly make sense of the degree of dispersion of data, we need to be able to take the average “distance” of each data point from the measure of central tendency.
The variance (s2) is almost exactly that!
s2 = S(X – M)2/n – 1
Let’s examine this in theory a bit

6. The Centroid and Data Points

7. Practical Experience: Calculating Variance by Hand Part 1: Sum of Squares10 12 13 15 18 20 22 25 M
16.875
X - M
(X – M)2
-6.875
Note: S(X-M)2 is also referred to as the Sum of Squares (SS)
-4.875
-3.875
3.5156
-1.875
1.125
1.2656
3.125
9.7656
5.125
8.125
S = 135
S =
S = 0.00

8. Calculating Variance Part 2: Plugging in the Numbers
Remember the equation: s2 = S(X – M)2/n – 1 (0r s2 = SS/n – 1)
s2 = /8-1
s2 = /7
s2 =

9. Why n-1? Remember, this is a sample, not a population.
The true variance of the population is probably giong to be a bit larger than that of the sample we’ve collected.
By dividing by a slightly smaller number (n-1), we raise the variance to more accurately reflect the population.

10. So What?
Now we know that the variance of the scores on this inventory is
What does this tell us?
Does the number have much of a connection to the original data set?
Yes, but sadly, not intuitively.
If you have a bit of experience with variances, you could tell that it’s a moderate to large variance for that data set.
If not, the number is simply too big to be very useful.

11. The Standard Deviation in One Easy Step
Is a crucial part of most statistics
Provides us with a sense of dispersion in a number that’s easier to understand.
SD (or s) = √s2
s = √
s = 5.25
No, Seriously!

12. Back to So What?
Now we know that the average change from score to score from the central tendency is a score of 5.25
Try this:

14. Correlation Does Not Imply Cause
The degree to which to phenomena co-occur
We do this all the time intuitively
Tooth decay and getting kicked out of 3rd grade health class
Not Useless
Provides ideas
Can lead to tentative conclusions that might be tested in more rigorous settings
Useful when experimental research is not feasible or unethical

15. Pearson's Product Moment Correlation
Galton the mathematically illiterate
-1.0  0.0  +1.0
Magnitude (0.03 vs 0.98)
Valence (-0.68 vs 0.68)
Singularity (+1.0 or -1.0)
Assumptions
Normality
Linearity
Interval level data (usually)

16. Correlation Matrix
A survey of 234 nursing students completed a survey indicating (among other things) their approach to learning and the nature of their control beliefs.
From Cantwell, R.H. (1997). Cognitive dispositions and student nurses' appraisals of their learning environment. Issues in Educational Research, 7(1), 1997, 19-36

17. Pearson's Product Moment Correlation Computational Equation
This is the “official” equation for the Pearson’s r. It’s not as unwieldy as it looks, but it can be simplified with a simple algebraic manipulation to this one…

18. Let's Compute a Correlation!
Though not the only way to hand-calculate Pearson’s r, it’s the easiest equation available
Note that denominator (AKA the covariate) is always going to be a positive number, whereas the numerator may be either positive or negative. This allows the numerator to dictate the valence of the coefficient.

19. IQ (X)
110
112
118
119
122
125
127
130
132
134
136
138
GPA (Y)
1.0
1.6
1.2
2.1
2.6
1.8
2.0
3.2
3.0
3.6 X2 Y2 XY
12,100
1.00
110.0
12,544
179.2
2.56
13,924
1.44
141.6
14,161
249.9
4.41
14,884
6.76
317.2
15,625
3.24
225.0
16,129
6.76
330.2
16,900
4.00
260.0
17,424
10.24
422.4
17,956
6.76
348.4
18,496
9.00
408.0
19,044
12.96
496.8
1503
27.3
189,187
69.13
3488.7

20. Now For Some Calculations

21. Coefficient of Determination
Degree to which the variance on one variable is explained by the other. r2
= 0.732
73% of GPA can be accounted for by IQ
OR 73% of IQ can be accounted for by GPA
Remember, correlation does not imply cause

22. Getting Our Hands Dirty
Import the provided data set and calculate the range, variance and standard deviation for each variable
Construct a correlation matrix using all variables and interpret the results
Calculate the coefficient of determination for at least three of the correlation coefficients in the matrix

23. Which of the following describes a negative correlation (AKA an inverse relationship)?
When high scores on one variable tend to be associated with high scores on another variable
When high scores on one variable tend to be associated with low scores on another variable
When scores on one variable appear to be completely unrelated to the sores on another variable.
Both b and c are correct
When high scores on one variable tend to be associated with low scores on another variable

24. Standardized distribution
Which of the following terms best describes the average of squared deviations from the mean?
Standard deviation
Variance
Mode
Standardized distribution
Variance

25. Questions? Thoughts?