Three Broad Purposes of
Quantitative Research
• 1. Description
• 2. Theory Testing
• 3. Theory Generation

Four Things to Know About
Statistics
• What statistical methods are used to analyze quantitative data.
• When to use these statistical methods (with what kinds of data).
• How to use these statistical methods (the calculations).
• What the results of statistical tests mean.

Whenever a researcher has a large number of test scores, it is advisable to describe the many scores with a few simple indicators that provide some important information about the set of scores.

3 Measures of Central
Tendency:
• Mean : the arithmetic average in a distribution of scores.
• Median : the midpoint in a distribution of scores (most typical score).
• Mode : the most frequently-occurring score in a distribution of scores.

Three Measures of Variability
•
Range : the difference between the highest and lowest scores in a distribution of scores.
•
Variance : a measure of dispersion indicating the degree to which scores cluster around the mean score.
•
Standard deviation : index of the amount of variation in a distribution of scores.

Calculating a Mean Score
Scores:
79
81
82
86
86
88
91
93
95
97 total = 878
Divide by n = 10 scores
Mean = 87.8

Computing a Median Value in a Distribution of Scores

Two distributions of scores
Distribution 1
• 24
• 24
Distribution 2
• 16
• 19
• 25
• 25
• 26
• 26
– Mean = 25
–
Range = 3
• 22
• 25
• 28
• 30
• 35
–
Mean = 25
– Range = 20

COMPUTING DEVIATION SCORES
9
10
10
10
Raw Mean DEV. SQUARED score score deviation score
4
8
- 10 = -6
- 10 = -2
36
4
- 10 = -1
- 10 = 0
- 10 = 0
- 10 = 0
1
0
0
0
12
13
- 10 = 2
- 10 = 3
14 - 10 = 4
90/9 = 10.00 = MEAN
4
9
16
70/9 = 7.77 = Variance
STANDARD DEVIATION: (Square Root of Variance) = 2.79

Statistical Tests and Related
Procedures
• t-test
– independent groups
– non-independent
• Analysis of variance
• chi-square
• Correlation
–
Regression
–
Multiple regression
• Factor analysis

Let’s conduct an educational experiment!

Compare two methods for teaching 6th grade science
Students randomly assigned to:
Method A: “creative exploration” or
Method B: “interactive collaboration”

Results:
Mean scores on “
Science Achievement
Test
”:
Method A = 90.3 (s.d.= 2.89)
Method B = 84.9 (s.d.= 3.77)

Must interpret this observed difference in mean scores:
(1) Method A caused the difference; or
(2) The difference between the groups occurred by chance (the null hypothesis).

The null hypothesis:
Ho : There will be no significant difference in mean science test performance between
6th grade students taught by Method A and those taught by Method B .

We need to choose between the chance explanation ( null hypothesis ) and the alternative hypothesis that there is a relationship between teaching method and test performance.

Two potential errors!
• TYPE I ERROR:
– occurs when a null hypothesis is rejected, but null hypothesis is true.
– Practical result is that changes may be made that are
not warranted.
• TYPE II ERROR
– occurs when null hypothesis is accepted, but null is false .
– Practical result is that educators may fail to make needed changes.

Calculating the two-group t-test statistic: t = Mean group 1
– Mean group 2 standard error
Standard error =>
1. Divide standard deviation for Group 1 by n of Group 1
2. Divide s.d. for Group 2 by n of Group 2.
3. Sum.
4. Compute square root of this sum.

What do you do with this t-value?
If calculated t value is equal to or greater than the critical t value (found in a t-table) based on (1) alpha level and (2) degrees of freedom, then reject the null hypothesis that there is no difference between the groups.

What’s an alpha level?
The predetermined “level of significance,” usually p = .05
, meaning that the null hypothesis (no difference) occurs by chance alone no more than five times out of 100 hypothetical studies.

What are degrees of freedom?
df = n
1
+ n
2
- 2 n
1
= number of subjects in group 1 n
2
= number of subjects in group 2

What is a t-table?

One-Way Analysis of Variance
(F-test) variation between groups
F = ______________________ variation within groups

What do you do with the derived F value?
If derived F value is equal to or greater than the critical F value
(found in F-table, based on sample size, alpha level, and degrees of freedom), then reject the null hypothesis.

What does an F table look like?

The X 2 (chi-square) Statistic
X 2 =

(observed count – expected count)2 expected count

What do you do with the calculated X 2 statistic?
If derived value is equal to or greater than the critical value (found in a X2 table, based on alpha level and degrees of freedom), then reject the null hypothesis.

What does a X 2 table look like?