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:

79818286868891939597

total = 878Divide by n = 10 scoresMean = 87.8

Computing a Median Value in a Distribution of Scores

Two distributions of scoresDistribution 1 Distribution 2• 24• 24• 25• 25• 26• 26

– Mean = 25

– Range = 3

• 16• 19• 22• 25• 28• 30• 35

– Mean = 25

– Range = 20

COMPUTING DEVIATION SCORES

Raw Mean DEV. SQUAREDscore score deviation score 4 - 10 = -6 36 8 - 10 = -2 4 9 - 10 = -1 110 - 10 = 0 010 - 10 = 0 010 - 10 = 0 012 - 10 = 2 413 - 10 = 3 914 - 10 = 4 1690/9 = 10.00 = MEAN

70/9 = 7.77 = VarianceSTANDARD 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 12. 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 = n1 + n2 - 2

n1= number of subjects in group 1

n2 = number of subjects in group 2

What is a t-table?

One-Way Analysis of Variance(F-test)

variation between groupsF = ______________________

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 X2 (chi-square) Statistic

X2 = (observed count – expected count)2

expected count

What do you do with the calculated X2 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 X2 table look like?