Statistics

Statistics

• Definition• Methods of organizing

and analyzing quantitative data

• Types

• Descriptive statistics– Central tendency,

variability, etc.

• Inferential statistics– Chi-squares, t-tests,

F-ratios, etc.

Descriptive Statistics

• Measurement scales– Nominal (male/female, yes/no)– Ordinal (high to low)– Interval(1, 2, 3. . .)– Ratio

Visual Depictions of Data

• Frequency distributions

• Histograms and Bar Graphs

Visual Depictions of Data

0

10

20

30

40

50

60

70

80

90

1st Qtr 2ndQtr

3rd Qtr4th Qtr

First gradersSecond gradersThird graders

Measures of Central Tendency

• Mean• Median• Mode

Measures of Central Tendency

• Normal distribution• Skewed distributions

A B

C

Measures of Variability

• How spread out the distribution of scores is from the mean of the distribution

• Range• Standard deviation• Percentile Rank• Standard Scores

0 +1 +2-1-2 +3-3

34.13%34.13%

13.59%13.59% 2.14%2.14%

.13%.13%

Descriptive Statistics

• Measures of relationship– Correlation coefficient

• High (.70 or above) Moderate (.40-.60) Low (.20)

• Pearson product moment coefficient ®• Correlation Matrix

Correlation MatrixIntercorrelations Among Teacher Practices

1 2 3 41 Choice 1.0

2 Directives 0.01 1.0

3 Coach 0.06 0.34 1.0

4 Positive -0.34 0.55 0.28 1.0

Descriptive Statistics• Measures of relationship

– Multiple regression• Shows the relationship of several independent

variables with a dependent variable

• Path Analysis

Application Activity?• Your students earned the following scores on your

recent test:• 89, 95, 78, 50, 80, 89, 76• What scale of measurement are you using?• What was the range?• Compute the mean, median and mode.• What is your distribution like?

• The Null Hypothesis

• Level of Significance– Type I Error– Type II Error

• Effect Size

Let’s apply this!

Application Activity• If the null hypothesis is rejected, we can say that

the observed results proves that the theory we are testing is true.

• The null hypothesis will be rejected if the sample data turn out to be _____________ consistent/inconsistent with what one would expect if it were true.

• Which level of significance offers the greater protection against Type I errors, .05 or .01?

• Is it possible for a researcher to make both a Type I error and a Type II error at the same time?

Application Activity

• A researcher tells the superintendent at the local school that the new curriculum is statistically significantly superior to the old curriculum. What question(s) should the superintendent ask the researcher?

• T-Test: • Used to compare 2 means

• Analysis of Variance (ANOVA)– One Way– Two Way– Repeated Measures

• Post Hoc and Planned Comparisons– E.g., Bonferoni, Tukey, Duncan’s, etc.

One-Way ANOVA

StimulationLow High

AD/HD

Two-Way ANOVA

AD/HD

Control

Treatment 1

Treatment 2

Treatment 1

Treatment 2

Interaction

Treatment 1 Treatment 2

ADHD

Control

Behavior

• Mixed Design

AD/HD

Control

Group Repeated FactorSchedule Puzzle 1 Puzzle 2 Puzzle 3

• Multivariate Analysis of Variance

• Analysis of Covariance (ANCOVA

Nonparametric Tests

• Frequencies, percentages, or proportions

• Chi-square test (X2)

Race Ignored jobs Mentioned jobsMinorities 59.1 40.9Caucasian 73.1 26.9

Chi square significance = .00787

Importance of jobs by ethnicity