Summary of Measurement
NOMINAL
Used as a name
Makes no mathematical assumption.
0, 12 and 1 have no preference.
Examples:
The # on a race car
Bank ID number
Airplane model #
Part numbers
The # on the side of your horse
ORDINAL
Used to report rank or order.
Assumes the numbers can be arranged in order. Allows descriptions of 1st, 2nd and 3rd place but steps need not be the same size. Winning a close race receives the same score as an easy win.
Examples:
Finish order in contest
College sports ranking
Rating scales
The finish order of your horse
INTERVAL
Used to count conceptual characteristics (IQ, aggression, etc.)
Assumes numbers indicate equal units. Allows distinctions to be made between difficult and easy races but does not allow “twice as much” comparisons. O does not mean lack of intelligence, etc.
Examples:
The # of test items passed.
Temperature in Fahrenheit
Temperature in Celsius
The # of hurdles your horse jumps
RATIO
Used to measure physical characteristics.
Assumes 0 is absolute (indicates lack of entity being measured). Allows 2:1, 3:2, “twice as much” and “half as much” comparisons. 0 means no time has elapsed or no distance has been traveled, etc.
Examples:
Distance, time and weight
Temperature in Kelvin
Miles per gallon
How fast your horse runs
NOW YOU CHOOSE:
Day 1: Measurement
A Bit More About Measurement
Even More About Measurement 1
Even More About Measurement 2
Even More About Measurement 3
Basic Facts About Measurement
Vocabulary
Quiz 1
Summary
Summary of Central Tendency
ITEM A
11
3
12
1
3
6
4
3
Mean
Median
Mode
Positivelyskewed distribution. Mean will be higher than median and mode. Median is better representative of where most scores are located.
ITEM B
9
8
8
7
6
5
8
1
6
Mean
Median
Mode
Negativelyskewed distribution. Mean will be lower than median and mode. Median is better representative of where most scores are located.
ITEM C
5
5
5
5
5
5
5
Mean
Median
Mode
This is a constant. Everyone has the same score.
NOW YOU CHOOSE:
Day 2: Central Tendency
A Bit More About Central Tendency
Even More About Central Tendency
More Examples
More Mean Examples
More Median Examples
Median Is Middle Of Distribution
More Mode Examples
Impact of Outlying Scores
On The Mean
On The Median
On The Mode
How To Calculate Central Tendency
Calculating The Mean
Calculating The Median
When There’s No MiddleMost Score
Calculating The Mode
Formulas For Central Tendency
Basic Facts About Central Tendency
Vocabulary
Quiz 2
Summary
Summary of Dispersion
If everyone has the same score, there is no dispersion from the mean. If everyone has a different scores, dispersion is at it’s maximum but there is no commonality in the scores. In a normal distribution, there are both repeated scores (height) and dispersion (width).
Percentiles, quartiles and stanines imply that distributions look like plateaus. Scores are assumed to be spread out evenly, like lines on a ruler. People are nicely organized in equalsized containers.
SS, variance and standard deviation imply that distributions look like a mountain. Scores are assumed to be clustered in the middle, people are more alike than different. People are mostly together at the bottom on the bowl with a few sticking to the sides.
You can describe an entire distribution as 3 steps (standard deviations) to the left and 3 steps to the right of the mean. The percentages go 2, 14, 34, 34, 14, and 2. This is believed to be true of all normally distributed variables, regardless of what it measures.
NOW YOU CHOOSE:
Day 3: Dispersion
A Bit More About Dispersion
Even More About Dispersion
Range
MAD
Sum of Squares
Variance
Standard Deviation
How To Calculate
Range
MAD
Sum of Squares
Variance
Standard Deviation
Formulas For Dispersion
Practice Problems
More Practice Problems
Basic Facts About Dispersion
Vocabulary
Quiz 3
Summary
Summary of zScore
A zscore indicates how many steps a person is from the mean. A raw score below the mean corresponds to negative z score; a score which is above the mean would have a positive z. The standard deviation indicates how big each step is. Approximately 68% of the scores lie within one standard deviation of the mean. That is, a majority of the distribution is from z = 1 to z = +1.
There are 5 primary applications of zscores:
a. locating an individual score
b. using z as a standard. Individual raw scores are converted to zscores and compared to a set standard. Two common standards are z = 1.65, which represents a 1tailed area of 95%, and z = + 1.96 or – 1.96 (between which is a 2tailed area of 95%).
c. standardizing a distribution and smoothing its data.
d. making a linear transformations of variables; converting the mean and standard deviation to numbers that easier to remember or handle.
e. comparing 2 raw score distributions with different means and standard deviations.
Day 4: zScore
A Bit More About zScores
Even More About zScores
How To Calculate zScores
Practice Problems
Basic Facts About zScores
Vocabulary
Formulas For zScores
Quiz 4
Summary
Summary of Correlation

To measure the strength of relationship between two variables, it would be best to use a correlation

A correlation can only be between 1 and +1.

The closer the correlation coefficient is to 1 (either + or ), the stronger the relationship.

The sign indicates the direction of relationship.

The coefficient of determination is calculated by squaring r. The coefficient of determination shows how much area the two variables share; the percentage of variance explained (accounted for).

The coefficient of nondetermination is calculated by subtracting the coefficient of determination from 1. The coefficient of nondetermination shows how much the two variables don’t share; the percentage of unexplained variance.

To calculate the correlation between two continuous variables, the Person productmoment coefficient is used. To calculate the correlation between two discrete variables, the phi coefficient is used. To calculate the correlation between one discrete and one continuous variable, the point biserial coefficient is used.

Correlations are primarily a measure of consistency, reliability, and repeatability.

Correlations are based on two pairedobservations of the same subjects.

A causeeffect relationships has a strong correlation but a strong correlation doesn’t guarantee a causeeffect relationship. In a correlation, A can cause B or B can cause A or both A and B can be caused by another variable. Inferences of causeeffect based on correlations are dangerous. A correlation shows that a relationship is not likely to be due to chance but it cannot indicate which variable was cause and which effect.

Testretest coefficients are correlations.

In order to make good predictions between two variables, a strong correlation is necessary.
NOW
Summary of Regression
The variable with the smallest standard deviation is the easiest to predict. The less dispersion, the easier to predict.
Without knowing anything else about a variable, the best predictor of it is its mean.
The angle of a regression line is called the slope. Slope is calculated by dividing the Sxy by the SSx.
The point where the regression line crosses the criterion axis is called the intercept.
Predicting the future based on past experience is best done with a regression.
Predicting scores between known values is called interpolation.
Predicting scores beyond known values is called extrapolation.
Regression works best when a relationship is strong and linear.
Regression works best when the correlation is strong.
The error around a line of prediction is consistent along the whole line. It doesn’t vary or waver along the line, so there is only 1 standard error of estimate for the entire line.
The error around a line of prediction can be estimated with the standard error of estimate.
Plus or minus one SEE accounts for 68% of the prediction errors.
A regression is based on pairedobservations on the same subjects.
Pre and Posttest performance is best analyzed by using a regression.
NOW YOU CHOOSE:
Day 6: Regression
Bit More About Regression
Even More About Regression
Calculate Regression
Practice Problems
More Practice Problems
Word Problems
Sim1 Sim2 Sim3
Sim4 Sim5 Sim6
Sim7 Sim8 Sim9
Basic Facts About Regression
Vocabulary
Formulas
Quiz 6
Summary
Summary of Advanced Procedures
The General Linear Model is “general” because it includes a broad group of procedures, including correlation, regression and the more complex linear models. And it includes both continues and discrete variables. It’s linear because it assumes that the relationship between model components is consistent. When one variable goes up (has larger numbers), the other variable consistent reacts. The reaction can be go the same way (positive) or go the opposite way (negative). But the assumption is that changes in one variable will be accompanied by changes in the other variable.
Another assumption is that causation may not be proved but it can be inferred. Although random assignment might increase one’s confidence in causeeffect conclusions, causation can be inferred based simply on consistency. Such an assumption can be risky but we do it all the time. We assume that the earth gets warm because the sun rises. We’ve never randomly assigned the sun to rising and now rising conditions. But we feel quite confident is our conclusion that the sun causes the heat, and no the other way around.
Here are nine applications of the General Linear Model
Continuous Models compare:
a. frequency distribution One variable (predictor or criterion)
b. correlation Two regressions
c. regression Single predictor; single criterion Same as F test or tsquared
d. multiple regression Multiple predictors; single criterion Same as ANOVA
e. multivariate analysis Multiple predictors; multiple criteria
f. causal modeling Multiple measures of a factor
Discrete Models Compare:
a. ttest 2 means; 1 independent variable
b. oneway ANOVA 3 or more means; 1 independent variable
c. factorial ANOVA Multiple means on 2+ independent variables
NOW YOU CHOOSE:
Day10: Advanced Procedures
Bit More About Advanced Procedures
Even More About Advanced Procedures
Basic Facts About Advanced Procedures
Vocabulary
Quiz 10
Summary
Summary of 1Way ANOVA
 Although there are multiple groups, they vary on one independent variable.
 The dependent variable is what the numbers measure. The numbers are dependent on the performance of the subjects.
 The independent variable is what the experimenter manipulates. It is independent of the subjects performance.
 The F is a ratio of the variance between the groups to the variation within the groups.
 The F test assumes that the variation within a group is due to ability and chance.
 The F test assumes that the variation between groups is due to ability, chance, and manipulation of the independent variable.
 The F test assumes that variance due to ability and chance (between and with subjects) will cancel each other out, so that what remains is a measurement of the effect of the independent variable on the dependent variable.
 Mean Squares is a variance term.
 Mean Squares equals Sum of Squares divided by its appropriate degrees of freedom (SS/df)
 BetweenSs degrees of freedom equals the number of groups minus 1 (k1).
 WithinSs degrees of freedom equals the total number of people minus the number of groups (Nk).
 Total degrees of freedom equals the total number of people minus 1 (N1).
 ANOVA stands for ANalysis Of VAriance.
NOW YOU CHOOSE
Day9: 1Way ANOVA
Bit More About 1Way ANOVA
Even More About 1Way ANOVA
Calculate 1Way ANOVA
Practice Problems
More Practice Problems
Word Problems
Sim1 Sim2 Sim3
Sim4 Sim5 Sim6
Sim7 Sim8 Sim9
Vocabulary
Formulas
Quiz 9
Summary
Summary of tTest
 There are two kinds of ttests: independent and correlated.
 Correlated ttests use paired observations on one group of subjects. This is called a withinsubjects design.
 Independent ttest use subjects which have been randomlyassigned to two groups.
 The degrees of freedom for a correlated ttest equals n1.
 The degrees of freedom for an independent ttest equals N2.
 Ttests are not used with more than two groups because of the likelihood of Type I error.
 Ttests measure the differences between means.
 Ttests are like zscores.
 The independent ttest pools the variance of the subgroups.
NOW YOU CHOOSE:
Day 8: Student’s tTest
Bit More About tTest
Even More About tTest
How to Calculate tTest
Practice Problems
More Practice Problems
Word Problems
Sim1 Sim2 Sim3
Sim4 Sim5 Sim6
Sim7 Sim8 Sim9
Basic Facts About tTest
Vocabulary
Formulas
Quiz 8
Summary
Summary Of Probability & ANOR
 The probability of X and Y occurring at the same time equals the probability of X times the probability of Y.
 The probability of X or Y occurring (either one) is the addition of the probability of X and the probability of Y.
 Analysis of Regression tests the likelihood that the linear relationship between the two variables is due to chance. A significant F indicates that X predicts Y well and that the relationship between the two variables is not likely to be due to chance.
 Analysis of Regression does not prove cause and effect. X may cause Y or Y may cause X, or both could be caused by another variable.
NOW YOU CHOOSE:
Day 7: Probability
Bit More About Probability
Even More About Probability
Even More About ANOR
Calculate ANOR
Practice Problems
More Practice Problems
Word Problems
Sim1 Sim2 Sim3
Basic Facts About Probability
Vocabulary
Formulas
Quiz 7
Summary