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

 

 

Summary of Central Tendency

     ITEM A

11
  3
12
  1
  3
  6
  4
  3

Mean
Median
Mode

Positively-skewed 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

Negatively-skewed 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 Middle-Most 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 equal-sized 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 z-Score

A z-score 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 z-scores:

a. locating an individual score 

b. using z as a standard. Individual raw scores are converted to z-scores and compared to a set standard. Two common standards are z = 1.65, which represents a 1-tailed area of 95%, and z = + 1.96 or – 1.96 (between which is a 2-tailed 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.

 

 

 

 

 

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 product-moment 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 paired-observations of the same subjects.
• A cause-effect relationships has a strong correlation but a strong correlation doesn’t guarantee a cause-effect 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 cause-effect 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.
• Test-retest 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 paired-observations on the same subjects.

Pre- and Post-test 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 cause-effect 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 t-squared
  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. t-test                                 2 means; 1 independent variable 
  b. one-way ANOVA            3 or more means; 1 independent variable 
  c. factorial ANOVA             Multiple means on 2+ independent variables

Summary of 1-Way 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)
• Between-Ss degrees of freedom equals the number of groups minus 1 (k-1).
• Within-Ss degrees of freedom equals the total number of people minus the number of groups (N-k).
• Total degrees of freedom equals the total number of people minus 1 (N-1).
• ANOVA stands for ANalysis Of VAriance.

 

Summary of t-Test

• There are two kinds of t-tests: independent and correlated.
• Correlated t-tests use paired observations on one group of subjects. This is called a within-subjects design.
• Independent t-test use subjects which have been randomly-assigned to two groups.
• The degrees of freedom for a correlated t-test equals n-1.
• The degrees of freedom for an independent t-test equals N-2.
• T-tests are not used with more than two groups because of the likelihood of Type I error.
• T-tests measure the differences between means.
• T-tests are like z-scores.
• The independent t-test pools the variance of the subgroups.

 

 

NOW YOU CHOOSE:
    Day 8: Student’s t-Test
    Bit More About t-Test
    Even More About t-Test
    How to Calculate t-Test
    Practice Problems
    More Practice Problems
    Word Problems
        Sim1           Sim2            Sim3
        Sim4           Sim5            Sim6
        Sim7           Sim8            Sim9
    Basic Facts About t-Test
    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