Nominal: When numbers are simply names as in Group 1 and Group 2. No math can be performed with nominal numbers. The Groups could have been named A & B.

Ordinal: Numbers that show relative standing in a distribution. The distance between numbers (ranks) is not necessarily equal. As an example, the tennis players ranked #1 and #2 may be very close in their skills, and far ahead of the player ranked #3. Percentile ranks are ordinal numbers. The absolute distance between the 60th and 61st percentiles is greater than the absolute distance between the 90th and 91st percentiles. Nonparametric statistics are typically used with ordinal scale numbers where the median is used as the measure of central tendency.

Interval: Numbers that have equal distance between units (2 is half of 4 ... 500 half of 1,000). We assume that the measures obtained on standardized tests are interval scale numbers or at least closely approximate the interval scale. Statistical procedures with interval scale numbers are called Parametric and use the mean as the measure of central tendency.

Ratio: Numbers with a true 0 and equal distance between units (as in interval numbers). As we measure constructs of human creation in the social and behavioral sciences (e.g., intelligence, anxiety, depression, aptitudes) we do not have true ratio scale numbers. There is no such thing, for example, as zero intelligence. There is only comparatively more or less of this construct.

Mean: Arithmetic average of a distribution (group of numbers). Generally the best central tendency measure for interval scale numbers.

Median: 50th percentile or midpoint of any distribution. In the first distribution within parentheses the median is 6 (2, 4, 6, 6, 8, 9) and in the second 6.5 (2, 4, 6, 7, 8, 9). Generally the best central tendency measure for ordinal scale numbers.

Mode: Most frequently appearing number in a distribution. In the first distribution above the mode is 6. The second distribution does not have a mode. The following distribution has two modes, and is called a bimodal distribution (3, 4, 4, 5, 8, 9, 9, 10).

Range: The Inclusive Range is the highest score minus the lowest score in a distribution plus 1. If the highest score on an examination is 97 and the lowest score 65, the range is 33. The plus 1 correction captures the values from 97.49 to 64.50. The Exclusive Range is just the highest score minus the lowest score. In the above example 32.

Variance: Conceptually the average of the squared deviation scores from the mean of the distribution ... divided by the number of observations (N). When estimating the variance of a population from a very small sample we divide by N-1 because very small samples tend to underestimate the variance of the larger population they are drawn from. The N-1 correction obviously becomes less important and even irrelevant as sample size increases.

Standard Deviation: Square root of the variance. If the variance of a distribution is 25, the standard deviation is 5.

Raw Score: The number of actual points a person scores on a test or one of its scales. Because tests and even scales within the same test have different numbers of items, it is very difficult to compare raw scores. Raw scores mean nothing unless you also know the mean and standard deviation of the distribution they come from.

Standard Scores: A converted score (formerly a raw score) where the mean and standard deviation of a distribution have been set at certain values, and scores are expressed along that scale.

Z-Scores: A standard score where the mean of the distribution is set at 0 and standard deviation at 1. A subject with a z-score of 2.5 scored 2.5 standard deviations above the mean. Any raw score can be converted to a z-score by first subtracting it from the mean and then dividing by the standard deviation.

T-Scores: A standard score where the mean of the distribution is 50 and standard deviation 10. A subject with a T-score of 65 is 1.5 standard deviations above the mean. Many tests convert subject raw scores to T-scores so that scales with different numbers of items can be easily compared, and T-scores are superior to z-scores in that they avoid negative numbers (it is easier to tell a subject they scored 45 than -.5). Z-scores are converted to T-scores by multiplying them by 10 and then adding 50.

Percentile Ranks: Test results are often reported as a percentile standing in a distribution. A percentile rank of 85 means that the subject's performance was equal to or better than 85% of the people taking the test.

Normal Probability Curve: This is the famous bell shaped curve that is perfectly symmetrical on both sides. The mean falls directly at the center, and standard deviation bands fall at precisely known percentiles along the curve. This curve is at the very foundation of statistical inference because observations from large data sets tend to approximate it.

Bimodal Distribution: This is often called the camel back distribution because of its two high points. When it occurs, the data may be reflecting the performances of two different groups. The actual mean of a bimodal distribution falls somewhere between the two high points.

Skewed Distributions: The direction of the tail defines whether a distribution is positively or negatively skewed. A positively skewed distribution has its tail extended to the right (toward higher values), and a negatively skewed distribution has its tail going off to the left (toward lower values). When the observations are correct answers on a test, a strong negative skew is said to show a ceiling effect. The items were too easy for most of those who took the test, and the test could only discriminate among those with poor performances. The opposite of this is sometimes called a basement effect.

When a subject takes a test the resulting score they obtain is called the Observed Score. It consists of both a "Signal" which is a true and valid reading of what the test is trying to measure, and "Noise or Error". The quantification of the noise or error is the test's Standard Error of Measurement. If we could factor out all the noise and error in the observed score, and be left with only a valid signal, that would be the subject's True Score.

A Standard Error of Measurement (SEM) is a number, reported in the test manual, for each scale on the test and frequently for different groups of subjects. As an example, on the WAIS-R the overall SEM averages across age groups for Verbal, Performance and Full Scale IQs are reported as 2.74, 4.14 and 2.53, respectively. SEMs for the nine different age groups the WAIS-R is normed to are also reported for all three IQs.

SEMs are understood by using the normal probability curve. Instead of dividing the curve with standard deviation bands, we now divide it with SEM bands at the same percentile standings used with whole number standard deviations. The observed score for the subject is placed at the center of the curve. There is then approximately a 68% probability that the true score will fall within one SEM of the observed score (from -1 to 1 SEM). There is approximately a 95% probability that the true score falls within two SEMs of the observed score (-2 to 2). These are called the 68% and 95% Confidence Bands for the true score.

As examples, the following are 68% and 95% confidence bands for three subjects earning different scores on a test with an SEM of 4.

For those interested, the SEM of a test is calculated by multiplying the standard deviation of the test by the square root of 1 - r (where r is the test's reliability).

Correlations are how we measure the extent of a relationship between two sets of paired numbers. It is how the validity and reliability of tests are reported in their manuals. There are special correlations for dichotomous variables (where only a value of zero or one exists) and for ordinal numbers, but with interval scale numbers the correlation used almost universally is the Pearson Product-Moment Correlation.

Correlations can range in value from negative one (-1) to positive one (+1). A correlation of -1.0 means a perfect inverse relationship between two sets of data, and a correlation of +1.0 a perfect positive relationship. In a perfect inverse or negative relationship as every score drops in one distribution, every matched score proportionately rises in the other. In a perfect positive relationship the paired numbers move proportionally up or down in a perfectly matched relationship.

Correlation will show an association between two sets of data, but not cause and effect. A correlation of 0.0 means there is no association or relationship between the two groups of paired numbers. Any positive or negative correlation "technically" shows some degree of association, but as a general rule of thumb correlations between -.30 and +.30 are not significant (i.e. essentially the same as zero). In relation to criterion validity, tests start to have some minimal degree of predictive power at .40, and reliability correlations should exceed .80.

To obtain a VERY conservative estimate of the percent of shared variance between the two sets of data one squares the correlation. A correlation of .60 would then suggest that at least 36% of what contributes to variation in one group of numbers contributes to variation in the other.

When we know the correlation between two sets of data we also know their means and standard deviations (or we could not have calculated the correlation), and this information is all that is needed to plot a regression line within an XY axis that allows us to predict one measure (number) from the other. The accuracy of our ability to predict one number from the other will increase as the correlation between the two sets of data becomes stronger. Multiple Regression is a statistical procedure designed to predict a single dependent variable using corelations obtained with a number of other variables. Stepwise Multiple Regression explains as much variance as possible by the best predictor, then moves to the second until its predictive power is exhausted, the third, and so forth.

In Factor Analysis you are looking to find a much smaller number of underlying factors or qualities that are the essence of many separate and distinct measurements you have obtained. You first need a correlation matrix of all the measures you are looking at for underlying structure (i.e. natural groupings of the different measures).

Imagine a correlation matrix between five different variables: A, B, C, D, and E. These might, for example, be items on an experimental test. Assume next that A & B are highly correlated with each other, C & D highly correlated with each other (but not with A or B), and E not highly correlated with any of the other four variables. Factor Analysis would likely deduce that A & B are tapping into the same underlying factor or quality, C & D the same with a second factor or quality, and E reflecting still a third factor or quality. By this method a large number of items on a test might be reduced to four or five underlying factors. These might then be turned into Scales.

Usually in psychology, factor analysis is a way to group (sort together) intercorrelated measures to identify a smaller number of underlying factors present within the data being analyzed.