# APAS/Wizard Statistics

## Alpha

Statisticians use the Greek letter alpha () to indicate the probability of rejecting the statistical hypothesis tested when in fact, that hypothesis is true. Before conducting any statistical test, it is important to establish a value for alpha. For most psychologists, and for many other scientists, it is customary to set alpha at 0.05.

This is the equivalent of asserting that you will reject the hypothesis tested if the obtained statistic is among those that would occur only 5 out of 100 times that random samples are drawn from a population in which the hypothesis is true. If your obtained statistic leads you to reject the hypothesis tested, it's not because you believe that the obtained statistic could not have occurred by chance.

It's that you are asserting that the odds of obtaining that statistic by chance only are sufficiently low (one out of twenty) that it reasonable to conclude that your results are not due to chance. Could you be in error? Of course you could, but at least you know the probability of such an error. It is exactly equal to the value you have previously established for alpha.

## Alternative Hypothesis

The test of a given statistical hypothesis entails an assessment of whether or not our sample (or samples) have yielded a statistic that is among those cases that would only occur alpha proportion of the time if the hypothesis tested is true.

In these circumstances we know the probability of rejecting the hypothesis tested when it is true (that probability is equal to alpha) but unless we have also specified an alternative hypothesis to the hypothesis tested, we have no idea of the probability of being in error, if our test has failed to yield a value that enables us to reject the hypothesis tested.

## Beta

Statisticians use the Greek letter beta () to indicate the probability of failing to reject the hypothesis tested when that hypothesis is false and a specific alternative hypothesis is true. For a given test, the value of beta is determined by the previously elected value of alpha, certain features of the statistic that is being calculated (particularly the sample size) and the specific alternative hypothesis that is being entertained. While it is possible to carry out a statistical test without entertaining a specific alternative hypothesis, neither beta nor power can be calculated if there is no specific alternative hypothesis. It is relevant to note here that power ( the probability that the test will reject the hypothesis tested when a specific alternative hypothesis is true ) is always equal to one minus beta (i.e. Power = 1 - beta).

## Coefficient of Variation (cov) *

Standard deviation / mean...

## Consistency of Effort (coe) *

## % Deficit to Norm (def0) *

## % Deficit to Opposing (def1)

## Degrees of Freedom (df)

Statisticians use the terms "degrees of freedom" to describe the number of values in the final calculation of a statistic that are free to vary. Consider, for example the statistic s-square.

To calculate the s-square of a random sample, we must first calculate the mean of that sample and then compute the sum of the several squared deviations from that mean. While there will be n such squared deviations only (n - 1) of them are, in fact, free to assume any value whatsoever. This is because the final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample. All of the other (n - 1) squared deviations from the mean can, theoretically, have any values whatsoever. For these reasons, the statistic s-square is said to have only (n - 1) degrees of freedom.

## Examinee's Best Effort (max)

TODO...

## Examinee Sample (mean)

The mean of a random sample is an unbiased estimate of the mean of the population from which it was drawn. Another way to say this is to assert that regardless of the size of the population and regardless of the size of the random sample, it can be shown (through The Central Limit Theorem) that if we repeatedly took random samples of the same size from the same population, the sample means would cluster around the exact value of the population mean.

## Examinee's Worst Effort (min)

TODO...

## H0: Examinee = Norm

The null hypothesis being tested is that the Examinee performs "at least as good" as the Norm.

The null hypothesis is a term that statisticians often use to indicate the statistical hypothesis tested. The purpose of most statistical tests, is to determine if the obtained results provide a reason to reject the hypothesis that they are merely a product of chance factors. For example, in an experiment in which two groups of randomly selected subjects have received different treatments and have yielded different means, it is always necessary to ask if the difference between the obtained means is among the differences that would be expected to occure by chance whenever two groups are randomly selected. In this example, the hypothesis tested is that the two samples are from populations with the same mean. Another way to say this is to assert that the investigator tests the null hypothesis that the difference between the means of the populations from which the samples were drawn, is zero. If the difference between the means of the samples is among those that would occur rarely by chance when the null hypothesis is true, the null hypothesis is rejected and the investigator describes the results as statistically significant.

## H1: Side = Opposing Side

The null hypothesis being tested is that the Examinee performs "at least as good" on the tested side as on the opposing side. For more information see H0.

## Level of Confidence

The level of confidence ... = 1 - alpha.

## Norm (mean)

The mean is one of several indices of central tendency that statisticians use to indicate the point on the scale of measures where the population is centered.

The mean is the average of the scores in the population. Numerically, it equals the sum of the scores divided by the number of scores. It is of interest that the mean is the one value which, if substituted for every score in a population, would yield the same sum as the original scores, and hence it would yield the same mean.

## Population

Statisticians define a population as the entire collection of items that is the focus of concern. The branch of Statistics called "Descriptive Statistics" provides us with ways to describe the characteristics of a given population by measuring each of its items and then summarizing the set of measures in various ways.

The branch of Statistics called "Inferential Statistics" consists of procedures to make educated inferences about the characteristics of a population by drawing a random sample and appropriately analyzing the information it provides.

A population can be of any size and while the items need not be uniform, the items must share at least one measurable feature. For example here is a population of 9 persons. While no two of the persons are identical they have many features in common. Each of the persons in this population has a weight, a height, a hat size and a shoe size, among many other potential features. The set of 9 measurements of any one of these features would, in statistical terms, be defined as a population.

The critical difference between a population and a sample, is that with a population our interest is to identify its characteristics whereas with a sample, our interest is to make inferences about the characteristics of the population from which the sample was drawn.

## Power

For a statistician, the power of a test is the probability that the test will reject the hypothesis tested when a specific alternative hypothesis is true. To calculate the power of a given test it is necessary to specify alpha (the probability that the test will lead to the rejection of the hypothesis tested when that hypothesis is true) and to specify a specific alternative hypothesis.

## Probability (p)

## Range of Effort (roe)

## Range of Motion (rom)

## Standard Deviation (sd)

The standard deviation is one of several indices of variability that statisticians use to characterize the dispersion among the measures in a given population.

To calculate the standard deviation of a population it is first necessary to calculate that population's variance. Numerically, the standard deviation is the square root of the variance. Unlike the variance, which is a somewhat abstract measure of variability, the standard deviation can be readily conceptualized as a distance along the scale of measurement.

## Student T-Test

The Student T-Test employs the statistic (t) to test a given statistical hypothesis about the mean of a population (or about the means of two populations). A one-sided t-test is used to test if the sample mean is significantly different from a particular value:

With s:

Check here for more information

## Trial Consistency

An indication of how consistent the examinee was in performing the test.

## T Statistic

This statistic is a measure on a random sample (or pair of samples) in which a mean (or pair of means) appears in the numerator and an estimate of the numerator's standard deviation appears in the denominator. The later estimate is based on the calculated s square or s squares of the samples.

If these calculations yield a value of (t) that is sufficiently different from zero, the test is considered to be statistically significant.

## Type I Error

You have committed a Type One error if you have rejected the hypothesis tested when it was true.

In a given statistical test, the probability of a type 1 error is equal to the value you have set for alpha.

## Type II Error

You have committed a Type II error if you failed to reject the hypothesis tested when a given alternative hypothesis was true.

In a given statistical test, the probability of a type II error is equal to the value calculated for Beta.