Stats Chapter 4: Measures of Variability

Standard Deviation and Variance for a Population

The standard deviation is the most commonly used and the most important measure of variability. Standard deviation uses the mean of the distribution as a reference point and measures variability by considering the distance each score and the mean. It provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered around the mean or are widely scattered.

The Process of Computing Standard Deviation: Step 4

Remember that our goal is to compute a measure of the standard distance from the mean. Variance, which measures the average squared distance from the mean, is not exactly what we want. The final step takes the square root of the variance to obtain the *standard deviation*, which measures the standard distance from the mean.

The Range

The distance covered by the scores in a distribution, from the smallest score to the largest. When the scores are measurements of a continuous variable, the range can be defined as the difference between the upper real limit (URL) or the largest score (Xmax) and the lower real limit (LRL) for the smallest score (Xmin). When the scores are whole numbers, this definition of range is also a measure of the number of measurement categories which also works for discrete variables that are measured with numerical scores.

Deviation

The distance from the mean. There are two parts to a deviation score: the sign (+ or -) and the number. The sign tells the direction of the mean (whether the score is above (+) or below (-) the mean). The number gives the actual distance from the mean.

The Problem with using the Range

The problem with using the range as a measure of variability is that it is completely determined by two extreme values and ignore the other scores in the distribution. Thus, a distribution with one unusually large score has a large range even if the other scores are all clustered together. Therefore, it often does not give an accurate description of the variability for the entire distribution. For this reason, the range is considered to be a crude and unreliable measure of variability.

Standard Deviation

The square root of the variance and provides a measure of the standard, or average, distance from the mean. Because the standard deviation and variance are in terms of distance from the mean, these measures of variability are used only with numerical scores that are obtained from measurements on an interval or ratio scale. Because the mean is a critical component in the calculation of standard deviation and variance, the same restrictions that apply to the mean also apply the these two measures of variability. Specifically, the mean, the standard deviation, and the variance should be used only wit numerical scores from interval or ordinal scales of measurement.

Variability

Variability provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out to clustered together. The purpose for measuring variability is to obtain an objective measure of how the scores are spread out in a distribution. There are three different measures of variability: the range, standard deviation, sonf the variance. Of these three, the standard deviation and the related measure of variance are the most important.

The Process of Computing Standard Deviation: Step 2

Because out goal is the compute a measure of the standard distance from the mean, the next step is to calculate the mean of the deviation scores. To compute this mean, you add up the deviation scores and divide by N. Note that the deviation scores add up to 0 since the mean serves as a balance point for the distribution and so is no measure of variability.

Population Variance

Equals the mean squared deviation. Variance is the average squared distance from the mean.

Q: Briefly explain what is measured by the standard deviation and variance.

Standard deviation measures the standard distance from he mean, and variance measures the average squared distance from the mean.

The Process of Computing Standard Deviation: Step 3

The average of the deviation scores does not work as a measure or variability because it is always zero as a result of the positive and negative values cancelling each other out. The solution is to get rid of the signs accomplished by squaring the values, then you compute the *mean square deviation* (also called *population variance*).

Q: Formulas for Population Variance and Standard Deviation

The concepts of standard deviation and variance are the same for both populations and samples however, details in the calculations differ slightly depending on whether you have data from a sample or a complete population.

The Process of Computing Standard Deviation

1. Determine the deviation, or distance from the mean, for each individual score. 2. Calculate the mean of the deviation scores (Add up the deviation scores and divide by N). 3. Square each deviation score to compute the mean square deviation (or population variance) 4. Take the square root of the variance to obtain the standard deviation, which measures the standard distance from the mean.

Purpose of Good Measures of Variability

1. Variability describes the distribution telling whether the scores are clustered together or are spread out. Usually variability is defined in terms of *distance*. 2. Variability measures how well an individual score (or group of scores) represents the entire distribution and provides information about how much error to expect if you are using a sample to represent a population.

Alternative Range Definition

A commonly used alternative definition of the range simply measures the distance between the largest score (Xmax) and the smallest score (Xmin), without any reference to real limits. For discrete variables, which do not have any real limits, this definition is considered inappropriate. This definition also works for variables with precisely defined upper and lower boundaries. Using either definition, the range is the most obvious way to describe ho spread out scores are.