Chapter 4 Measures of Variability

4.1 Overview Variability can be defined several ways Purposes of Measure of Variability

Figure 4.1 Population Distributions FIGURE 4.1 Population distribution of adult heights and weights.

Three Measures of Variability The Range The Variance The Standard Deviation

4.2 The Range The distance covered by the scores in a distribution URL=Upper Real Limit; LRL=Lower Real Limit. Some instructors may want to point out that for discrete variables, the range is more accurately defined as the maximum score minus the minimum score.

4.3 Standard Deviation and Variance for a Population Most common and most important measure of variability is the standard deviation A measure of the standard, or average, distance from the mean Describes whether the scores are clustered closely around the mean or are widely scattered Calculation differs for population and samples Variance is a necessary companion concept to standard deviation but not the same concept The twin concepts of variance and standard deviation are among the most challenging concepts in a basic statistics course to communicate and to learn. Instructors will almost certainly want to invest special care in the preparation of materials to help communicate these very difficult concepts.

Defining the Standard Deviation Step One: Determine the Deviation Deviation is distance from the mean Step Two: Find a “sum of deviations” to use as a basis of finding an “average deviation” Two problems Deviations sum to 0 If sum always 0, “Average Deviation” will always be 0. Need a new strategy! Deviation score = Having students try to come up with an intuitive method for developing a measure of variability based on deviation scores is a great way to get them thinking about what a dead end strategy averaging deviations is. Several teams working on it in a classroom exercise often results in a valuable insight about the issue (averaging absolute value of deviations) and might produce the one we use—squaring the deviations to eliminate the negative values.

Defining the Standard Deviation (continued) Step Two Revised: Remove negative deviations First square each deviation score Then sum the Squared Deviations (SS) Step Three: Average the squared deviations Mean Squared Deviation is known as “Variance” Variability is now measured in squared units The concept is sum of squared deviations (SS) is absolutely vital to efficient understanding fo statistical tests presented in the remainder of the text. The authors have reduced the computational complexity and the cognitive load required of students—contingent upon grasping and retaining the concept of SS presented in this chapter. The authors also lay the foundation for efficiently learning the fundamentals of ANOVA—contingent upon grasping and retaining the concept of variance presented in this chapter. Consequently, this chapter is essential to success in the remainder of the course.

Defining the Standard Deviation (continued) Step Four: Goal: to compute a measure of the “standard” (average) distance of the scores from the mean Variance measures the average squared distance from the mean—not quite our goal Adjust for having squared all the differences by taking the square root of the variance Variance (in squared distance units) is not intuitively easy to grasp despite being a measure of average squared distance of scores from the mean. Consequently, it is important to emphasize the need to take the square root of the variance to return it to the same distance unit used in the original measurement procedure.

Figure 4.2 FIGURE 4.2 The calculation of variance and standard deviation.

Population Variance Formula sum of squared deviations Variance = number of scores Some instructors may want to provide an introduction to the next slide by emphasizing that the two formulas produce the same answer only when there is no rounding error.

Two formulas for SS Definitional Formula Computational Formula Find each deviation score (X–μ) Square each deviation score, (X–μ)2 Sum up the squared deviations Square each score and sum the squared scores Find the sum of scores, square it, divide by N Subtract the second part from the first Some instructors may want to flag these formulas as particularly likely to produce confusion and that after students learn the definition of SS and practice using it on a very simple set of numbers, they might want to make a note that the definitional formula is particularly time-consuming and error-prone when analyzing real data.

Population Variance: Formula and Notation Variance is the average of squared deviations, so we identify population variance with a lowercase Greek letter sigma squared: σ2 Standard deviation is the square root of the variance, so we identify it with a lowercase Greek letter sigma: σ

Learning Check Decide if each of the following statements is True or False. T/F The computational & definitional formulas for SS sometimes give different results If all the scores in a data set are the same, the Standard Deviation is equal to 1.00

4.4 Standard Deviation and Variance for a Sample Goal of inferential statistics: Samples differ from the population Samples have less variability Computing the Variance and Standard Deviation in the same way as for a population would give a biased estimate of the population values

Figure 4.4 Population of Adult Heights FIGURE 4.4 The population of adult heights forms a normal distribution. If you select a sample from this population, you are most likely to obtain individuals who are near average in height. As a result, the scores in the sample will be less variable (spread out) than the scores in the population.

Sample Variance and Standard Deviation Sum of Squares (SS) is computed as before Formula for Variance has n-1 rather than N in the denominator Notation uses s instead of σ

Learning Check A sample of four scores has SS = 24. What is the variance? A The variance is 6 B The variance is 7 C The variance is 8 D The variance is 12

Variance and Inferential Statistics Goal of inferential statistics is to detect meaningful and significant patterns in research results Variability in the data influences how easy it is to see patterns