Outline

Introduce the concept of variability

Discuss measures of variability, and how they are calculated

            Range     

Interquartile Range

Variance

Standard deviation

VARIABILITY

A score by itself is meaningless.  It takes on meaning only when it is compared with other scores.

If we know the mean of a distribution of a given variable we can determine:

·        Whether a particular score is higher or lower than the mean.

• How much higher or lower

The mean provides a limited amount of information.

To describe a distribution more fully we require additional information concerning the dispersion or variability of scores about the measure of central tendency.

VARIABILITY: The extent to which scores deviate from the measure of central tendency (usually the mean)

Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.

MEASURES OF SPREAD

RANGE

• The simplest and least informative measure of spread.

• Scores between extremes are not taken into account.

• Very sensitive to extreme scores.

Calculation: URLx_max – LRLx_min  (the difference between the upper real limit of the highest value and the lower real limit of the lowest value)

SEMI-Interquartile range (semi-IQR)

• Less sensitive than the range to extreme scores.

• Used when you want a simple, rough estimate of spread.

• Typically used when the median is reported as a measure of central tendency.

Calculation:       1. Put the data in numerical order

2. Divide the data into two equal groups at the median

3.       Find the median of the low group.  This is the first quartile, or Q1.

4.       Find the median of the high group.  This is the third quartile or Q2.

The semi-Interquartile range is the distance between them divided by 2

Semi-IQR = Q3 – Q1 / 2

Figure 4-3  (p. 107)
Frequency distribution for a population of N = 16 scores. The first quartile is Q1 = 4.5. The third quartile is Q3 = 8.0. The interquartile range is 3.5 points. Note that the third quartile (Q3) divides the two boxes at X = 8 exactly in half, so that a total of 4 boxes are above Q3 and 12 boxes are below it.

VARIANCE

• Average squared deviation of scores from the mean.

STANDARD DEVIATION

• The square root of the variance.  This is the most widely used measure of spread.

• Standard deviation uses the mean (M) of the distribution as a reference point and measures variability by considering the distance between each score and the mean.

• It approximates the average distance from the mean

The standard deviation describes the typical distance of scores from the mean.  It measures the standard, or typical, deviation score. It provides a measure of the typical distance.

This is primarily a descriptive measure. 

Figure 4-4  (p. 114)
A frequency distribution histogram for a population of N = 5 scores. The mean for this population is µ = 6. The smallest distance from the mean is 1 point, and the largest distance is 5 points. The standard distance (or standard deviation) should be between 1 and 5 points.

CONCEPTUAL STEPS IN THE CALCULATION OF THE VARIANCE AND STANDARD DEVIATION

1.       Find the average distance from the mean by determining the deviation of each score from the mean.

2.       Since we want average distance, we need to calculate something equivalent to the average of deviation scores. In an average, typically we add up the scores and divide.

PROBLEM: Deviation scores add up to zero (because the mean is the center of gravity)

3.       SOLUTION: Get rid of the positive and negative signs by squaring each deviation score. That gives us what we call sums of squares, or SS.  Then compute the mean squared deviation (mean square, or MS).  So variance = SS / N

4.       Since variance is not exactly what we want (average distance from the mean), we correct for having squared all the distances by taking the square root of the variance.  This is the STANDARD DEVIATION.

Figure 4-5  (p. 116)
The graphic representation of a population with a mean of µ = 40 and a standard deviation of σ = 4.