Measures of Variability Vernon Enage Reyes Review Measures

Measures of Variability Vernon Enage Reyes

Review • Measures of central tendency 1. Mean 2. Median 3. Mode However, measures of central tendency shows an incomplete picture

Example • Honolulu, Hawaii Mean daily temperature = 75 degrees • Phoenix, Arizona Mean daily temperature = 75 degrees However: Honolulu’s temperature range between 70 – 80 degrees Arizona’s temperature range between 40 degrees (January) to 100 degrees (July and August)

Measures of Variability An index of how the scores are scattered around the center of the distribution (measure of central tendency – usually the mean) Variability is also called: spread, width or dispersion Well know measures of variability: the range, mean deviation, the variance and standard deviation

1. The RANGE • A quick and easy way to get a rough measure of variability. The Range (R) is the difference between the highest and lowest score of distribution • Honolulu hottest temp = 89 degrees • Honolulu coldest temp = 65 degrees Range = 24 degrees Formula is. . . R = H - L

Disadvantage • Although the range is quick-and-easy, it is DEPENDENT only on two scores (the highest and lowest), thus it shows inaccurate data. Example: Prof A: 18, 19, 20, 22 and 23 Prof B: 18, 19, 20, 21 and 43 Thus scores are only affected only by ONE student.

2. The Mean Deviation • The mean deviation was discussed last time. • Remember: the deviation is the distance of any given raw score from the mean. (X – X) • Remember also, if we add the deviations it will equal to zero

Example ---------------X X–X --------------9 +3 +5 8 +2 6 0 5 -1 -5 2 -4 X=6 • Notice that if we add all the deviations it will always equal to zero! • (+)5 + (-)5 = 0

2. The Mean Deviation • The mean deviation was discussed last time. • Remember: the deviation is the distance of any given raw score from the mean. (X – X) • Remember also, if we add the deviations it will equal to zero • So what we do is convert the deviations into “absolute deviations”(we just ignore the plus and minus) then add them.

Mean deviation • Formula MD = Σ |X – X| N MD = mean deviation Σ |X – X| = sum of the absolute deviation (ignoring +/- signs) This is important to not get a Zero N = total number of scores

To get the mean deviation • Formula MD = Σ |X – X| N = 16 6 MD = 2. 67

The Variance and Standard Devition • The Mean deviation is NOT anymore widely used because it can not be manipulated algebraically • Also it can not be used to advanced statistical procedures (inferential statistics) • Thus what we do is just square the deviation instead of converting it into absolute values This also removes the negative and positive sign but can be algebraically manipulated

Variance • Formula s 2 = Σ (X – X)2 N

What for? Distribution A -----------------X |X – X | (X – X)2 3 2 4 5 0 0 7 2 4 X=5 MD = 1 S 2 = 2 Distribution B -----------------X |X – X | (X – X)2 4 1 1 6 1 1 X=5 MD = 1 S 2 = 1

What for? Graph form Distribution A Distribution B _____________________ 2 3 4 5 6 7 8

Standard deviation • However, since the values has been squared, we need to turn it back to its original unit of measurement, that is, we need to take the square root of the variance • Formula Variance s 2 = Σ (X – X)2 N Standard deviation s = Σ (X – X)2 N √

Standard deviation an easier formula Standard deviation s = s 2 = Σ X 2 / N – X 2 s= √ Σ X 2 / N - X 2 √ Σ (X – X)2 N X 2 = sum of squared raw scores ( each raw score is squared FIRST, then summed) N = total number of score X 2 = mean squared

In the end • We shall discuss further how the standard deviation is useful next week. • Note however that deviations, are just like mean, median and mode • Mean/median/mode gives a single score to describe a group • Variance/standard deviations gives a single score to describe (how close or how far are from the mean) the scores of a group.