Measures of VariationB-324Aragon, Prince AllenArcangel, Flitzer GlennBernabe, KristianCaspe, Phamela JaneDichoso, Lhen TopicsIntroduction and InformationThe RangeThe Inter-Quartile RangeThe Semi-Interquartile Range or Quartile DeviationThe Mean Deviation or Average DeviationThe VarianceThe Standard Deviationmeasures of variationQuantities that express the amount of variation in arandom variable(comparemeasures of location). Variation is sometimes described asspreadordispersionto distinguish it from systematic trends or differences. Measures of variation are either properties of aprobability distributionor sample estimates of them.

Therangeof a sample is the difference between the largest and smallest value. Theinterquartile rangeis potentially more useful. If the sample is ranked in ascending order of magnitude two values ofxmay be found, the first of which is exceeded by 75% of the sample, the second by 25%; their difference is the interquartile range. An analogous definition applies to a probability distribution.

Introduction and InformationThevarianceis the expectation (or mean) of the square of the difference between arandom variableand its mean; it is of fundamental importance in statistical analysis. The variance of a continuous distribution with mean

The variance of a discrete distribution

The sample variance of a sample ofnobservations with meanxis

and is denoted bys2. The value (n 1) corrects forbias.

Thestandard deviationis the square root of the variance, denoted by (for a distribution) ors(for a sample). The standard deviation has the same units of measurement as the mean, and for a normal distributionabout 5% of the distribution lies beyond about two standard deviations each side of the mean. The standard deviation of the distribution of an estimated quantity is termed thestandard error.

Themean deviationis the mean of the absolute deviations of the random variable from the mean.

The RangeThe Range is the simplest to compute, is the difference between the largest and the lowest value of numerical data.The Range for :Ungrouped data is obtained by finding the difference between the largest value and the lowest value.Grouped data is determined by subtracting the lower boundary of the lowest class interval from the upper boundary of the highest class interval of a frequency distribution.The class boundaries are considered the true limits.

The RangeExample 1: (Ungrouped Data)The scores obtained by 10 students in History Class are 87,80,78,93,85,75,90,79,98,76. Find the range.Solution:R = HV-LV= 98-75R = 23Example 2: (Grouped Data)Find the range of a given frequency distribution whose highest class interval is 98-94 and lowest class interval is 60-65.Solution:R = UB(hci) LB (lci)= 98.5-59.5R = 39

The Inter-Quartile RangeWe learned in the preceding chapter that the quartile divides the distribution numerical data into 4 equal parts. The first or lower quartile lies on the 25% of the total number of values, while the third or the upper quartile is on the 75%.

The quartile range (IQR) is found by finding the difference between the values of the third quartile (Q3) upper and the first quartile (Q1) or the lower quartile.we have IQR = Q3 Q1

With an Even Sample Size:

For the sample (n=10) the median diastolic blood pressure is 71 (50% of the values are above 71, and 50% are below).The quartiles can be determined in the same way we determined the median, except we consider each half of the data set separately.There are 5 values below the median (lower half), the middle value is 64 which is the first quartile.There are 5 values above the median (upper half), the middle value is 77 which is the third quartile.The interquartile range is 77 64 = 13; the interquartile range is the range of the middle 50% of the data.

With an Odd Sample Size:When the sample size is odd, the median and quartiles are determined in the same way.Suppose in the previous example, the lowest value (62) were excluded, and the sample size was n=9. The median and quartiles are indicated below.

When the sample size is 9, the median is the middle number 72.The quartiles are determined in the same way looking at the lower and upper halves, respectively. There are 4 values in the lower half, the first quartile is the mean of the 2 middle values in the lower half ((64+64)/2=64). The same approach is used in the upper half to determine the third quartile ((77+81)/2=79).

What is the Inter-Quartile for the following set of numbers:13,18,6, 20, 25, 11, 9, 18, 3, 30, 16, 9, 8, 23, 26, 17 Q1=9 Q2=16 Q3=21.5

Q1 is the mean of 9 and 9 = (9+9)/2 = 9Q2 is the mean of 16 and 17 = (16+7)/2 = 16Q3 the mean of 20 and 23 = (20+23)/2 = 21.5The Semi-Interquartile Range or Quartile DeviationIndicates the variation or dispersion of the values covering the middle 50% of the distribution of the data.It is found by getting half of the value or distance between the third quartile or upper quartile and the first quartile or lower quartile.Appropriate measure of variation only if the median is the one that is used as the measure of central tendency, and specially if the distribution is skewed.

The Mean Deviation or Average DeviationTakes into account the Deviations of the individual values from the mean.Calculate using the absolute values.

Example:The number of Cellphone units sold by an Gadget store for a 10-day period are: 7,10,8,13,8,12,5,13,9 and 11.

THE MEAN DEVIATION OR AVERAGE DEVIATIONThe VarianceThe use of the absolute values for the mean deviation is particularly done to avoid having negative deviations.Two most important measures of variability, called the variance and its corresponding square root. The variance is defined as the average of the squared deviations from the mean.The square root of this variance is known as the standard deviation.

The Standard DeviationThe most important measure of variation.We will be able to determine the Position of the scores in a frequency distribution in relation to the mean.A standard deviation of a small value means that the values, in a distribution are scattered or spread out near the mean and vice versa.

Example: The 10 sample data, showing values under x-x and (x-x)x = x n= 5 + 6 + 7 + 8 + 9 + 9 + 10 + 11 + 12 + 13 = 90 10 10

x = 9And the variance,S = (x-x)n-1 = 81.5 10-1S = 9.05THE VARIANCETHE STANDARD DEVIATION