QUANTITATIVE ANALYSIS

Quantitative Analysis



Quantitative Analysis

Measures of Central Tendency

Several different measures of central tendency are defined below.

* The mode is the most frequently appearing value in the population or sample. Suppose we draw a sample of five women and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds. Since more women weigh 100 pounds than any other weight, the mode would equal 100 pounds.

* To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Thus, in the sample of five women, the median value would be 130 pounds; since 130 pounds is the middle weight.

* The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five women, the mean weight would equal (100 + 100 + 130 + 140 + 150)/5 = 620/5 = 124 pounds.

Dispersion (Measures of):

Measures of dispersion express quantitatively the degree of variation or dispersion of values in a population or in a sample . Along with measures of central tendency , measures of dispersion are widely used in practice as descriptive statistics . Some measures of dispersion are the standard deviation , the average deviation , the range , the interquartile range .

For example, the dispersion in the sample of 5 values (98,99,100,101,102) is smaller than the dispersion in the sample (80,90,100,110,120), although both samples have the same central location - "100", as measured by, say, the mean or the median . Most measures of dispersion would be 10 times greater for the second sample than for the first one (although the values themselves may be different for different measures of dispersion).

It is important from a practical standpoint that measures of dispersion are normally constructed to be shift invariant and scale invariant . If a measure is not scale invariant, for example, then the value of dispersion might depend on the units of measurement. For example, say the value of dispersion of prices of a particular CD-player model across a country is $10. If the measure of dispersion is scale-invariant and you convert all the prices from dollars to cents by multiplying them by 100, then the measure of dispersion will change from 10 (dollars) to 1000 (cents).

Quartiles

If we divide a cumulative frequency curve into quarters, the value at the lower quarter is referred to as the lower quartile, the value at the middle gives the median and the value at the upper quarter is the upper quartile.

A set of numbers may be as follows: 8, 14, 15, 16, 17, 18, 19, 50. The mean of these numbers is 19.625 . However, the extremes in this set (8 and 50) distort the range. The inter-quartile range is a method of measuring the spread of the ...