Ratio Statistics: Descriptive Statistics for the Ratio of Two Variables, Study notes of Mathematical Statistics

Download Ratio Statistics: Descriptive Statistics for the Ratio of Two Variables and more Study notes Mathematical Statistics in PDF only on Docsity! 1 Ratio Statistics This procedure provides a variety of descriptive statistics for the ratio of two variables. Notations The following notation is used throughout this chapter unless otherwise stated: n Number of observations Ai Numerator of the I-th ratio (i = 1, …, n). This is usually the appraisal roll value. Si Denominator of the i-th ratio (i = 1, …, n). This is usually the sale price. Ri The i-th ratio (i = 1, …, n). Often called the appraisal ratio. fi Case weight associated with the i-th ratio (i = 1, …, n). Data This procedure requires for i = 1, …, n that: • 0>iA , • 0>iS , • 0>if , and • iw is a whole number. If the SPSS Weight variable contains fractional values, then only the integral parts are used. A case is considered valid if it satisfies all four requirements above. This procedure will use only valid cases in computing the requested statistics. Ratio Statistics Ratio R A S i ni i i = =, , ,1 K Minimum The smallest ratio and is denoted by minR . 2 Maximum The largest ratio and is denoted by maxR . Range The difference between the largest and the smallest ratios. It is equal to minmax RR − . Median The middle number of the sorted ratios if n is odd. The mean (average) of the two middle ratios if the n is even. The median is denoted as R ~ . Average Absolute Deviation (AAD) ∑∑ == −= n i i n i ii fRRfAAD 11 ~ Coefficient of Dispersion (COD) R AAD COD ~%100 ×= Coefficient of Concentration (COC) Given a percentage 100% × g, the coefficient of concentration is the percentage of ratios falling within the interval ( ) ( )[ ]RgRg ~1,~1 +− . The higher this coefficient, the better uniformity. Mean ∑∑ == == n i i n i ii fRfRSA 11 Standard Deviation (SD) ( ) ( )∑= − − = n i ii RRfF s 1 2 1 1 where ∑ = = n i ifF 1 .