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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. S (^) i 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. f (^) i Case weight associated with the^ i -th ratio^ ( i^ = 1, …,^ n ).
Data
This procedure requires for i = 1, …, n that:
- Ai > 0 ,
- Si > 0 ,
- f (^) i > 0 , and
- wi 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 i i^ n i
= , = 1 , K,
Minimum
The smallest ratio and is denoted by R min.
Maximum
The largest ratio and is denoted by R max.
Range
The difference between the largest and the smallest ratios. It is equal to R max (^) − R min.
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
AAD fi Ri R f 1 1
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 [( g ) R ( g ) R ]
1 − +. The higher this coefficient, the better uniformity.
Mean
∑ ∑ = =
n
i
i
n
i
AS R fi Ri f 1 1
Standard Deviation (SD)
( ) ∑ (^ ) =
n
i
fi Ri R F
s 1
2 1
where (^) ∑
n
i
F fi 1
An equivalent formula is
( ) (^) ∑
−
=
1
0
- 5 2
r
k
n (^) k
n I n r r α .
Since the rightmost term is the cumulative Binomial distribution and it is discrete, r is solved as the largest value such that
∑
−
=
1
(^20)
r
k
n (^) k
α n .
Thus the confidence interval has coverage probability of at least 1 − α.
Normal Distribution
Assuming the ratios follow a normal distribution, a two-sided 100%× ( 1 −α) confidence interval for the median of a normal distribution is
( R + g (α 2 ; 0. 5 , d ) × s , R + g ( 1 − α 2 ; 0. 5 , d )× s )
where g ( (^) γ ; p , d )are values defined in Table 1 of Odeh and Owen (1980).
The value g ( (^) γ ; p , d )is, in fact, the solution to the following equations:
Pr^ ( T^ d ≤ g n |δ = Kp n )=γ
with Td follows a noncentral Student t -distribution where d is degrees of freedom associated with the standard deviation s , δ is noncentrality parameter, γ is the probability, n is the sample size, and K (^) p is the upper p percentile point of a standard normal distribution.
Confidence Interval for the Mean
The normal distribution is used to approximate the distribution of the ratios. The 100%× ( 1 −α)confidence interval for the mean is:
R t s f F
n
i
F ∑ i
± − × ×
1
2 α 2 ; 1
where t α 2 ; F − 1 is the upper α 2 percentage point of the t distribution with F − 1 degrees of
freedom, and where (^) ∑
n
i
F fi 1
Confidence Interval for the Weighted Mean
Using the Delta method, variance of the weighted mean is approximated as
( ) ( ) ( ) 4
2 2 3
var 2 cov , var var S
A S
S
A AS
S
A
S
A
where
( ) ( )
( ) 2 1
2 1
2 1
var f A A f F F
A
n
i
i
n
i
∑ i^ i ∑ = =
− ×
( ) ( )
( ) 2 1
2 1
2 1
var f S S f F F
S
n
i
i
n
i
∑ i^ i ∑ = =
− ×
= , and
( ) ( )
( )( ) 2 1
2 (^11)
cov , f A A S S f F F
AS
n
i
i
n
i
∑ i^ i i ∑ = =
− − ×
References
International Association of Assessing Officers (1990). Property Appraisal and Assessment Administration. International Association of Assessing Officers: Chicago, Illinois. Odeh, Robert E., and Owen, D. B. (1980). Tables for Normal Tolerance Limits, Sampling Plans, and Screening. Marcel Dekker: New York.
