Relative Variation, Normality, and a Sales Statistics Case

Relative Variation, Normality, and a Sales Statistics Case

The Normal Distribution and the Empirical Rule

The Normal Distribution (Gaussian bell curve) is a symmetrical, continuous probability distribution where data clusters symmetrically around the mean.

For a normally distributed dataset, the Empirical Rule (68-95-99.7 Rule) establishes fixed mathematical boundaries:

• 68.27% of all observations lie within $\pm 1$ standard deviation of the mean ($\bar{x} \pm 1s$).
• 95.45% of all observations lie within $\pm 2$ standard deviations of the mean ($\bar{x} \pm 2s$).
• 99.73% of all observations lie within $\pm 3$ standard deviations of the mean ($\bar{x} \pm 3s$).

Appraisal Outlier Detection

Appraisers use the empirical rule to identify and scrutinize market outliers:

• Any closed sale falling beyond $\pm 2$ standard deviations from the sample mean represents an atypical transaction (occurring in less than 5% of cases under normality).
• Transactions exceeding $\pm 3$ standard deviations are statistical outliers that warrant rigorous investigation. Appraisers must verify whether such sales involved non-arm's-length motivations, undisclosed seller concessions, unrecorded physical defects, substantial personal property, or erroneous recording data. If non-arm's-length conditions exist, the sale must be discarded or adjusted.

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Coefficient of Variation (CV)

While standard deviation quantifies absolute dispersion, it cannot evaluate relative dispersion across different price levels. Comparing a $25,000 standard deviation in a $200,000 starter-home market to a $25,000 standard deviation in a $1,500,000 luxury-home market is misleading.

The Coefficient of Variation (CV) solves this by expressing standard deviation as a percentage of the arithmetic mean:

$\text{CV} = \frac{s}{\bar{x}} \times 100\%$

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Sample Size, Degrees of Freedom, and Paired Sales Verification

Degrees of Freedom ($df = n - 1$)

When calculating sample variance ($s^2$) and sample standard deviation ($s$), statisticians divide the sum of squared deviations by $n - 1$ rather than $n$.

• Why? The sample mean ($\bar{x}$) is already calculated from the sample itself, imposing one mathematical restriction: the deviations $(x_i - \bar{x})$ must sum to exactly zero. Consequently, only $n - 1$ deviations are free to vary.
• Dividing by $n$ would systematically underestimate the true population variance (a phenomenon known as downward bias). Dividing by $n - 1$ (Bessel's correction) provides an unbiased estimator of population variance.

Sample Size Considerations in Valuation

• Small Samples ($n < 30$): Residential appraisers frequently work with small samples (e.g., 5 to 15 closed sales in a specific neighborhood over 6 months). In small samples, individual outliers exert dramatic distortion, standard errors are larger, and parametric statistical assumptions must be applied cautiously.
• Paired Sales Verification: Paired sales analysis relies on isolating a single variance between two properties. However, if market noise (random price variation) exceeds the contributory value of the feature, the extracted adjustment is invalid. Use repeated evidence and an appropriate statistical model when making an inferential claim. A difference between two sales is not automatically significant merely because it exceeds a broadly calculated standard error.

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Step-by-Step Numerical Walkthrough: Central Tendency & Dispersion

An appraiser collects eight recent closed sales of detached single-family homes in the subject's competitive subdivision:

$\text{Closed Sales: } \$310,000, \; \$315,000, \; \$320,000, \; \$320,000, \; \$330,000, \; \$335,000, \; \$340,000, \; \$450,000$

Step 1: Arrange the Ranked Array

Order the sales from lowest to highest:

1. $310,000
2. $315,000
3. $320,000
4. $320,000
5. $330,000
6. $335,000
7. $340,000
8. $450,000 (Sale 8 represents a fully renovated home with an in-ground swimming pool)

Step 2: Calculate the Measures of Central Tendency

• Sum: $\sum x = 310,000 + 315,000 + 320,000 + 320,000 + 330,000 + 335,000 + 340,000 + 450,000 =$ $2,720,000
• Arithmetic Mean: $\bar{x} = \frac{\$2,720,000}{8} = \mathbf{\$340,000}$
• Median: Because $n = 8$ is even, average the 4th and 5th values: $\text{Median} = \frac{\$320,000 + \$330,000}{2} = \mathbf{\$325,000}$
• Mode: The value $320,000 appears twice; all other values appear once. $\text{Mode} = \mathbf{\$320,000}$

Step 3: Calculate the Range

$R = \$450,000 - \$310,000 = \mathbf{\$140,000}$

Step 4: Calculate Deviations, Variance, and Standard Deviation

1. Sum of Squared Deviations: $14,550,000,000$
2. Sample Variance ($s^2$): $s^2 = \frac{14,550,000,000}{8 - 1} = \frac{14,550,000,000}{7} \approx \mathbf{2,078,571,429}$
3. Sample Standard Deviation ($s$): $s = \sqrt{2,078,571,429} \approx \mathbf{\$45,591}$
4. Coefficient of Variation (CV): $\text{CV} = \frac{\$45,591}{\$340,000} \times 100\% \approx \mathbf{13.41\%}$

Step 5: Impact of Removing the Outlier

If the appraiser investigates Sale 8 and determines that its premium pool and custom renovation make it non-comparable to typical neighborhood sales, removing it leaves $n = 7$ sales:

• Sum = $2,270,000
• New Mean $\bar{x}$ = $2,270,000 ÷ 7 = $324,286
• New Median = $320,000
• New Standard Deviation $s$ ≈ $10,965
• New $\text{CV} = \frac{10,965}{324,286} \times 100\% \approx \mathbf{3.38\%}$

Removing the non-comparable outlier reduces relative market dispersion from 13.41% to a narrow 3.38%, describing a narrower price distribution. Whether the retained sales are competitive and representative still requires independent analysis; do not remove a valid sale simply to reduce dispersion.