Measures of Dispersion and Shape: Inferential Statistics in Real Estate Appraisal

Chapter: Measures of Dispersion and Shape: Inferential Statistics in Real Estate Appraisal

This chapter delves into measures of dispersion and shape, crucial components of inferential statistics that are essential for real estate appraisal. We will explore how these measures help us understand the variability and distribution of data, and how this knowledge informs our ability to make inferences about larger populations based on sample data.

1. Introduction to Measures of Dispersion

Measures of dispersion quantify the amount of variation or spread within a dataset. They provide insights into how closely the data points cluster around the central tendency (e.g., mean or median). Comparing dispersion measures against known distributions like the normal distribution is vital for determining the appropriateness of parametric statistical methods.

• Importance of Dispersion:
  • Indicates the homogeneity or heterogeneity of data.
  • Helps assess the reliability of measures of central tendency.
  • Allows comparison of variability between different datasets.
  • Informs the selection of appropriate statistical tests (parametric vs. non-parametric).

2. Standard Deviation and Variance

These are the two fundamental measures of dispersion, considering the distribution of all data points. Standard deviation (SD) is particularly important because it allows for further statistical manipulation and inference, helping us understand the uncertainty associated with our estimations.

2.1. Definitions and Formulas

• Variance: A measure of the average squared deviation of data points from the mean.

  • Population Variance (σ²):

    σ² = Σ[(xᵢ - μ)²] / N

    Where:
    * xᵢ represents each individual data point.
    * μ is the population mean.
    * N is the population size.
    * Σ indicates summation across all data points.
    * Sample Variance (S²):

    S² = Σ[(xᵢ - X)²] / (n - 1)

    Where:
    * xᵢ represents each individual data point in the sample.
    * X is the sample mean.
    * n is the sample size.
    * The denominator (n-1) represents the degrees of freedom.
    * Standard Deviation: The square root of the variance. It represents the average distance of data points from the mean in the original units of measurement.
    * Population Standard Deviation (σ):

    σ = √[Σ[(xᵢ - μ)²] / N]
    * Sample Standard Deviation (S):

    S = √[Σ[(xᵢ - X)²] / (n - 1)]

2.2. Practical Application and Interpretation

A larger standard deviation indicates greater variability in the dataset. In real estate appraisal, a high standard deviation of comparable sales prices suggests a less reliable indication of value for the subject property, meaning that there is a high level of variance in the market that can affect the predictability of sales price.

2.3 Example

Let’s use the garden-level apartment rents data given by the PDF file. We have the following sample:

Monthly Rent: \$600, \$650, \$695, \$710, \$715, \$730, \$735, \$735, \$760, \$760, \$785, \$800, \$800, \$805, \$815, \$820, \$820, \$825, \$825, \$825, \$825, \$850, \$850, \$850, \$850, \$850, \$850, \$860, \$860, \$890, \$890, \$920, \$920, \$930, \$970, \$995

1. Calculate the Sample Mean (X):
   X = \$29,370/36 = \$815.83

2. Calculate the Sample Standard Deviation (S):
   S = √[Σ[(xᵢ - X)²] / (n - 1)] = \$84.71

3. Coefficient of Variation

The coefficient of variation (CV) is a relative measure of dispersion. It expresses the standard deviation as a percentage of the mean. This is particularly useful when comparing the variability of datasets with different units or substantially different means.

3.1. Formula

CV = (S / X) * 100%

Where:

• S is the sample standard deviation.
• X is the sample mean.

3.2. Interpretation

A higher CV indicates greater relative variability.

3.3 Example

Using the apartment rent data, the coefficient of variation is:

CV = (\$84.71 / \$815.83) * 100% = 10.38%

This allows to directly compare the variabily with other samples.

4. Range

The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in the dataset.

4.1. Formula

Range = Maximum Value - Minimum Value

4.2. Limitations

• Sensitive to outliers.
• Only considers the extreme values, ignoring the distribution of the rest of the data.

4.3. Example

For the apartment rent data, the range is:
Range = \$995 - \$600 = \$395

5. Interquartile Range

The interquartile range (IQR) measures the spread of the middle 50% of the data. It is more robust to outliers than the range.

5.1. Quartiles

Quartiles divide the ordered dataset into four equal parts:

• Q1 (First Quartile): The value below which 25% of the data falls.
• Q2 (Second Quartile): The median, the value below which 50% of the data falls.
• Q3 (Third Quartile): The value below which 75% of the data falls.

5.2. Calculation

IQR = Q3 - Q1

5.3. Decision Rules

• Rule 1: If the position point calculation is an integer, then the ordered observation occupying that position point is the quartile boundary.
• Rule 2: If the position point is halfway between two integers, then the midpoint between the next-largest and next-smallest ordered observation is the quartile boundary.
• Rule 3: If the position point is neither an integer nor halfway between two integers, then the position point is rounded to the nearest integer and the corresponding ordered observation is the quartile boundary.

5.4. Example

Using the apartment rent data (36 observations):

1. Calculate the positions of Q1, Q2, and Q3:

   • Q1 position = (37 + 4) = 9.25, rounded down to 9th ordered observation
   • Q2 = median
   • Q3 position = 3(37 + 4) = 27.75, rounded up to the 28th ordered observation

2. Identify the corresponding values:

   • Q1 = \$760
   • Q2 = \$825
   • Q3 = \$860

3. Calculate the IQR:

   • IQR = \$860 - \$760 = \$100

6. Measures of Shape

Measures of shape describe the overall form of a data distribution. These are essential to determine how close a data distribution is to normal and the extent to which extreme values are distorting the difference between the median and the mean. The most important measures of shape are skewness and kurtosis.

6.1. Skewness

Skewness refers to the asymmetry of a distribution.

• Symmetrical Distribution: Mean = Median, Skewness = 0.
• Left-Skewed (Negatively Skewed): Mean < Median, Skewness < 0 (tail extends to the left).
• Right-Skewed (Positively Skewed): Mean > Median, Skewness > 0 (tail extends to the right).

Formula:

Skewness = {n / [(n-1)(n-2)]} * Σ[(xᵢ - X) / S]³

Where:
* X = sample mean
* n = sample size
* S = sample standard deviation

6.2. Kurtosis

Kurtosis describes the “peakedness” or “tailedness” of a distribution.

• Mesokurtic: Kurtosis ≈ 3 (normal distribution).
• Leptokurtic: Kurtosis > 3 (more peaked, heavier tails).
• Platykurtic: Kurtosis < 3 (flatter, lighter tails).

6.3 Example

The Skewness for the apartment rent data is -0.312 (Excel formula: SKEW.P). Kurtosis = 0.42 (Excel formula: KURT). It is important to consider if the data is close to normal.

7. Assessing Normality

Many statistical tests rely on the assumption of normality. Several methods can be used to assess whether a dataset is approximately normally distributed:

1. Visual Inspection:
   • Histograms: Check for a bell-shaped curve.
   • Box Plots: Examine symmetry and outliers.
   • Normal Probability Plots (Q-Q Plots): Data points should fall close to a straight line.
2. Numerical Measures:
   • Skewness and Kurtosis: Values should be close to zero (for skewness) and three (for kurtosis).
3. Statistical Tests:
   • Kolmogorov-Smirnov Test: Tests the null hypothesis that the data comes from a specified distribution (e.g., normal).
   • Shapiro-Wilk Test: Another test for normality, often more powerful than Kolmogorov-Smirnov for small sample sizes.
4. Standard deviation ranges of normality:
   • Approximately 68% of the observations are expected to lie within ± 1 standard deviation of the mean.
   • Approximately 80% within ± 1.28 standard deviations of the mean.
   • Approximately 95% within ± 2 standard deviations of the mean.

7.1. Interpreting Normality Test Results

The p-value from normality tests indicates the probability of observing the data if it were drawn from a normal distribution. A small p-value (typically less than 0.05) suggests that the data is not normally distributed.

8. Parametric vs Nonparametric Statistics

• Parametric statistics: are based on the assumption that the data follows a specific distribution, such as the normal distribution. These tests are more powerful and sensitive than nonparametric tests, but they require that the underlying assumptions of the test are met.

• Nonparametric statistics: do not rely on any assumptions about the distribution of the data. These tests are useful when the data does not follow a normal distribution, when the sample size is small, or when the data is measured on a nominal or ordinal scale.

9. Central Limit Theorem and Inference

The Central Limit Theorem (CLT) states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. This is crucial for inferential statistics, as it allows us to make inferences about population parameters (e.g., mean) based on sample data, even if the population distribution is not normal.

• Importance of CLT in Appraisal: Even if the distribution of property values is not perfectly normal, the distribution of sample means of comparable sales will tend to be normal, allowing us to use parametric statistical tests for inference.

10. Conclusion

Measures of dispersion and shape are critical tools for understanding the characteristics of real estate data. By carefully analyzing these measures, appraisers can:

• Assess the reliability of their estimates.
• Select appropriate statistical methods.
• Make sound inferences about property values and market trends.
• Understand the uncertainty associated with these estimations.

This knowledge is essential for making informed decisions and providing credible appraisal opinions.