Descriptive Statistics in Real Estate Appraisal

Descriptive Statistics in Real Estate Appraisal

This chapter will delve into the crucial role of descriptive statistics in real estate appraisal. We will explore how these statistical tools help appraisers summarize, analyze, and interpret market data, ultimately leading to more informed and reliable property valuations.

1. Introduction to Descriptive Statistics

Descriptive statistics are used to describe the basic features of data in a study. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.

• Definition: Descriptive statistics are methods used to quantitatively summarize or describe the salient features of a collection of data.

• Purpose in Appraisal: In real estate appraisal, descriptive statistics help:

  • Summarize sales data (prices, sizes, locations).
  • Identify market trends (price increases, inventory levels).
  • Analyze property characteristics (square footage, number of bedrooms).
  • Compare properties (subject vs. comparables).
  • Provide support for adjustments in the sales comparison approach.

• Distinction from Inferential Statistics: Unlike inferential statistics, which draw conclusions about a population based on a sample, descriptive statistics focus solely on describing the characteristics of the observed data. Inferential statistics support conclusions about the population data while descriptive statistics only reflect the characteristics of the sample data set. (Reference: The Appraisal of Real Estate, 14th edition, p. 276)

2. Measures of Central Tendency

Measures of central tendency describe the ‘typical’ or ‘average’ value in a dataset. They provide a single, representative value that summarizes the entire distribution.

• Mean (Arithmetic Mean): The sum of all values divided by the number of values.

  • Formula:

    Mean (x̄) = (Σxᵢ) / n

    Where:

    • x̄ = Sample mean
    • xᵢ = Individual data points
    • n = Number of data points

  • Application: Calculating the average sales price of comparable properties. For example, given the following sales prices: $300,000, $320,000, $310,000, $330,000, and $315,000, the mean sales price is ($300,000 + $320,000 + $310,000 + $330,000 + $315,000) / 5 = $315,000.

  • Sensitivity: Sensitive to extreme values (outliers). The mean is affected by extreme values. (Reference: The Appraisal of Real Estate, 14th edition, p. 280)

• Median: The middle value in a dataset when the values are arranged in ascending order.

  • Calculation:

    • Odd number of values: The median is the middle value.
    • Even number of values: The median is the average of the two middle values. The median is calculated by finding the middle value of the ordered array of data values (Reference: The Appraisal of Real Estate, 14th edition, p. 280)

  • Application: Determining the typical rent per square foot in an apartment complex. (See exercise 22 in the provided document). In the apartment rent example given, with the rents listed and ground living area in square feet, one would divide each rent by its corresponding square footage to arrive at rent per sq ft. The median of these values would be the median rent per square foot.

  • Robustness: Less sensitive to extreme values than the mean. This makes it a better measure of central tendency when outliers are present.

• Mode: The value that appears most frequently in a dataset.

  • Application: Identifying the most common architectural style in a neighborhood.

  • Limitations: A dataset may have no mode (if all values occur only once), one mode (unimodal), or multiple modes (bimodal, trimodal, etc.). Of the three measures of central tendency, the least practical for making inferences is the mode. (Reference: The Appraisal of Real Estate, 14th edition, p. 281). In the provided document example, no value occurs more than once in the array (page 149, question 8).

• Practical Example and Experiment

  • You gather sales data on 15 similar properties in a neighborhood.
  • The sales prices are: $250,000, $260,000, $255,000, $270,000, $255,000, $265,000, $250,000, $280,000, $255,000, $260,000, $255,000, $275,000, $260,000, $255,000, $400,000 (outlier).
  • Calculate the mean: Summing the values and dividing by 15 yields $272,333.33. The outlier significantly inflated the mean.
  • Calculate the median: Arrange the data in ascending order; the middle value is $260,000. The median provides a more representative central value.
  • Calculate the mode: The value $255,000 appears most often (5 times), so the mode is $255,000.

3. Measures of Dispersion

Measures of dispersion describe the spread or variability of data points around the central tendency. They indicate how tightly the data is clustered.

• Range: The difference between the highest and lowest values in a dataset.

  • Formula:

    Range = Maximum Value - Minimum Value

  • Application: Determining the price range of houses in a particular subdivision.

  • Limitations: Very sensitive to outliers and provides no information about the distribution of values within the range.

• Variance: The average of the squared differences between each value and the mean.

  • Formula (Population Variance):

    σ² = Σ(xᵢ - μ)² / N

    Where:

    • σ² = Population variance
    • xᵢ = Individual data points
    • μ = Population mean
    • N = Number of data points in the population

  • Formula (Sample Variance):

    s² = Σ(xᵢ - x̄)² / (n - 1)

    Where:

    • s² = Sample variance
    • xᵢ = Individual data points
    • x̄ = Sample mean
    • n = Number of data points in the sample

  • Application: Assessing the variability of property sizes in a neighborhood.

  • Interpretation: A higher variance indicates greater variability.

• Standard Deviation: The square root of the variance.

  • Formula (Population Standard Deviation):

    σ = √(σ²) = √[Σ(xᵢ - μ)² / N]

  • Formula (Sample Standard Deviation):

    s = √(s²) = √[Σ(xᵢ - x̄)² / (n - 1)]

  • Application: Quantifying the dispersion of sales prices around the average.

  • Interpretation: A higher standard deviation indicates a wider spread of data.

• Coefficient of Variation (CV): A relative measure of dispersion, expressed as a percentage of the mean. It allows for comparing the variability of datasets with different units or different means.

  • Formula:

    CV = (Standard Deviation / Mean) * 100%

  • Application: Comparing the price variability of condos versus single-family homes. Which measure of dispersion is the best indicator of which of two data sets is more variable? The coefficient of variance (Reference: The Appraisal of Real Estate, 14th edition, pp. 285-286). (See exercise 23 in provided document).

  • Interpretation: A higher CV indicates greater relative variability.

• Interquartile Range (IQR): The difference between the 75th percentile (Q3) and the 25th percentile (Q1).

  • Calculation: IQR = Q3 - Q1

  • Application: Measuring the spread of home values excluding extreme outliers.

  • Advantages: Robust to outliers, providing a more stable measure of variability.

• Practical Example and Experiment

  • Consider the following rent per square foot data for 5 apartment buildings: $1.50, $1.60, $1.70, $1.80, $1.90.
  • Calculate the mean: ($1.50 + $1.60 + $1.70 + $1.80 + $1.90) / 5 = $1.70
  • Calculate the standard deviation: Approximately $0.158.
  • Calculate the coefficient of variation: ($0.158 / $1.70) * 100% = 9.29%. This indicates that the rents per square foot have a relatively low variability compared to the mean.

4. Frequency Distributions and Histograms

• Frequency Distribution: A table that shows the number of times each value or range of values occurs in a dataset.

• Histogram: A graphical representation of a frequency distribution, where the height of each bar represents the frequency of values within a specific interval.

  • Application: Visualizing the distribution of property ages in a neighborhood.

  • Interpretation: Histograms can reveal patterns in the data, such as:

    • Symmetry: A symmetrical histogram indicates a normal distribution. (Reference: The Appraisal of Real Estate, 14th edition, p. 279)
    • Skewness: A skewed histogram indicates that the data is not evenly distributed.
      • Left Skewness: The tail is longer on the left side; the mean is less than the median. The mean will be greater than the median when a data set is left skewed (Reference: The Appraisal of Real Estate, 14th edition, p. 288)
      • Right Skewness: The tail is longer on the right side; the mean is greater than the median.
    • Modality: The number of peaks in the histogram (unimodal, bimodal, etc.).

• Practical Example and Experiment

  • Collect data on the square footage of 100 houses in a neighborhood.
  • Create a frequency distribution table, grouping the square footage into intervals (e.g., 1000-1200 sq ft, 1200-1400 sq ft, etc.).
  • Construct a histogram based on the frequency distribution.
  • Analyze the shape of the histogram to determine if the data is normally distributed, skewed, or has multiple modes.

5. Applications in Appraisal Practices

• Sales Comparison Approach: Descriptive statistics are essential for analyzing comparable sales data. Appraisers use measures of central tendency (mean, median) and dispersion (standard deviation, range) to identify adjustments for differences between the subject property and comparable properties.

• Cost Approach: Descriptive statistics can be used to analyze land values and construction costs, providing a basis for estimating the reproduction or replacement cost of the subject property.

• Income Capitalization Approach: Descriptive statistics are useful for analyzing rental income and operating expenses of comparable properties, providing support for estimating the net operating income (NOI) of the subject property.

• Market Analysis: Descriptive statistics provide a quantitative foundation for market analysis, helping appraisers to identify trends, assess market conditions, and forecast future values. Market analysis is used in the income capitalization approach to estimate market rents and expenses and extract capitalization rates from market data.

• Automated Valuation Models (AVMs): AVMs rely heavily on statistical analysis of real estate data. Understanding descriptive statistics is crucial for interpreting the results and evaluating the reliability of AVM outputs. Automated valuation models (AVMs) are currently perceived as a technology designed to help appraisers increase efficiency and cut costs. (Reference: The Appraisal of Real Estate, 14th edition, pp. 296-298)

6. Cautions and Considerations

• Data Quality: The accuracy and reliability of descriptive statistics depend on the quality of the underlying data. Appraisers should carefully scrutinize data sources and verify the accuracy of the information.
• Sample Size: Small sample sizes may not accurately represent the population. Larger sample sizes generally lead to more reliable results.
• Outliers: Outliers can significantly distort descriptive statistics, particularly the mean and range. Appraisers should identify and carefully consider the potential impact of outliers. The sample data should include values at each extreme. (Reference: The Appraisal of Real Estate, 14th edition, p. 280)
• Misinterpretation: It is crucial to interpret descriptive statistics in the context of the specific appraisal assignment and avoid drawing unwarranted conclusions.
• Population vs. Sample: The term population refers to the complete data set from which the sample data set is derived (Reference: The Appraisal of Real Estate, 14th edition, p. 278)

7. Conclusion

Descriptive statistics are fundamental tools for real estate appraisers. By understanding and applying these techniques, appraisers can effectively summarize and analyze market data, leading to more reliable and credible property valuations. This chapter has provided a comprehensive overview of the key concepts and applications of descriptive statistics in real estate appraisal, equipping you with the knowledge and skills necessary to master this essential aspect of market analysis.