Income Approach and Statistics Example
You will encounter situations where knowledge of statistics is needed for real estate appraising and the Income Approach. For example, you will need to analyze the sales comparables for the Income Approach on an appraisal assignment. Using statistics will enable you to determine if the sales comps you have are what is considered a "tight fit". When you have a "tight fit", then you can assume your data is an acceptable representation of the market. Meaning that the sales you're analyzing are probably good comps to use and accurately reflect the local market. A tight fit from an appraisal standpoint is generally one in which the Percentage Deviation is 10% or less.
When someone refers to population in statistics, it doesn't have to mean that they are talking about "people". It refers to whatever units of measurement or values you're analyzing. For example, it could refer to cars, houses, rents, dollar amounts, etc. In statistics an absolute number is the number without any negative or positive signs before it. If you're analyzing a set of numbers, your first step should always be to set them up in an ascending or descending order. This order is based on the values of the numbers. For example, let's say you're analyzing some recent comparable sales of houses for an appraisal assignment. You're analyzing the sales prices (SP) of the houses; their gross monthly rents (GMR) for unfurnished apartments; and their gross rent multipliers (GRM) in your area; to assist you on the appraisal. GRM (Gross Rent Multiplier) is equal to the SP (Sales Price) divided by the monthly rent of unfurnished nits. The Gross Rent Multiplier is used for the Direct Sales Comparison Approach for small residential income properties. You use unfurnished apartment rental figures.
Your field work for this appraisal assignment has found the following information:

The Sales Prices of the houses are: $125,000, $92,000, $87,000, $116,000, $147,000, $138,000.

The GMR of the houses are: $370, $290, $285, $355, $405, $390.

GRM of the houses are: 337.84, 317.24, 305.26, 326.76, 362.96, 353.85.
Your first step is to place this information in an order that makes it easier to analyze the data statistically.
Analysis Figures:  
Sales Number  Sales Price  Gross Monthly Rents  Gross Rent Multiplier 
1 2 3 4 5 6 
$ 147,000 138,000 125,000 116,000 92,000 87,000 
$ 405 390 370 355 290 285 
$ 362.96 353.85 337.84 326.76 317.24 305.26 
Totals:  $705,000  $2,095  $2,003.91 
Mean  refers to the number that is the average of all of the values. We find the Mean by dividing the total amount of the values, by the total number of values we are analyzing. In our example the Mean for the Sales Prices, the GMR, and the GRM would be calculated with the following equations:
$705,000/6 = $117,500 is the Mean or average Sales Price
$2,095/6 = $349.17'is the Mean or average GMR
2,003.91/6 = 333.99 is the Mean or average GRM
Median  refers to the number that is the midpoint of all of the values. When you have an even number of figures, for example the six that we are using, you have to add the two middle figures and then divide them by two. Think of the Median as a position on the scale of values that you're dealing with. In our example the Median for the Sales Prices, the GMR, and GRM would be calculated with these equations:
($125,000 + $116,000)/2 = $120,500 is the Median or midpoint Sales Price
($370 + $355)/2 = $362.50 is the Median or midpoint GMR
(337.84 + 326.76)/2 = 332.3 is the Median or midpoint GRM
Mode  refers to the number that is repeated most often in all of the values. For example, let's say sale #1 and sale #2 both sold for $138,000. Since the two sales prices are identical then the Mode would be $138,000 for the Sales Prices. It's the same thing for the GMR and the GRM, or any other values your analyzing. If you have values that have several different numbers that are identical, then the Mode would be the number that is found the most frequently. For example, let's say sale #1 and sale #2 both sold for $138,000 but sales #4, #5 and #6 all sold for $92,000. The Mode in this situation would be $92,000 because there are 3 values with that identical number. In our example there is no Mode because none of the values are repeated.
Range  refers to the number that is the difference of all of the values. You simply subtract the lowest value from the highest value to get the Range. In our example the Range for the Sales Prices, the GMR, and the GRM would be calculated with the following equations:
$147,000  $87,000 = $60,000 is the Range or difference of Sales Prices
$405  $285 = $120'is the Range or difference of GMR
362.96  305.26 = 57.7 is the Range or difference of GRM
Absolute Deviation  refers to the number that is the difference from the Mean for each of the individual values. Remember that an absolute number disregards any plus or minus signs for the values. First we have to set up another table to be able to analyze the values clearly. Then you simply subtract each value from the Mean number, and then add those amounts together to get the Absolute Deviation. This would be calculated as follows:
Analysis Figures:  
Sales Number  Sales Price  Mean  Absolute Deviation 
1 2 3 4 5 6 
$ 147,000 138,000 125,000 116,000 92,000 87,000 
$ 117,500 117,500 117,500 117,500 117,500 117,500 
$ 29,500 20,500 7,500 1,500 25,500 30,500 
Totals:  $705,000  .  $115,000 
The total of the Absolute Deviation from the Mean of the Sales Prices is determined to be $115,000.
Now we'll find the Absolute Deviation from the Mean of the Gross Monthly Rent values:
Analysis Figures:  
Sales Number  Gross Monthly Rent  Mean  Absolute Deviation 
1 2 3 4 5 6 
$ 405 390 370 355 290 285 
$ 349.17 349.17 349.17 349.17 349.17 349.17 
$ 55.83 40.83 20.83 5.83 59.17 64.17 
Totals:  $2,095  .  $246.66 
The total of the Absolute Deviation from the Mean of the Gross Monthly Rent values in our example is determined to be $246.66.
Now we'll find the Absolute Deviation from the Mean of the Gross Rent Multiplier values:
Analysis Figures:  
Sales Number  Gross Rent Multiplier  Mean  Absolute Deviation 
1 2 3 4 5 6 
$ 362.96 353.85 337.84 326.76 317.24 305.26 
$ 333.99 333.99 333.99 333.99 333.99 333.99 
$ 28.97 19.86 3.85 7.23 16.75 28.73 
Totals:  $2,003.91  .  $105.39 
The total of the Absolute Deviation from the Mean of the Gross Rent Multiplier values in our example is determined to be 105.39.
Now we'll find the Arithmetic Deviation for all three sets of values that we're analyzing:
Arithmetic Deviation  refers to the number that is the average of the total of the Absolute Deviation values. You simply divide the total Absolute Deviation value by the number of values to get the Arithmetic Deviation. In our example the Arithmetic Deviation for the Sales Prices, the GMR, and the GRM would be calculated with the following equations:
$115,000/6 = $19,166.67 is the Arithmetic Deviation for the Sales Prices
$246.66/6 = $41.11'is the Arithmetic Deviation for the GMR
105.39/6 = 17.57 is the Arithmetic Deviation for the GRM
Now we'll use all these calculations, to show you how statistics math equations can help you to do an actual appraisal report. As we said earlier, we'll assume that you're analyzing these recent sales prices (SP) of houses; their gross monthly rents (GMR) or unfurnished apartments; and their gross rent multipliers (GRM) in your area; to assist you on an appraisal assignment.
We've found the Arithmetic Deviation for all three sets of values. Now we will use this data to decide whether these sales are good comparables which accurately reflect the local market. We learn this by finding out the percentage difference of the Arithmetic Deviation from the Mean of each set of values. We calculate this percentage by dividing the Arithmetic Deviation by the Mean of each set of values. Therefore, in our example the percentage difference for the Sales Prices, the GMR, and the GRM would be calculated with the following equations:
$19,166.67/$117,500 = .16 or 16% is the Percentage Deviation of the Sales Prices
$41.11/$349.17 = .12 or 12% is the Percentage Deviation of the GMR
17.57/333.99 = .05 or 5% is the Percentage Deviation of the GRM
Now, I'll repeat what I said earlier in this section so you see why we had to go through all these statistics examples for an Income Approach appraisal. When you have what is considered a "tight fit " then you can assume that your data is an acceptable descriptive representation of the market. Meaning that the sales you're analyzing are probably good comparables to use and accurately reflect the local market. A tight fit from an appraisal standpoint is generally one in which the Percentage Deviation is 10% or less. Using the example sales, we have learned that the only "tight fit" for the three different sets of values we're analyzing is in the Gross Rent Multiplier amounts. The Gross Rent Multiplier amounts only deviate from the Mean by 5%. One way to estimate market value using the Income Approach is to multiply the Monthly Total Gross Estimated Rent (GMR) by the Gross Rent Multiplier amount (GRM). An example is shown in the sample MultiFamily appraisal report in this book.