HOA case study in respect, Reality, and Reason
Part 2: Homeowners' opinions of the proposed CC&R amendment
Links to the other top-level parts of this discussion:
In
addition to exceptionally strong quantitative data showing 74% support
in Waterford for permission of high quality dimensional composition
roofing, results of the 2007 survey had a fair sample of opinions
written by owners as comments on the returned survey forms. Those
opinions are reported verbatim on this report of written comments.
The general reaction of those responsible for Waterford's "Roofers Rebellion" has assumed that the board is guilty of misconduct by conducting its own agenda on roofing. This is false. When there is a very clear majority of owners calling for a CC&R change on a matter of preference it is a the duty of the board to honor that majority's expressed opinions and desires.
Click on
either image below to look at high resolution images of the graphs
below showing sampled opinion of the homeowners and the two ways
that have been proposed to interpret it. This web page will present
very limited mathematical results and will cite references to the two
most directly applicable. Wikipedia pages on concepts in
probability and statistics.
Opponents
in the 5% core minority group claim that any owner opinion not sampled
represents an opinion in opposition to the proposed CC&R amendment.
This is a remarkable example of cognitive bias which produces
mathematical bias in evaluating the collective opinions of all
homeowners.
In statistics "the sample mean is the usual estimator of a population mean." (See for example, the Wikipedia page on Standard Error (statistics),
referring to standard error of the mean. This is another way to say
that it is reasonable to expect that unsampled data is likely to be
similar to sampled data. In this case the data represent homeowner
approval of dimensional composition roofing in the Lake
Forest-Waterford Owners Association.
Original data with extrapolations
up to the full HOA membership
of 393 homeowners
Graph plots number of responses in favor of CC&R amendment as a function of number of opinion samples from:
1) 2007 survey
2) 2009 vote
showing two extrapolations up to all 393 homeowners
Red line:
Projection up
to the full population of 393 owners. This line assumes unknown
opinions are like known opinions: 74% in support of an amendment
Statistical margin of error for projecting data samples forward:
-- Gray area is margin of error based on the 2007 sample.
-- Yellow area is margin of error based on the 2009 sample
Green lines (minority extrapolation):
Projection up to the full population of 393 owners under minority assertion that any lack of response represents opposition.
|  |
Same data graphed as percent of total number of homeowners (393)
The
green line is drawn through the 2007 and 2009 sample points, which both
showed 74% support for a CC&R amendment, then extended down to 0
responses and up to 393 responses.
The red line represents the minority assertion that any unknown opinion is an opposing opinion, based on the 2007 sample.
The dark blue line represents the minority assertion that any unknown opinion is an opposing opinion, based on the 2009 sample. |  |
Here's
a tabular summary of the two data samples. Projections to 393 owners
are based on the finding of 74% support among all owners who responded
to the survey and to the vote for the amendment . Stated slightly more formally, to two significant figures the Probability
Distribution Function for owners supporting an amendment is a constant (0.74).
Tabular summary of actual data
and valid projection to full population
| Data Sample | Count of responses | Owners opposed | Owners in favor | Expected value in support, for 393 owners | Margin of error of expected value for 393 owners |
| 2007 survey | 100 | 26 (26%) | 74 (74%) | 291 (74%) | 8.47% |
| 2009 vote | 216 | 56 (26%) | 160 (74%) | 291 (74%) | 4.48% |
Comparison of valid projection and core-minority projection
from 2007 survey results to 2009 vote results
| Projection | Projected "Yes" votes in 2009 | Actual "Yes" votes in 2009 | Error,
number of votes | Error as
percent of actual "Yes" votes**
| Error as
percent of estimated "Yes" votes
|
| Valid* | 160 | 160 | 0 | 0% | 0% |
| Core-minority | 74 | 86 | 53.75% | 116% |
* "Valid" refers to the principle in probability and statistics that the sample mean is the estimator of the population mean.
** Error as percent of actual "Yes" votes is defined in probability and statistics as the relative standard error (RSE). |
The
error summary above involves the core-minority assumption that any
opinion not received is an opinion
opposing a CC&R amendment. When that assumption is evaluated
mathematically it
produces the maximum possible mathematical bias and error in
extrapolations to sample sets larger than the actual samples. The
area of bias is the triangle shaded orange in the image below (click on
the image to see a high resolution copy).
Experience
has shown that very few, if any, opponents of a CC&R amendment are
familiar with the mathematics of probability and statistics. This case
is so
simple that it should be unnecessary to actually use those mathematical
disciplines: The only essential principle to reach mathematically valid
conclusions is to accept the notion that unknown opinion probably is
like known opinion.
At least one opponent rejects statistics because it "can
be used to lie": This is cognitive bias: The implicit logical fallacy is "if something can be used to lie, then use of it will produce lies". Incorrect use of tools does not preclude correct use of tools.
Mathematics provides
a tool kit which at worst is precisely as biased as its user.
More normally it produces unbiased analysis, as well as often
being the only analytical tool set possible to use in areas such as
physical sciences.
Margin
of error was computed for a 95% confidence interval, which is generally used in public opinion polls. This means
that 95% of the time a random sample will produce an observed value
within the margin of error percentage of the expected value for the
full statistical population. For example, the 2009 sample forecasts
that 95% of the time a random sample of the full population of 393
would produce a result in the range of 74% ± 4.48% in favor.
A good reference for understanding margin of error is http://en.wikipedia.org/wiki/Margin_of_error.
A further reference on statistical estimators, expected value,
estimator error, sampling deviation, variance, and mathematical bias is http://en.wikipedia.org/wiki/Estimator. Other Wikipedia web pages provide further discussion on details of probability and statistics.
If you want detailed data used to for the graphs above click here to retrieve it as an Excel spreadsheet.
