Compared with an equal-observation arithmetic mean, a weighted mean with specified nonnegative unequal weights:
Correct Answer
B) Assigns relative influence according to the specified weights
A weighted mean is sum(w_i x_i)/sum(w_i). Its weights determine each observation’s relative influence; they may reflect frequencies, a defined statistical design or justified analytical judgments. A simple arithmetic mean uses equal weights. Unequal weights need not raise the result, do not use only extremes and do not require normality.
Why This Is the Correct Answer
Option B is correct because a weighted mean equals the sum of w times x divided by the sum of the weights, so each observation's influence is proportional to its weight. In a ratio study, for example, the weighted mean ratio uses weights proportional to sale prices. A simple arithmetic mean is the special case in which all weights are equal.
Why the Other Options Are Wrong
Option A: Produces a higher result than the simple mean in each case
Option A, saying it produces a higher result than the simple mean in each case, is wrong because the weighted mean can be higher, lower or equal depending on which observations receive more weight.
Option C: Uses the largest and smallest observations and drops the rest
Option C, saying it uses the largest and smallest observations and drops the rest, describes something like a midrange. A weighted mean includes every observation according to its weight.
Option D: Requires the observations to be normally distributed first
Option D, saying it requires normally distributed observations first, is incorrect; a weighted mean is an arithmetic calculation that does not depend on the data's distribution.
Heavier weights pull harder
In a weighted mean, observations with heavier weights pull the result toward themselves more strongly.
How to use: Use this image to predict how a weighted mean will move relative to a simple mean.
Exam Tip
Be wary of options saying a weighted mean is higher in each case; direction depends on which values get the larger weights.
Common Mistakes to Avoid
- -Assuming a weighted mean is always higher than a simple mean
- -Thinking a weighted mean ignores some observations
- -Believing weighted means require normally distributed data
Concept Deep Dive
Analysis
This question tests what a weighted mean does differently from a simple mean. A weighted mean is the sum of each weight times its observation, divided by the sum of the weights. The weights set each observation's relative influence, whereas a simple mean gives every observation equal weight.
Background Knowledge
Weighted means are used when observations should count unequally, such as weighting ratios by sale price to reflect dollar value. The weights may reflect frequencies, a statistical design or justified analytical judgment.
Real-World Application
An appraiser analyzing an assessor's ratio study notices the weighted mean ratio is lower than the simple mean. Because the weights are sale prices, this tells the appraiser that higher-priced properties tend to have lower ratios.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Land or Site Valuation
4.5% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam
Previous Question
Assessment-to-sale ratios are 0.88, 0.94, 0.98, 1.02, 1.08, 1.22. Compute the arithmetic mean. Report as a decimal ratio to three places.
Next Question
In a set of sale prices, the third quartile is $700,000, the median is $480,000 and the first quartile is $300,000. Skewness is measured as the distance from the median up to the third quartile minus the distance from the first quartile up to the median, divided by the interquartile range. What is this skewness measure, to two decimals?