When variance and standard deviation are calculated for the same data on the same sample or population basis, what is the variance?
Correct Answer
C) The square of the standard deviation of that data
For a consistently defined sample or population statistic, variance is the square of its corresponding SD. It has squared units, while SD has original units. The range is maximum minus minimum and the median is an ordered central value. CV requires dividing SD by a suitable mean; merely labeling an SD as a percentage is not the same definition.
Why This Is the Correct Answer
Option C is correct because the standard deviation is defined as the square root of the variance, which means the variance equals the standard deviation squared. Variance is expressed in squared units, while the standard deviation is in the original units of the data.
Why the Other Options Are Wrong
Option A: The standard deviation expressed as a percentage
Option A describes the standard deviation expressed as a percentage. That loosely resembles the coefficient of variation, which divides the standard deviation by the mean; it is not the variance.
Option B: The difference between the highest and lowest value
Option B describes the difference between the highest and lowest value, which is the range. Variance averages squared deviations from the mean and is not based only on the two extremes.
Option D: The middle observation once the data is ordered
Option D describes the middle observation once the data are ordered, which is the median, a measure of central tendency. Variance is a measure of dispersion.
Square the SD, get variance
Standard deviation squared is variance; the square root of variance is the standard deviation.
How to use: Use this to move between variance and standard deviation in any calculation.
Exam Tip
Match each definition to its measure: range is max minus min, median is the middle value, CV is SD divided by the mean, and variance is SD squared.
Common Mistakes to Avoid
- -Thinking variance is the square root of the standard deviation
- -Confusing variance with the range
- -Mixing sample and population formulas when converting
Concept Deep Dive
Analysis
This question tests the relationship between variance and standard deviation. The standard deviation is the square root of the variance, so the variance is the square of the standard deviation. The two must use the same sample or population convention for the relationship to hold.
Background Knowledge
Variance measures the average squared deviation from the mean, and the standard deviation is its square root. Sample and population versions differ in their divisors, so the square relationship holds only when both use the same convention. Because variance is in squared units, the standard deviation is usually easier to interpret.
Real-World Application
A statistical report shows a variance of price per square foot in squared dollars, which is hard to read. The appraiser takes the square root to report the standard deviation in dollars per square foot.
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
Ordered sale prices are $300,000, $320,000, $340,000, $360,000, $380,000, $400,000, $420,000, $440,000, $460,000, $480,000, $500,000, $520,000, $540,000, $560,000, $580,000, $600,000, $620,000, $640,000, $660,000, $680,000. Using the nearest-rank method, which takes the lowest price at or below which at least 5% of the sales fall, what is the 5th percentile?
Next Question
Which activity best describes responsible data cleaning before analysis?