A sale price is $450,000. The sample mean is $420,000 and the standard deviation is $15,000. What is the z-score of this sale price, to two decimals?
Correct Answer
C) 2.00
(450000−420000)/15000=2. The dollar units cancel, so this standardized score is not a dollar amount.
Why This Is the Correct Answer
Option C is correct because ($450,000 - $420,000) / $15,000 = $30,000 / $15,000 = 2.00. The observation lies two standard deviations above the mean, and the dollar units cancel, leaving a unitless score.
Why the Other Options Are Wrong
Option A: 0.07
Option A, 0.07, divides the difference by the mean instead of the standard deviation: $30,000 / $420,000 = 0.07, which is a percentage deviation from the mean, not a z-score. The correct calculation is $30,000 / $15,000 = 2.00.
Option B: 0.50
Option B, 0.50, comes from inverting the ratio, $15,000 / $30,000 = 0.50, dividing the standard deviation by the difference. The correct calculation is $30,000 / $15,000 = 2.00.
Option D: 30.00
Option D, 30.00, divides the sale price itself by the standard deviation: $450,000 / $15,000 = 30.00, leaving out the mean. The z-score measures distance from the mean: ($450,000 - $420,000) / $15,000 = 2.00.
Distance Over Spread
A z-score is the distance from the mean divided by the spread: how far away, measured in standard deviations.
How to use: Use this phrase to keep the difference on top and the standard deviation on the bottom.
Exam Tip
Divide the distance from the mean by the standard deviation; dividing by the mean or leaving the mean out of the numerator gives a different measure, not a z-score.
Common Mistakes to Avoid
- -Dividing the sale price by the standard deviation without subtracting the mean
- -Dividing the standard deviation by the difference
- -Dividing the difference by the mean instead of the standard deviation
Concept Deep Dive
Analysis
This question tests the z-score formula, (observation - mean) / standard deviation. The order of subtraction sets the sign, and the standard deviation, not the mean, must be the divisor. Because the dollar units cancel, the result is a unitless number of standard deviations.
Background Knowledge
A z-score expresses how many standard deviations an observation lies from the mean, with positive values above and negative values below. It allows comparisons across different scales and is often used to flag potentially unusual observations for further investigation.
Real-World Application
An appraiser screening a set of sales finds one at $450,000 when the mean is $420,000 and the standard deviation is $15,000. Its z-score of 2.00 prompts the appraiser to verify the sale details before deciding how to treat it.
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.
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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?
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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.
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