What is the harmonic mean of the values 80 and 120, to three decimals?
Correct Answer
B) 96.000
H=2/(1/80+1/120)=96. The arithmetic and geometric means use different definitions. Common errors: computing the arithmetic mean gives 100.000; computing the geometric mean gives 97.980; not multiplying by n = 2 gives 48.000.
Why This Is the Correct Answer
Option B is correct because H = 2 ÷ (1/80 + 1/120) = 2 ÷ (0.0125 + 0.008333) = 2 ÷ 0.020833 = 96.000. The harmonic mean divides the number of observations by the sum of their reciprocals. It differs from the arithmetic and geometric means, which use different definitions.
Why the Other Options Are Wrong
Option A: 48.000
Option A, 48.000, comes from 1 ÷ (1/80 + 1/120) = 48, which omits multiplying by n = 2. The correct calculation is 2 ÷ (1/80 + 1/120) = 96.000.
Option C: 97.980
Option C, 97.980, is the geometric mean, (80 × 120)^0.5 = 9,600^0.5 ≈ 97.980. The harmonic mean is 2 ÷ (1/80 + 1/120) = 96.000.
Option D: 100.000
Option D, 100.000, is the arithmetic mean, (80 + 120) ÷ 2 = 100. The harmonic mean uses reciprocals: 2 ÷ (1/80 + 1/120) = 96.000.
Harmonic is the humble one
For unequal positive values, the order is harmonic below geometric below arithmetic, so the harmonic mean is the smallest of the three.
How to use: Use this to sanity-check a harmonic mean answer against the arithmetic and geometric means of the same data.
Exam Tip
Compute the arithmetic mean first as a quick ceiling; a correct harmonic mean of unequal positive values should come out below it.
Common Mistakes to Avoid
- -Forgetting to multiply by n after summing reciprocals
- -Reporting the arithmetic mean instead of the harmonic mean
- -Reporting the geometric mean instead of the harmonic mean
Concept Deep Dive
Analysis
This question tests computing the harmonic mean from its formula, H = n ÷ Σ(1/x). With two values, sum the reciprocals of 80 and 120, then divide the count, 2, by that sum. The distractors reflect other means or skipping the multiplication by n.
Background Knowledge
For positive values that are not all equal, the harmonic mean is less than the geometric mean, which is less than the arithmetic mean. The harmonic mean is useful when averaging rates or ratios where the reciprocal has a meaningful interpretation. It is one of several measures of central tendency, none of which is preferred for every problem.
Real-World Application
An analyst averaging price-per-unit rates across two market segments considers whether the harmonic mean better reflects the combined rate. Computing it shows a value below the simple average, illustrating how the choice of mean affects the reported figure.
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
Two numerical variables exhibit a strong Pearson correlation in the observed sample. What may be concluded from that fact alone?
Next Question
Assessment-to-sale ratios are 1.08, 0.86, 0.97, 1.14, 0.91, 1.02, 0.94. Compute the median; order the observations first and average the two middle values if n is even. Report as a decimal ratio to three places.