Question

Harmonic mean of the following twelve observations: \(15, 20, 20, 5, 20, 20, 15, 20, 15, 20, 20,\) and \(20\) is (rounded off to one decimal place).

Hint:

Harmonic mean is useful when observations are rates or ratios. It is calculated using reciprocals of observations.

Answer 1

Correct Answer: 15

Step 1: Recall the harmonic mean.
For $n$ numbers the harmonic mean is
\[ HM=\frac{n}{\sum \dfrac{1}{x_i}} \]
Here $n=12$.

Step 2: Group the repeats.
In the list, $15$ shows up $3$ times, $20$ shows up $8$ times, and $5$ shows up once. So
\[ \sum\frac{1}{x_i}=3\Big(\frac{1}{15}\Big)+8\Big(\frac{1}{20}\Big)+\frac{1}{5} \]

Step 3: Add the reciprocals.
These give $0.2+0.4+0.2=0.8$.

Step 4: Divide.
\[ HM=\frac{12}{0.8}=15 \]

Step 5: Round.
To one decimal place this is $15.0$.
\[ \boxed{15.0} \]