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 formula of harmonic mean.
For \(n\) observations \(x_1,x_2,x_3,\ldots,x_n\), the harmonic mean is given by
\[ HM=\frac{n}{\sum \frac{1}{x_i}} \]
Here, the total number of observations is
\[ n=12 \]

Step 2: Group the repeated observations.
The given observations are
\[ 15,20,20,5,20,20,15,20,15,20,20,20 \]
Here, \(15\) appears \(3\) times, \(20\) appears \(8\) times, and \(5\) appears \(1\) time.
Therefore,
\[ \sum \frac{1}{x_i} = 3\left(\frac{1}{15}\right)+8\left(\frac{1}{20}\right)+1\left(\frac{1}{5}\right) \]

Step 3: Calculate the reciprocal sum.
\[ 3\left(\frac{1}{15}\right)=\frac{3}{15}=0.2 \] \[ 8\left(\frac{1}{20}\right)=\frac{8}{20}=0.4 \] \[ 1\left(\frac{1}{5}\right)=0.2 \]
Thus,
\[ \sum \frac{1}{x_i}=0.2+0.4+0.2 \] \[ \sum \frac{1}{x_i}=0.8 \]

Step 4: Substitute in the harmonic mean formula.
\[ HM=\frac{12}{0.8} \] \[ HM=15 \]

Step 5: Round off to one decimal place.
Since the answer is exactly \(15\), rounded off to one decimal place it becomes
\[ 15.0 \]

Step 6: Final conclusion.
Hence, the harmonic mean of the given twelve observations is
\[ \boxed{15.0} \]