The question asks for the approximate harmonic mean of a given set of numbers: 8, 6, 12, and 4.

Understanding the Harmonic Mean

The harmonic mean is one of the types of average used in statistics. It is particularly useful when dealing with rates or ratios. For a set of positive numbers, the harmonic mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocals of the numbers.

The formula for the harmonic mean of a set of $n$ positive numbers $x_1, x_2, \dots, x_n$ is:

\begin{량을*} HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}} \end{량을*}

Calculating the Harmonic Mean for 8, 6, 12, 4

In this problem, we have $n=4$ numbers: 8, 6, 12, and 4.

Step 1: Calculate the reciprocal of each number.

• Reciprocal of 8 is $\frac{1}{8}$
• Reciprocal of 6 is $\frac{1}{6}$
• Reciprocal of 12 is $\frac{1}{12}$
• Reciprocal of 4 is $\frac{1}{4}$

Step 2: Sum the reciprocals.

Sum of reciprocals = $\frac{1}{8} + \frac{1}{6} + \frac{1}{12} + \frac{1}{4}$

To add these fractions, we find a common denominator, which is 24.

\begin{량을*} \frac{1}{8} + \frac{1}{6} + \frac{1}{12} + \frac{1}{4} &= \frac{1 \times 3}{8 \times 3} + \frac{1 \times 4}{6 \times 4} + \frac{1 \times 2}{12 \times 2} + \frac{1 \times 6}{4 \times 6} \\ &= \frac{3}{24} + \frac{4}{24} + \frac{2}{24} + \frac{6}{24} \\ &= \frac{3 + 4 + 2 + 6}{24} \\ &= \frac{15}{24} \end{량을*}

This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3.

\begin{량을*} \frac{15}{24} = \frac{15 \div 3}{24 \div 3} = \frac{5}{8} \end{량을*}

So, the sum of the reciprocals is $\frac{5}{8}$.

Step 3: Apply the Harmonic Mean formula.

Using the formula $HM = \frac{n}{\text{Sum of reciprocals}}$, where $n=4$ and the sum of reciprocals is $\frac{5}{8}$:

\begin{량을*} HM = \frac{4}{\frac{5}{8}} \end{량을*}

To divide by a fraction, we multiply by its reciprocal:

\begin{량을*} HM = 4 \times \frac{8}{5} \\ HM = \frac{32}{5} \end{량을*}

Step 4: Convert the result to a decimal.

\begin{량을*} HM = \frac{32}{5} = 6.4 \end{량을*}

The harmonic mean of 8, 6, 12, and 4 is 6.4.

Comparing this result to the given options, we find that 6.40 matches one of the options.

Revision Table: Key Statistical Means

Additional Information: Understanding Harmonic Mean Applications

The harmonic mean is particularly useful in situations where the average of rates is needed. For example:

• Average Speed: If you travel a fixed distance at different speeds, the average speed for the entire trip is the harmonic mean of the individual speeds. For instance, if you travel 100 km at 50 km/h and return the same 100 km at 75 km/h, the average speed is the harmonic mean of 50 and 75, not the arithmetic mean.
• Average Resistance in Parallel Circuits: In electrical circuits, the equivalent resistance of resistors connected in parallel is calculated using a formula related to the harmonic mean of the individual resistances.
• Average P/E Ratio in Stock Indices: Sometimes used in finance to average price-to-earnings (P/E) ratios.

The harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean for a set of positive numbers (AM ≥ GM ≥ HM).