If the three observations are 3, -6 and -6, then what is their harmonic mean?

Options

A. 0
B. `oo`
C. `-1//2`
D. `-3`

Explanation

To find the harmonic mean (HM) of the three observations 3, -6, and -6, we can use the formula for the harmonic mean of three numbers, which is given by:

\[ HM = \frac{3}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3}} \]

### Step 1: Identify the values
Let \( x_1 = 3 \), \( x_2 = -6 \), and \( x_3 = -6 \).

### Step 2: Calculate the reciprocals
Now, we need to calculate the reciprocals of these values:
\[ \frac{1}{x_1} = \frac{1}{3}, \quad \frac{1}{x_2} = \frac{1}{-6}, \quad \frac{1}{x_3} = \frac{1}{-6} \]

### Step 3: Sum the reciprocals
Next, we sum these reciprocals:
\[ \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} = \frac{1}{3} + \left(-\frac{1}{6}\right) + \left(-\frac{1}{6}\right) \]
\[ = \frac{1}{3} - \frac{1}{6} - \frac{1}{6} \]

### Step 4: Find a common denominator
To combine these fractions, we find a common denominator, which is 6:
\[ \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \frac{1}{3} - \frac{1}{6} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} - \frac{1}{6} = \frac{2 - 1 - 1}{6} = \frac{0}{6} = 0 \]

### Step 5: Substitute back into the HM formula
Now we substitute this sum back into the harmonic mean formula:
\[ HM = \frac{3}{\frac{1}{3} + \frac{1}{-6} + \frac{1}{-6}} = \frac{3}{0} \]

### Step 6: Conclusion
Since division by zero is undefined, the harmonic mean of the three observations 3, -6, and -6 is undefined.