Consider the frequency distribution of the monthly viewing times of a sample television audience
given in the table below:
TV Sample Audience
Monthly Viewing Times (hours) No. of Viewers (*100)
50 and less than 54 2
54 and less than 58 7
58 and less than 62 11
62 and less than 66 12
66 and less than 70 9
70 and less than 74 6
74 and less than 78 2
78 and less than 82 1
Calculate the:
(a) Mean, and
(b) Standard deviation.

To solve this problem, we need to calculate the mean and standard deviation of the monthly viewing times based on the given frequency distribution.

### Step 1: Organize the Data

Given the frequency distribution:

| Monthly Viewing Times (hours) | No. of Viewers (*100) |
|-------------------------------|-----------------------|
| 50 and less than 54            | 2                     |
| 54 and less than 58            | 7                     |
| 58 and less than 62            | 11                    |
| 62 and less than 66            | 12                    |
| 66 and less than 70            | 9                     |
| 70 and less than 74            | 6                     |
| 74 and less than 78            | 2                     |
| 78 and less than 82            | 1                     |

### Step 2: Calculate the Midpoint for Each Class

The midpoint (class mark) for each class interval is calculated as:

\[
\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}
\]

Let's calculate the midpoints for each class:

\[
\text{Midpoints} = \left(\frac{50 + 54}{2}, \frac{54 + 58}{2}, \frac{58 + 62}{2}, \frac{62 + 66}{2}, \frac{66 + 70}{2}, \frac{70 + 74}{2}, \frac{74 + 78}{2}, \frac{78 + 82}{2}\right)
\]

\[
\text{Midpoints} = \left(52, 56, 60, 64, 68, 72, 76, 80\right)
\]

### Step 3: Calculate the Mean

The mean is given by the formula:

\[
\text{Mean} (\mu) = \frac{\sum{f \cdot x}}{\sum{f}}
\]

where:
- \( f \) is the frequency of each class.
- \( x \) is the midpoint of each class.

Let's calculate the mean:

\[
\text{Mean} (\mu) = \frac{2 \times 52 + 7 \times 56 + 11 \times 60 + 12 \times 64 + 9 \times 68 + 6 \times 72 + 2 \times 76 + 1 \times 80}{2 + 7 + 11 + 12 + 9 + 6 + 2 + 1}
\]

### Step 4: Calculate the Standard Deviation

The standard deviation is given by the formula:

\[
\text{Standard Deviation} (\sigma) = \sqrt{\frac{\sum{f \cdot (x - \mu)^2}}{\sum{f}}}
\]

Let's calculate the standard deviation after computing the mean.

I will calculate these now.### Results:

(a) **Mean** of the monthly viewing times: \( \mu = 64.0 \) hours.

(b) **Standard deviation** of the monthly viewing times: \( \sigma \approx 6.30 \) hours.

These results indicate that on average, the sample audience views television for 64 hours per month, with a standard deviation of approximately 6.30 hours, reflecting the spread of viewing times around the mean.

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. How does the range of viewing times compare to the standard deviation?
2. What would the effect on the mean and standard deviation be if we had an additional class with more extreme viewing hours?
3. Can the skewness of this distribution be inferred from the mean and standard deviation?
4. How would you calculate the mode of this distribution?
5. What is the significance of a lower standard deviation in the context of television viewing times?

**Tip:** When dealing with grouped data, the midpoint represents the "typical" value for each class, and it's crucial for calculating accurate statistics like the mean and standard deviation.

Learn how to calculate the mean and standard deviation from a frequency distribution table of monthly television viewing times. This problem involves organizing data, calculating midpoints, and applying statistical formulas to find the average and spread of viewing times. Suitable for college-level students studying statistics.

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