To find the mean of the data in a frequency distribution, we will use the following formula:

$\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}$

Where:

• $f$ is the frequency for each interval
• $x$ is the midpoint of each class (the average of the lower and upper bounds of each interval)

Step 1: Calculate the midpoints

• For the interval 40–44: $x = \frac{40 + 44}{2} = 42$
• For the interval 45–49: $x = \frac{45 + 49}{2} = 47$
• For the interval 50–54: $x = \frac{50 + 54}{2} = 52$
• For the interval 55–59: $x = \frac{55 + 59}{2} = 57$
• For the interval 60–64: $x = \frac{60 + 64}{2} = 62$

Step 2: Multiply each midpoint by the corresponding frequency

• $42 \times 3 = 126$
• $47 \times 6 = 282$
• $52 \times 13 = 676$
• $57 \times 4 = 228$
• $62 \times 1 = 62$

Step 3: Sum the products

$126 + 282 + 676 + 228 + 62 = 1374$

Step 4: Sum the frequencies

$3 + 6 + 13 + 4 + 1 = 27$

Step 5: Calculate the mean

$\text{Mean} = \frac{1374}{27} \approx 50.9$

So, the mean of the frequency distribution is approximately 50.9 degrees.

Step 6: Compare with the actual mean

The computed mean of 50.9 degrees is very close to the actual mean of 50.5 degrees, with a difference of 0.4 degrees.

Would you like further details or explanations? Here are some follow-up questions to expand your understanding:

1. How are the midpoints chosen for frequency intervals?
2. What effect does increasing the frequency in one interval have on the mean?
3. Can we calculate the median or mode for this data?
4. Why is it necessary to compare the computed mean to the actual mean?
5. How would this calculation change with a different frequency distribution?

Tip: When rounding to the nearest tenth, always check the digit in the hundredths place.