Find the mean of the data summarized in the given frequency distribution. Compare the computed mean to the actual mean of 51.9 degrees. Low Temperature (°F) 40-44 45-49 50-54 55-59 60-64 Frequency 2 6 11 6 1 The mean of the frequency distribution is __ degrees. (Round to the nearest tenth as needed.) Which of the following best describes the relationship between the computed mean and the actual mean? A. The computed mean is not close to the actual mean because the difference between the means is less than 5% of the actual mean. B. The computed mean is not close to the actual mean because the difference between the means is more than 5% of the actual mean. C. The computed mean is close to the actual mean because the difference between the means is less than 5% of the actual mean. D. The computed mean is close to the actual mean because the difference between the means is more than 5% of the actual mean.
The Correct Answer and Explanation is:
To find the mean of a frequency distribution, we use the following steps:
Step 1: Find the midpoint of each class
The midpoint (class mark) of each class is calculated using:Midpoint=Lower limit+Upper limit2\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}Midpoint=2Lower limit+Upper limit
| Class Interval (°F) | Frequency (f) | Midpoint (x) | f × x |
|---|---|---|---|
| 40–44 | 2 | 42 | 2 × 42 = 84 |
| 45–49 | 6 | 47 | 6 × 47 = 282 |
| 50–54 | 11 | 52 | 11 × 52 = 572 |
| 55–59 | 6 | 57 | 6 × 57 = 342 |
| 60–64 | 1 | 62 | 1 × 62 = 62 |
Step 2: Compute the mean
xˉ=∑(f⋅x)∑f=84+282+572+342+622+6+11+6+1=134226≈51.6 degrees\bar{x} = \frac{\sum (f \cdot x)}{\sum f} = \frac{84 + 282 + 572 + 342 + 62}{2 + 6 + 11 + 6 + 1} = \frac{1342}{26} \approx 51.6 \text{ degrees}xˉ=∑f∑(f⋅x)=2+6+11+6+184+282+572+342+62=261342≈51.6 degrees
Step 3: Compare with actual mean
- Computed mean: 51.6 degrees
- Actual mean: 51.9 degrees
- Difference:
∣51.9−51.6∣=0.3|51.9 – 51.6| = 0.3∣51.9−51.6∣=0.3
Now calculate the percentage difference relative to the actual mean:0.351.9×100≈0.578%\frac{0.3}{51.9} \times 100 \approx 0.578\%51.90.3×100≈0.578%
Since 0.578% < 5%, the computed mean is close to the actual mean.
Answer
- The mean of the frequency distribution is 51.6 degrees.
- The best choice is:
C. The computed mean is close to the actual mean because the difference between the means is less than 5% of the actual mean.
Explanation
To approximate the mean of grouped data, the midpoint of each class is used as a representative value. By multiplying each midpoint by its corresponding frequency and summing the products, we obtain the total weighted value. Dividing by the total frequency gives the mean. Comparing the computed mean to the actual mean reveals how closely the grouped data represents the true average. A small percentage difference (under 5%) indicates a strong approximation, validating the accuracy of using midpoints in frequency distributions.
