Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
| Number of heartbeats per minute | 65 - 68 | 68 - 71 | 71 - 74 | 74 - 77 | 77 - 80 | 80 - 83 | 83 - 86 |
|---|---|---|---|---|---|---|---|
| Number of women | 2 | 4 | 3 | 8 | 7 | 4 | 2 |
To find the mean heartbeats per minute, we use the Direct Method by calculating the class marks for each interval. The class mark () is found using the formula: .
We construct the following table:
| Number of heartbeats | Number of women () | Class Mark () | |
|---|---|---|---|
| 65 - 68 | 2 | 66.5 | 133.0 |
| 68 - 71 | 4 | 69.5 | 278.0 |
| 71 - 74 | 3 | 72.5 | 217.5 |
| 74 - 77 | 8 | 75.5 | 604.0 |
| 77 - 80 | 7 | 78.5 | 549.5 |
| 80 - 83 | 4 | 81.5 | 326.0 |
| 83 - 86 | 2 | 84.5 | 169.0 |
| Total |
The Mean is calculated using the formula:
Substituting the values:
Thus, the mean heartbeats per minute for these women is 75.9.
Explanation
The provided context (Example 2) demonstrates finding the mean by first calculating the class marks () for each class interval. This corresponds to the Direct Method. The class marks are calculated as the midpoint of the intervals. The sum of the product of frequency and class marks () is divided by the total number of women () to find the mean. The calculation yields a mean of 75.9 heartbeats per minute.
Solution Steps
Step 1: Calculate class marks () for each interval using the midpoint formula: (Lower Limit + Upper Limit) / 2.
Step 2: Multiply the frequency () by the class mark () for each row to find .
Step 3: Sum the frequencies to get and sum the products to get .
Step 4: Divide the total sum of products by the total frequency: .