Approximate the mean for following GFDT.
Data	Frequency
40 - 44	1
45 - 49	1
50 - 54	4
55 - 59	5
60 - 64	5
65 - 69	10
70 - 74	10
75 - 79	24
80 - 84	15

mean =

To approximate the mean for a grouped frequency distribution table (GFDT), we use the following formula:

\[
\text{Mean} = \frac{\sum f_i \cdot x_i}{\sum f_i}
\]

Where:
- \( f_i \) is the frequency of the \( i \)-th group.
- \( x_i \) is the midpoint of the \( i \)-th class interval.

**Step 1**: Calculate the midpoints (\( x_i \)) for each class interval.

The midpoint is calculated as:

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

**Step 2**: Multiply each midpoint by its corresponding frequency.

**Step 3**: Calculate the sum of the frequency (\( \sum f_i \)) and the sum of the product of midpoint and frequency (\( \sum f_i \cdot x_i \)).

Now, let's compute this.

### Data:

| Class Interval | Frequency \( f_i \) | Midpoint \( x_i \) | \( f_i \cdot x_i \) |
|----------------|---------------------|--------------------|---------------------|
| 40 - 44        | 1                   | 42                 | 42                  |
| 45 - 49        | 1                   | 47                 | 47                  |
| 50 - 54        | 4                   | 52                 | 208                 |
| 55 - 59        | 5                   | 57                 | 285                 |
| 60 - 64        | 5                   | 62                 | 310                 |
| 65 - 69        | 10                  | 67                 | 670                 |
| 70 - 74        | 10                  | 72                 | 720                 |
| 75 - 79        | 24                  | 77                 | 1848                |
| 80 - 84        | 15                  | 82                 | 1230                |

### Step 4: Calculate the totals

\[
\sum f_i = 1 + 1 + 4 + 5 + 5 + 10 + 10 + 24 + 15 = 75
\]

\[
\sum f_i \cdot x_i = 42 + 47 + 208 + 285 + 310 + 670 + 720 + 1848 + 1230 = 5360
\]

### Step 5: Apply the formula for the mean

\[
\text{Mean} = \frac{5360}{75} = 71.47
\]

Thus, the approximate mean is:

\[
\boxed{71.47}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How is the midpoint of a class interval calculated?
2. What is the difference between a grouped and ungrouped frequency distribution?
3. How can we use the mean to describe data sets?
4. How would you calculate the median for a grouped frequency distribution?
5. How does the number of class intervals affect the mean?

**Tip**: Always ensure the class intervals are consistent when calculating a mean for grouped data to avoid calculation errors.

Approximate the mean for following GFDT.
Data Frequency
40 - 44 1
45 - 49 1
50 - 54 4
55 - 59 5
60 - 64 5
65 - 69 10
70 - 74 10
75 - 79 24
80 - 84 15

Learn how to approximate the mean for a grouped frequency distribution table with detailed steps, including calculating midpoints and applying the formula. Data includes frequency intervals from 40 - 44 to 80 - 84.

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