Approximate the sample mean for the following frequency distribution.
Data	Frequency
10 - 13	11
14 - 17	10
18 - 21	6
22 - 25	3
26 - 29	1

Sample mean =

To approximate the sample mean for a frequency distribution, we need to follow these steps:

1. Calculate the **midpoint** for each class interval. The midpoint \( x_i \) for each class is found by averaging the lower and upper boundaries of each interval.
   \[
   x_i = \frac{{\text{{Lower Boundary}} + \text{{Upper Boundary}}}}{2}
   \]
2. Multiply the midpoint by the corresponding frequency to get the **weighted value** for each class.
   \[
   f_i \times x_i
   \]
3. Sum up all the weighted values.
4. Divide the sum of the weighted values by the total number of data points (sum of the frequencies).

### Step-by-Step Calculation:

| Class Interval | Frequency \( f_i \) | Midpoint \( x_i \) | \( f_i \times x_i \) |
|----------------|---------------------|--------------------|----------------------|
| 10 - 13        | 11                  | \( \frac{10 + 13}{2} = 11.5 \) | \( 11 \times 11.5 = 126.5 \) |
| 14 - 17        | 10                  | \( \frac{14 + 17}{2} = 15.5 \) | \( 10 \times 15.5 = 155.0 \) |
| 18 - 21        | 6                   | \( \frac{18 + 21}{2} = 19.5 \) | \( 6 \times 19.5 = 117.0 \) |
| 22 - 25        | 3                   | \( \frac{22 + 25}{2} = 23.5 \) | \( 3 \times 23.5 = 70.5 \)  |
| 26 - 29        | 1                   | \( \frac{26 + 29}{2} = 27.5 \) | \( 1 \times 27.5 = 27.5 \)  |

#### Sum of frequencies:
\[
\text{Total Frequency } = 11 + 10 + 6 + 3 + 1 = 31
\]

#### Sum of weighted values:
\[
\text{Sum of } f_i \times x_i = 126.5 + 155.0 + 117.0 + 70.5 + 27.5 = 496.5
\]

### Sample Mean:
\[
\text{Sample Mean} = \frac{\sum f_i \times x_i}{\sum f_i} = \frac{496.5}{31} \approx 16.02
\]

Thus, the approximate sample mean is **16.02**.

Would you like further details or have any questions? Here are some related questions:

1. How do we calculate variance for a frequency distribution?
2. What are the effects of increasing the frequency on sample mean?
3. How can class width affect the accuracy of the sample mean?
4. What is the difference between sample mean and population mean?
5. How does the sample mean change if the frequencies are doubled?

**Tip:** In frequency distributions, the midpoints serve as representative values for each interval, so always calculate them carefully to avoid rounding errors in the mean.

Approximate the sample mean for the following frequency distribution.
Data    Frequency
10 - 13    11
14 - 17    10
18 - 21    6
22 - 25    3
26 - 29    1

Sample mean =

Learn how to approximate the sample mean for a frequency distribution with the data 10-13 (11), 14-17 (10), 18-21 (6), 22-25 (3), and 26-29 (1). This step-by-step solution covers calculating midpoints, weighted values, and the sample mean. Suitable for high school students.

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