The heights of frequency distribution of basketball players are shown in the table below:

Height (cm) No. of Players (Frequency)
20-22.        3
23-25.        6
26-28.       12
29-31.        9
32-34.        2
Tasks:

Find the values of the Mean, Variance, and Standard Deviation.
Comment on the shape of the distribution.

To solve the problem, we'll calculate the Mean, Variance, and Standard Deviation for the given frequency distribution. Here's how to do it step by step:

### Step 1: Calculate the Midpoint of Each Class
For each class interval, the midpoint ( \( x_i \) ) is calculated as:
\[
x_i = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}
\]

The midpoints for each class interval are:
- For 20-22: \( \frac{20+22}{2} = 21 \)
- For 23-25: \( \frac{23+25}{2} = 24 \)
- For 26-28: \( \frac{26+28}{2} = 27 \)
- For 29-31: \( \frac{29+31}{2} = 30 \)
- For 32-34: \( \frac{32+34}{2} = 33 \)

### Step 2: Calculate the Mean
The mean ( \( \mu \) ) is calculated using the formula:
\[
\mu = \frac{\sum{(x_i \times f_i)}}{\sum{f_i}}
\]
where \( f_i \) is the frequency of each class.

Let's calculate:
- \( \sum{f_i} \) (Total Frequency) = \( 3 + 6 + 12 + 9 + 2 = 32 \)
- \( \sum{(x_i \times f_i)} \) = \( (21 \times 3) + (24 \times 6) + (27 \times 12) + (30 \times 9) + (33 \times 2) \)

Now, compute these:
\[
\sum{(x_i \times f_i)} = 63 + 144 + 324 + 270 + 66 = 867
\]

Mean:
\[
\mu = \frac{867}{32} \approx 27.09
\]

### Step 3: Calculate the Variance
Variance ( \( \sigma^2 \) ) is calculated using:
\[
\sigma^2 = \frac{\sum{f_i \times (x_i - \mu)^2}}{\sum{f_i}}
\]

Let's calculate:
1. \( (x_i - \mu) \) for each class
2. \( (x_i - \mu)^2 \) for each class
3. \( f_i \times (x_i - \mu)^2 \) for each class

Finally, sum them and divide by total frequency \( \sum{f_i} \).

Let's calculate these for each class:

- For 21: \( (21 - 27.09)^2 \times 3 \approx 111.39 \)
- For 24: \( (24 - 27.09)^2 \times 6 \approx 57.27 \)
- For 27: \( (27 - 27.09)^2 \times 12 \approx 0.12 \)
- For 30: \( (30 - 27.09)^2 \times 9 \approx 74.59 \)
- For 33: \( (33 - 27.09)^2 \times 2 \approx 70.08 \)

Sum these up:
\[
\sum{f_i \times (x_i - \mu)^2} = 313.45
\]

Variance:
\[
\sigma^2 = \frac{313.45}{32} \approx 9.80
\]

### Step 4: Calculate the Standard Deviation
The standard deviation ( \( \sigma \) ) is the square root of the variance:
\[
\sigma = \sqrt{9.80} \approx 3.13
\]

### Step 5: Comment on the Shape of the Distribution
To comment on the shape of the distribution, consider the following:

- **Mean**: 27.09
- **Midpoint of the middle class (26-28)**: 27

The mean is very close to the midpoint of the middle class interval, suggesting that the distribution is **approximately symmetric**. However, the frequency is higher in the middle class, with fewer players in the extreme classes, indicating a slight bell shape.

### Summary:
- **Mean**: 27.09
- **Variance**: 9.80
- **Standard Deviation**: 3.13
- **Shape of Distribution**: Approximately symmetric with a slight bell shape.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How does the mean change if the highest class interval frequency increases?
2. What would be the impact on variance if all frequencies were equal?
3. How can you determine if a distribution is skewed using these statistics?
4. How do you interpret a standard deviation in real-world contexts?
5. What does the variance tell us about the spread of the data?
6. How would the mean change if an additional class interval was added?
7. Can you determine if the data is bimodal with this information alone?
8. How does the class width affect the mean and standard deviation?

**Tip:** Always check the class interval width consistency when calculating means and standard deviations to ensure accuracy.

Learn how to calculate the mean, variance, and standard deviation from a frequency distribution of basketball players' heights. Includes step-by-step calculations and analysis of distribution shape. Suitable for high school students studying statistics.

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