Question 1166623
This is a great exercise in statistical data grouping and analysis. Here are the steps to solve the problem.

## 📊 a) Grouping the Data

### 1. Find the Range ($R$)

The range is the difference between the highest and lowest values in the data set.

* **Highest Value ($H$):** 63 kg
* **Lowest Value ($L$):** 40 kg
* $$R = H - L = 63 - 40 = 23$$

### 2. Determine the Number of Classes ($k$) using Sturges' Formula

Sturges' formula is used to estimate the optimal number of class intervals.
$$k = 1 + 3.322 \log_{10}(n)$$
where $n$ is the total number of observations ($n = 35$).

$$k = 1 + 3.322 \log_{10}(35)$$
$$k \approx 1 + 3.322(1.544)$$
$$k \approx 1 + 5.132$$
$$k \approx 6.132$$

We round this number up to the nearest integer to get the number of classes.
$$\mathbf{k = 7 \text{ classes}}$$

### 3. Determine the Class Width ($w$)

The class width is the range divided by the number of classes.

$$w = \frac{R}{k} = \frac{23}{7} \approx 3.286$$

We must choose a convenient, round number slightly greater than 3.286 to ensure all data points are covered. Let's choose a class width of $\mathbf{4}$.

### 4. Create the Frequency Distribution Table

Starting the first class boundary at the lowest value, 40, and using a width of 4:

| Class Interval (Weight in kg) | Tally | Frequency ($f$) |
| :---: | :---: | :---: |
| 40 - 43 | IIII | 4 |
| 44 - 47 | IIII I | 6 |
| 48 - 51 | IIII IIII | 9 |
| 52 - 55 | IIII IIII | 9 |
| 56 - 59 | IIII | 5 |
| 60 - 63 | II | 2 |
| **Total** | | **35** |

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## 💻 b) Find Mean, Median, Standard Deviation, and Variance

To calculate these statistics for **grouped data**, we must use the **class midpoint ($x$)** to represent the values in each interval.

| Class Interval | Midpoint ($x$) | Frequency ($f$) | $f \cdot x$ | $|x - \bar{x}|$ | $f \cdot |x - \bar{x}|$ | $x - \bar{x}$ | $(x - \bar{x})^2$ | $f \cdot (x - \bar{x})^2$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 40 - 43 | 41.5 | 4 | 166.0 | 9.07 | 36.28 | -9.07 | 82.26 | 329.04 |
| 44 - 47 | 45.5 | 6 | 273.0 | 5.07 | 30.42 | -5.07 | 25.70 | 154.20 |
| 48 - 51 | 49.5 | 9 | 445.5 | 1.07 | 9.63 | -1.07 | 1.14 | 10.26 |
| 52 - 55 | 53.5 | 9 | 481.5 | 2.93 | 26.37 | 2.93 | 8.58 | 77.22 |
| 56 - 59 | 57.5 | 5 | 287.5 | 6.93 | 34.65 | 6.93 | 47.98 | 239.90 |
| 60 - 63 | 61.5 | 2 | 123.0 | 10.93 | 21.86 | 10.93 | 119.46 | 238.92 |
| **Total** | | $\sum f = 35$ | $\sum fx = 1776.5$ | | | | | $\sum f(x - \bar{x})^2 = 1049.54$ |

### 1. Mean ($\bar{x}$)

The mean for grouped data is calculated as:
$$\bar{x} = \frac{\sum f x}{\sum f}$$
$$\bar{x} = \frac{1776.5}{35} \approx \mathbf{50.76 \text{ kg}}$$

### 2. Median ($M$)

The median is the value that falls at the $n/2$ position.
$$\frac{n}{2} = \frac{35}{2} = 17.5$$
The median class is the first class whose cumulative frequency is $\geq 17.5$.
* Cumulative Frequencies: 4, 10, **19**, 28, 33, 35.
* The median class is **48 - 51** (Cumulative Frequency 19).

Median Formula:
$$M = L + \left(\frac{\frac{n}{2} - C_f}{f_m}\right) \cdot w$$
Where:
* $L$: Lower boundary of the median class ($48 - 0.5 = 47.5$)
* $n$: Total frequency (35)
* $C_f$: Cumulative frequency of the class before the median class (10)
* $f_m$: Frequency of the median class (9)
* $w$: Class width (4)

$$M = 47.5 + \left(\frac{17.5 - 10}{9}\right) \cdot 4$$
$$M = 47.5 + \left(\frac{7.5}{9}\right) \cdot 4$$
$$M = 47.5 + 0.833 \cdot 4$$
$$M = 47.5 + 3.332 \approx \mathbf{50.83 \text{ kg}}$$

### 3. Variance ($\sigma^2$)

The variance is calculated using the formula:
$$\sigma^2 = \frac{\sum f(x - \bar{x})^2}{n - 1}$$
Using $n=35$:
$$\sigma^2 = \frac{1049.54}{35 - 1} = \frac{1049.54}{34} \approx \mathbf{30.87}$$

### 4. Standard Deviation ($\sigma$)

The standard deviation is the square root of the variance:
$$\sigma = \sqrt{\sigma^2}$$
$$\sigma = \sqrt{30.87} \approx \mathbf{5.56 \text{ kg}}$$

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## 📝 Summary of Results

| Statistic | Value (Grouped Data) |
| :---: | :---: |
| **Mean** ($\bar{x}$) | $50.76$ kg |
| **Median** ($M$) | $50.83$ kg |
| **Variance** ($\sigma^2$) | $30.87$ |
| **Standard Deviation** ($\sigma$) | $5.56$ kg |