Given below are the diameters of circles (in mm) drawn in a design.
`{:("Diameter",33-36,37-40,41-44,45-48,49-52),("Number of Circles",15,17,21,22,25):}`
Calculate the mean diameter of the circles, variance and standard deviation.

Answer

To solve the problem, we need to calculate the mean diameter, variance, and standard deviation of the given data. Here’s a step-by-step solution:

### Step 1: Create a Table for the Data
We will create a table to organize the data, including the diameter intervals, mid-values, frequencies, and calculations for the mean.

| Diameter (mm) | Frequency (f_i) | Mid-value (x_i) | f_i * x_i |
|----------------|------------------|------------------|-----------|
| 33 - 36 | 15 | 34.5 | 15 * 34.5 = 517.5 |
| 37 - 40 | 17 | 38.5 | 17 * 38.5 = 654.5 |
| 41 - 44 | 21 | 42.5 | 21 * 42.5 = 892.5 |
| 45 - 48 | 22 | 46.5 | 22 * 46.5 = 1023 |
| 49 - 52 | 25 | 50.5 | 25 * 50.5 = 1262.5 |

### Step 2: Calculate the Mid-values
The mid-value for each class interval is calculated as:
- For 33 - 36: (33 + 36) / 2 = 34.5
- For 37 - 40: (37 + 40) / 2 = 38.5
- For 41 - 44: (41 + 44) / 2 = 42.5
- For 45 - 48: (45 + 48) / 2 = 46.5
- For 49 - 52: (49 + 52) / 2 = 50.5

### Step 3: Calculate the Total Frequency and Total f_i * x_i
Now, we sum the frequencies and the products of frequency and mid-values:

- Total frequency (N) = 15 + 17 + 21 + 22 + 25 = 100
- Total f_i * x_i = 517.5 + 654.5 + 892.5 + 1023 + 1262.5 = 3350

### Step 4: Calculate the Mean Diameter
The mean diameter (μ) is calculated using the formula:
\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{N} = \frac{3350}{100} = 33.5 \]

### Step 5: Calculate Variance
To calculate the variance, we first need to find \(f_i * (x_i - \text{mean})^2\):

| Diameter (mm) | Frequency (f_i) | Mid-value (x_i) | f_i * (x_i - μ)^2 |
|----------------|------------------|------------------|--------------------|
| 33 - 36 | 15 | 34.5 | 15 * (34.5 - 43.5)^2 = 15 * 81 = 1215 |
| 37 - 40 | 17 | 38.5 | 17 * (38.5 - 43.5)^2 = 17 * 25 = 425 |
| 41 - 44 | 21 | 42.5 | 21 * (42.5 - 43.5)^2 = 21 * 1 = 21 |
| 45 - 48 | 22 | 46.5 | 22 * (46.5 - 43.5)^2 = 22 * 9 = 198 |
| 49 - 52 | 25 | 50.5 | 25 * (50.5 - 43.5)^2 = 25 * 49 = 1225 |

Now, we sum \(f_i * (x_i - μ)^2\):
- Total = 1215 + 425 + 21 + 198 + 1225 = 3084

Now we can calculate the variance:
\[ \text{Variance} = \frac{\sum (f_i \cdot (x_i - \text{mean})^2)}{N} = \frac{3084}{100} = 30.84 \]

### Step 6: Calculate Standard Deviation
The standard deviation (σ) is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{30.84} \approx 5.55 \]

### Final Results
- Mean Diameter: 43.5 mm
- Variance: 30.84
- Standard Deviation: 5.55