The heights, to the nearest cm, of 30 men are given below:
`{:(159, 170, 174, 173, 175, 160, 161, 164, 163, 165), (164, 171, 162, 170, 177, 185, 181, 180, 175, 165), (186, 174, 168, 168, 176, 176, 165, 175, 167, 180):}`
Using class intervals 155 - 160, 160 - 165, ... draw up a grouped frequency distribution and use this to estimate the Arithmetic mean and standard deviation.

Answer

To solve the problem, we will follow these steps:

### Step 1: Organize the Data
We have the heights of 30 men given as follows:
```
159, 170, 174, 173, 175, 160, 161, 164, 163, 165,
164, 171, 162, 170, 177, 185, 181, 180, 175, 165,
186, 174, 168, 168, 176, 176, 165, 175, 167, 180
```

### Step 2: Create Class Intervals
We will use the specified class intervals:
- 155 - 160
- 160 - 165
- 165 - 170
- 170 - 175
- 175 - 180
- 180 - 185
- 185 - 190

### Step 3: Count Frequencies
Now, we will count the number of heights that fall into each class interval:

| Class Interval | Frequency (f) |
|----------------|---------------|
| 155 - 160 | 1 |
| 160 - 165 | 6 |
| 165 - 170 | 6 |
| 170 - 175 | 6 |
| 175 - 180 | 6 |
| 180 - 185 | 3 |
| 185 - 190 | 2 |
| **Total** | **30** |

### Step 4: Calculate Mid Values
Next, we calculate the mid values (x_i) for each class interval:

- For 155 - 160: Mid = (155 + 160) / 2 = 157.5
- For 160 - 165: Mid = (160 + 165) / 2 = 162.5
- For 165 - 170: Mid = (165 + 170) / 2 = 167.5
- For 170 - 175: Mid = (170 + 175) / 2 = 172.5
- For 175 - 180: Mid = (175 + 180) / 2 = 177.5
- For 180 - 185: Mid = (180 + 185) / 2 = 182.5
- For 185 - 190: Mid = (185 + 190) / 2 = 187.5

### Step 5: Create a Table with Mid Values
We will now create a table that includes the mid values and calculate f * x_i:

| Class Interval | Frequency (f) | Mid Value (x_i) | f * x_i |
|----------------|---------------|------------------|---------|
| 155 - 160 | 1 | 157.5 | 157.5 |
| 160 - 165 | 6 | 162.5 | 975 |
| 165 - 170 | 6 | 167.5 | 1005 |
| 170 - 175 | 6 | 172.5 | 1035 |
| 175 - 180 | 6 | 177.5 | 1065 |
| 180 - 185 | 3 | 182.5 | 547.5 |
| 185 - 190 | 2 | 187.5 | 375 |
| **Total** | **30** | | **4175**|

### Step 6: Calculate the Arithmetic Mean
The formula for the arithmetic mean (A) is given by:
\[ A = \frac{\sum (f \cdot x_i)}{n} \]
Where \( n \) is the total frequency.

Substituting the values:
\[ A = \frac{4175}{30} = 139.1667 \]

### Step 7: Calculate the Standard Deviation
To calculate the standard deviation, we need to find \( f \cdot (x_i - A)^2 \):

1. Calculate \( (x_i - A)^2 \) for each mid value.
2. Multiply by frequency \( f \).
3. Sum these values.

| Class Interval | Frequency (f) | Mid Value (x_i) | x_i - A | (x_i - A)^2 | f * (x_i - A)^2 |
|----------------|---------------|------------------|---------|--------------|------------------|
| 155 - 160 | 1 | 157.5 | -1.6667 | 2.7778 | 2.7778 |
| 160 - 165 | 6 | 162.5 | -1.1667 | 1.3611 | 8.1667 |
| 165 - 170 | 6 | 167.5 | -0.6667 | 0.4444 | 2.6667 |
| 170 - 175 | 6 | 172.5 | -0.1667 | 0.0278 | 0.1667 |
| 175 - 180 | 6 | 177.5 | 0.3333 | 0.1111 | 0.6667 |
| 180 - 185 | 3 | 182.5 | 1.3333 | 1.7778 | 5.3333 |
| 185 - 190 | 2 | 187.5 | 2.3333 | 5.4444 | 10.8889 |
| **Total** | **30** | | | | **30.6667** |

Now, we can calculate the standard deviation using the formula:
\[ \sigma = \sqrt{\frac{\sum f \cdot (x_i - A)^2}{n}} \]

Substituting the values:
\[ \sigma = \sqrt{\frac{30.6667}{30}} = \sqrt{1.0222} \approx 1.0111 \]

### Final Results
- **Arithmetic Mean**: 172.5 cm
- **Standard Deviation**: 1.0111 cm