Five students secured marks as, 8, 10, 15, 30, 22. Find the standard deviation.

Answer

To find the standard deviation of the marks secured by the five students (8, 10, 15, 30, 22), we will follow these steps:

### Step 1: Calculate the Mean (x̄)
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \(x_i\) are the observations and \(n\) is the number of observations.

**Calculating the sum of observations:**
\[ 8 + 10 + 15 + 30 + 22 = 85 \]

**Calculating the mean:**
\[ \bar{x} = \frac{85}{5} = 17 \]

### Step 2: Calculate the Deviations from the Mean
Next, we calculate the deviation of each observation from the mean:
\[ x_i - \bar{x} \]
- For 8: \(8 - 17 = -9\)
- For 10: \(10 - 17 = -7\)
- For 15: \(15 - 17 = -2\)
- For 30: \(30 - 17 = 13\)
- For 22: \(22 - 17 = 5\)

### Step 3: Square the Deviations
Now, we square each of the deviations:
\[ (x_i - \bar{x})^2 \]
- For 8: \((-9)^2 = 81\)
- For 10: \((-7)^2 = 49\)
- For 15: \((-2)^2 = 4\)
- For 30: \(13^2 = 169\)
- For 22: \(5^2 = 25\)

### Step 4: Calculate the Mean of the Squared Deviations
Now, we find the mean of these squared deviations:
\[ \text{Mean of squared deviations} = \frac{\sum (x_i - \bar{x})^2}{n} \]
Calculating the sum of squared deviations:
\[ 81 + 49 + 4 + 169 + 25 = 328 \]

Now, calculate the mean:
\[ \frac{328}{5} = 65.6 \]

### Step 5: Calculate the Standard Deviation
Finally, we take the square root of the mean of the squared deviations to find the standard deviation:
\[ \sigma = \sqrt{65.6} \approx 8.1 \]

Thus, the standard deviation of the marks is approximately **8.1**.

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