If the standard deviation of 1,2,3,4.......10 is `sigma`, then find the standard deviation of 11, 12,13,14,.....,20.

Answer

To find the standard deviation of the series 11, 12, 13, ..., 20, given that the standard deviation of the series 1, 2, 3, ..., 10 is `σ`, we can follow these steps:

### Step 1: Calculate the Mean of the First Series
The mean of the series 1, 2, 3, ..., 10 can be calculated as follows:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{1 + 2 + 3 + ... + 10}{10} \]
The sum of the first 10 natural numbers is:
\[ \text{Sum} = \frac{n(n + 1)}{2} = \frac{10 \times 11}{2} = 55 \]
Thus, the mean is:
\[ \text{Mean} = \frac{55}{10} = 5.5 \]

### Step 2: Understand the Transformation
The second series is obtained by adding 10 to each element of the first series. Therefore, if \( x_i \) is an element from the first series, the corresponding element in the second series can be represented as \( y_i = x_i + 10 \).

### Step 3: Calculate the Mean of the Second Series
The mean of the second series (11, 12, ..., 20) can be calculated as:
\[ \text{Mean of second series} = \text{Mean of first series} + 10 = 5.5 + 10 = 15.5 \]

### Step 4: Calculate the Standard Deviation
The standard deviation formula is:
\[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
For the first series, we know that the standard deviation is `σ`.

For the second series, the transformation does not affect the spread of the data; it only shifts the data. Therefore, the standard deviation of the second series remains the same as that of the first series.

### Conclusion
Thus, the standard deviation of the series 11, 12, 13, ..., 20 is also `σ`.