The mean of first 11 terms of the sequence : 1,1,2,3,5,8,13,21,34,55,89 is 21.1 . Calculate the S.D.

Answer

To calculate the standard deviation (S.D.) of the first 11 terms of the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, given that the mean (x̄) is 21.1, we will follow these steps:

### Step 1: List the Observations
The first 11 terms of the sequence are:
\[ x_i = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 \]

### Step 2: Calculate the Deviations from the Mean
We need to calculate the deviation of each observation from the mean (21.1):
\[ x_i - x̄ = \begin{align*} 1 - 21.1 & = -20.1 \\ 1 - 21.1 & = -20.1 \\ 2 - 21.1 & = -19.1 \\ 3 - 21.1 & = -18.1 \\ 5 - 21.1 & = -16.1 \\ 8 - 21.1 & = -13.1 \\ 13 - 21.1 & = -8.1 \\ 21 - 21.1 & = -0.1 \\ 34 - 21.1 & = 12.9 \\ 55 - 21.1 & = 33.9 \\ 89 - 21.1 & = 67.9 \\ \end{align*} \]

### Step 3: Calculate the Squared Deviations
Now, we square each of the deviations calculated in Step 2:
\[ (x_i - x̄)^2 = \begin{align*} (-20.1)^2 & = 404.01 \\ (-20.1)^2 & = 404.01 \\ (-19.1)^2 & = 364.81 \\ (-18.1)^2 & = 327.61 \\ (-16.1)^2 & = 259.21 \\ (-13.1)^2 & = 171.61 \\ (-8.1)^2 & = 65.61 \\ (-0.1)^2 & = 0.01 \\ (12.9)^2 & = 166.41 \\ (33.9)^2 & = 1145.21 \\ (67.9)^2 & = 4604.41 \\ \end{align*} \]

### Step 4: Sum of Squared Deviations
Now, we sum all the squared deviations:
\[ \text{Sum} = 404.01 + 404.01 + 364.81 + 327.61 + 259.21 + 171.61 + 65.61 + 0.01 + 166.41 + 1145.21 + 4604.41 = 5909.9 \]

### Step 5: Calculate the Variance
The variance (σ²) is calculated using the formula:
\[ \sigma^2 = \frac{\text{Sum of squared deviations}}{n} \]
Where \( n \) is the number of observations (11 in this case):
\[ \sigma^2 = \frac{5909.9}{11} = 536.36 \]

### Step 6: Calculate the Standard Deviation
Finally, the standard deviation (σ) is the square root of the variance:
\[ \sigma = \sqrt{536.36} \approx 23.2 \]

### Final Answer
The standard deviation of the first 11 terms of the sequence is approximately **23.2**.