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

Answer

To calculate the standard deviation of the first 11 terms of the Fibonacci sequence, we will follow these steps:

### Step 1: List the Fibonacci Sequence
The first 11 terms of the Fibonacci sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89

### Step 2: Calculate the Mean (Given)
The mean \( \bar{x} \) is given as 21.1.

### Step 3: Calculate Each \( x_i - \bar{x} \)
We will subtract the mean from each term in the sequence:
- \( 1 - 21.1 = -20.1 \)
- \( 1 - 21.1 = -20.1 \)
- \( 2 - 21.1 = -19.1 \)
- \( 3 - 21.1 = -18.1 \)
- \( 5 - 21.1 = -16.1 \)
- \( 8 - 21.1 = -13.1 \)
- \( 13 - 21.1 = -8.1 \)
- \( 21 - 21.1 = -0.1 \)
- \( 34 - 21.1 = 12.9 \)
- \( 55 - 21.1 = 33.9 \)
- \( 89 - 21.1 = 67.9 \)

### Step 4: Calculate \( (x_i - \bar{x})^2 \)
Now we will square each of the results from Step 3:
- \( (-20.1)^2 = 404.01 \)
- \( (-20.1)^2 = 404.01 \)
- \( (-19.1)^2 = 364.81 \)
- \( (-18.1)^2 = 327.61 \)
- \( (-16.1)^2 = 259.21 \)
- \( (-13.1)^2 = 171.61 \)
- \( (-8.1)^2 = 65.61 \)
- \( (-0.1)^2 = 0.01 \)
- \( (12.9)^2 = 166.41 \)
- \( (33.9)^2 = 1149.21 \)
- \( (67.9)^2 = 4614.41 \)

### Step 5: Sum of Squared Differences
Now we will sum all the squared differences:
\[ 404.01 + 404.01 + 364.81 + 327.61 + 259.21 + 171.61 + 65.61 + 0.01 + 166.41 + 1149.21 + 4614.41 = 7967.91 \]

### Step 6: Calculate Variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\text{Sum of squared differences}}{n} = \frac{7967.91}{11} \approx 724.35 \]

### Step 7: Calculate Standard Deviation
Finally, the standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{724.35} \approx 26.91 \]

### Final Answer
The standard deviation of the first 11 terms of the Fibonacci sequence is approximately **26.91**.

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