To find the standard deviation of the set {10, 11, 9, 11, 9}, we will follow these steps:
### Step 1: Organize the Data
First, we will arrange the data in ascending order.
- The given set is: {10, 11, 9, 11, 9}
- In ascending order, it becomes: {9, 9, 10, 11, 11}
### Step 2: Calculate the Mean
Next, we calculate the mean (average) of the data set.
- Mean = (Sum of all observations) / (Number of observations)
- Sum = 9 + 9 + 10 + 11 + 11 = 50
- Number of observations (n) = 5
- Mean = 50 / 5 = 10
### Step 3: Calculate the Deviations from the Mean
Now, we will find the deviation of each observation from the mean.
- Deviation for 9: 9 - 10 = -1
- Deviation for 9: 9 - 10 = -1
- Deviation for 10: 10 - 10 = 0
- Deviation for 11: 11 - 10 = 1
- Deviation for 11: 11 - 10 = 1
### Step 4: Square the Deviations
Next, we square each of the deviations calculated in the previous step.
- Squared deviation for -1: (-1)² = 1
- Squared deviation for -1: (-1)² = 1
- Squared deviation for 0: (0)² = 0
- Squared deviation for 1: (1)² = 1
- Squared deviation for 1: (1)² = 1
### Step 5: Sum of Squared Deviations
Now, we will sum up all the squared deviations.
- Total = 1 + 1 + 0 + 1 + 1 = 4
### Step 6: Calculate the Variance
The variance is the average of the squared deviations.
- Variance = (Sum of squared deviations) / (Number of observations)
- Variance = 4 / 5 = 0.8
### Step 7: Calculate the Standard Deviation
Finally, we take the square root of the variance to find the standard deviation.
- Standard Deviation = √(Variance) = √(0.8)
### Step 8: Simplifying the Result
- Standard Deviation = √(0.8) = √(4/5) = 2/√5
Thus, the standard deviation of the set {10, 11, 9, 11, 9} is \( \frac{2}{\sqrt{5}} \).
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