Find the mean variance and standard deviation for the following data: 2,4,5,6,8,17

Explanation

To find the mean, variance, and standard deviation for the given data set \(2, 4, 5, 6, 8, 17\), we will follow these steps:

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

1. **Sum the observations**:
\[ 2 + 4 + 5 + 6 + 8 + 17 = 42 \]
2. **Count the number of observations**:
\[ n = 6 \]
3. **Calculate the mean**:
\[ x̄ = \frac{42}{6} = 7 \]

### Step 2: Calculate the Variance
The variance is calculated using the formula:
\[ \text{Variance} (σ^2) = \frac{1}{n} \sum_{i=1}^{n} (x_i - x̄)^2 \]

1. **Calculate the deviations from the mean and their squares**:
- For \(x_1 = 2\): \((2 - 7)^2 = (-5)^2 = 25\)
- For \(x_2 = 4\): \((4 - 7)^2 = (-3)^2 = 9\)
- For \(x_3 = 5\): \((5 - 7)^2 = (-2)^2 = 4\)
- For \(x_4 = 6\): \((6 - 7)^2 = (-1)^2 = 1\)
- For \(x_5 = 8\): \((8 - 7)^2 = (1)^2 = 1\)
- For \(x_6 = 17\): \((17 - 7)^2 = (10)^2 = 100\)

2. **Sum the squared deviations**:
\[ 25 + 9 + 4 + 1 + 1 + 100 = 140 \]

3. **Calculate the variance**:
\[ σ^2 = \frac{140}{6} \approx 23.33 \]

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ σ = \sqrt{σ^2} = \sqrt{23.33} \approx 4.83 \]

### Final Answers
- Mean: \(7\)
- Variance: \(23.33\)
- Standard Deviation: \(4.83\)

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