Calculate the standard deviation from the following following set of observations:
8, 9, 15, 23, 5, 11, 19, 8, 10, 12

Answer

To calculate the standard deviation from the given set of observations (8, 9, 15, 23, 5, 11, 19, 8, 10, 12), we will follow these steps:

### Step 1: Calculate the Mean (x̄)
The mean (x̄) is calculated using the formula:
\[ \bar{x} = \frac{\sum x}{n} \]
where \( \sum x \) is the sum of all observations and \( n \) is the number of observations.

**Calculating the sum:**
\[ \sum x = 8 + 9 + 15 + 23 + 5 + 11 + 19 + 8 + 10 + 12 = 120 \]

**Calculating the number of observations:**
\[ n = 10 \]

**Calculating the mean:**
\[ \bar{x} = \frac{120}{10} = 12 \]

### Step 2: Calculate the Deviations from the Mean
Next, we calculate the deviations of each observation from the mean and then square these deviations.

\[ \begin{align*} 8 - 12 & = -4 \quad \Rightarrow (-4)^2 = 16 \\ 9 - 12 & = -3 \quad \Rightarrow (-3)^2 = 9 \\ 15 - 12 & = 3 \quad \Rightarrow (3)^2 = 9 \\ 23 - 12 & = 11 \quad \Rightarrow (11)^2 = 121 \\ 5 - 12 & = -7 \quad \Rightarrow (-7)^2 = 49 \\ 11 - 12 & = -1 \quad \Rightarrow (-1)^2 = 1 \\ 19 - 12 & = 7 \quad \Rightarrow (7)^2 = 49 \\ 8 - 12 & = -4 \quad \Rightarrow (-4)^2 = 16 \\ 10 - 12 & = -2 \quad \Rightarrow (-2)^2 = 4 \\ 12 - 12 & = 0 \quad \Rightarrow (0)^2 = 0 \\ \end{align*} \]

### Step 3: Sum of Squared Deviations
Now, we sum all the squared deviations:
\[ \sum (x_i - \bar{x})^2 = 16 + 9 + 9 + 121 + 49 + 1 + 49 + 16 + 4 + 0 = 274 \]

### Step 4: Calculate the Variance (σ²)
The variance is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]
Substituting the values we have:
\[ \sigma^2 = \frac{274}{10} = 27.4 \]

### Step 5: Calculate the Standard Deviation (σ)
Finally, the standard deviation is the square root of the variance:
\[ \sigma = \sqrt{27.4} \approx 5.23 \]

### Final Answer:
The standard deviation of the given set of observations is approximately **5.23**.

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