Find the standard deviation of the following set of numbers:
25, 50, 45, 30, 70, 42, 36, 48, 34, 50

Answer

To find the standard deviation of the given set of numbers: 25, 50, 45, 30, 70, 42, 36, 48, 34, and 50, we will follow these steps:

### Step 1: Calculate the Mean (Average)

The mean \( \bar{x} \) is calculated using the formula:

\[ \bar{x} = \frac{\sum x_i}{n} \]

Where:
- \( \sum x_i \) is the sum of all data points.
- \( n \) is the number of data points.

**Calculation:**
\[ \sum x_i = 25 + 50 + 45 + 30 + 70 + 42 + 36 + 48 + 34 + 50 = 430 \]
\[ n = 10 \]
\[ \bar{x} = \frac{430}{10} = 43 \]

### Step 2: Calculate the Deviations from the Mean

Next, we calculate the deviation of each data point from the mean and then square these deviations.

**Deviations:**
- \( 25 - 43 = -18 \) → \( (-18)^2 = 324 \)
- \( 50 - 43 = 7 \) → \( (7)^2 = 49 \)
- \( 45 - 43 = 2 \) → \( (2)^2 = 4 \)
- \( 30 - 43 = -13 \) → \( (-13)^2 = 169 \)
- \( 70 - 43 = 27 \) → \( (27)^2 = 729 \)
- \( 42 - 43 = -1 \) → \( (-1)^2 = 1 \)
- \( 36 - 43 = -7 \) → \( (-7)^2 = 49 \)
- \( 48 - 43 = 5 \) → \( (5)^2 = 25 \)
- \( 34 - 43 = -9 \) → \( (-9)^2 = 81 \)
- \( 50 - 43 = 7 \) → \( (7)^2 = 49 \)

### Step 3: Sum of Squared Deviations

Now, we sum all the squared deviations calculated in the previous step.

**Calculation:**
\[ \sum (x_i - \bar{x})^2 = 324 + 49 + 4 + 169 + 729 + 1 + 49 + 25 + 81 + 49 = 1480 \]

### Step 4: Calculate the Variance

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]

**Calculation:**
\[ \sigma^2 = \frac{1480}{10} = 148 \]

### Step 5: Calculate the Standard Deviation

Finally, the standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{148} \approx 12.17 \]

### Final Answer
The standard deviation of the given set of numbers is approximately **12.17**.

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