Find the standard deviation for the following data:
10, 20, 30, 40, 50, 60, 70, 80, 90

Answer

To find the standard deviation of the given data set: 10, 20, 30, 40, 50, 60, 70, 80, 90, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.

\[ \text{Mean} (x̄) = \frac{\text{Sum of all values}}{n} \]

Where \( n \) is the number of values.

\[ \text{Sum} = 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 = 450 \]

\[ n = 9 \]

\[ x̄ = \frac{450}{9} = 50 \]

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

\[ \text{Deviations} = (x_i - x̄) \]

\[ \begin{align*} 10 - 50 & = -40 \\ 20 - 50 & = -30 \\ 30 - 50 & = -20 \\ 40 - 50 & = -10 \\ 50 - 50 & = 0 \\ 60 - 50 & = 10 \\ 70 - 50 & = 20 \\ 80 - 50 & = 30 \\ 90 - 50 & = 40 \\ \end{align*} \]

Now, squaring these deviations:

\[ \begin{align*} (-40)^2 & = 1600 \\ (-30)^2 & = 900 \\ (-20)^2 & = 400 \\ (-10)^2 & = 100 \\ 0^2 & = 0 \\ 10^2 & = 100 \\ 20^2 & = 400 \\ 30^2 & = 900 \\ 40^2 & = 1600 \\ \end{align*} \]

### Step 3: Sum of Squared Deviations
Now, we sum the squared deviations:

\[ \text{Sum of squared deviations} = 1600 + 900 + 400 + 100 + 0 + 100 + 400 + 900 + 1600 \]

Calculating this gives:

\[ \text{Sum} = 1600 + 900 + 400 + 100 + 0 + 100 + 400 + 900 + 1600 = 5000 \]

### Step 4: Calculate the Variance
The variance is calculated by dividing the sum of squared deviations by the number of values.

\[ \text{Variance} (\sigma^2) = \frac{\text{Sum of squared deviations}}{n} \]

\[ \sigma^2 = \frac{5000}{9} \approx 555.56 \]

### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance.

\[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{5000}{9}} \approx 23.57 \]

### Final Answer
The standard deviation of the given data set is approximately **23.57**.

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