Calculate Standard Deviation for Data

To find the standard deviation for the dataset {4, 7, 9, 10, 15}, follow these steps:

Step 1: Calculate the Mean ($\bar{x}$)

The mean is the sum of the data points divided by the count of data points.

Sum = 4 + 7 + 9 + 10 + 15 = 45

Number of data points ($n$) = 5

Mean ($\bar{x}$) = $\frac{45}{5} = 9$

Step 2: Calculate Squared Deviations from the Mean

Find the difference between each data point and the mean, then square the result.

• $(4 - 9)^2 = (-5)^2 = 25$
• $(7 - 9)^2 = (-2)^2 = 4$
• $(9 - 9)^2 = (0)^2 = 0$
• $(10 - 9)^2 = (1)^2 = 1$
• $(15 - 9)^2 = (6)^2 = 36$

Step 3: Calculate Variance ($\sigma^2$)

Sum the squared deviations and divide by the number of data points ($n$).

Sum of Squared Deviations = 25 + 4 + 0 + 1 + 36 = 66

Variance ($\sigma^2$) = $\frac{66}{5} = 13.2$

Step 4: Calculate Standard Deviation ($\sigma$)

The standard deviation is the square root of the variance.

Standard Deviation ($\sigma$) = $\sqrt{13.2} \approx 3.633$

The calculated standard deviation is approximately 3.633.