Calculate Sample Standard Deviation for Data

To calculate the sample standard deviation for the dataset {8, 7, 9}, we follow these steps:

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

The mean is the average of the data points. We sum the values and divide by the number of values ($n$). Here, $n=3$.

Calculation: $ \bar{x} = \frac{8 + 7 + 9}{3} = \frac{24}{3} = 8 $

The mean ($\bar{x}$) is 8.

Step 2: Calculate Deviations from the Mean

Next, we find the difference between each data point and the mean.

Deviations ($x_i - \bar{x}$):

• For 8: $8 - 8 = 0$
• For 7: $7 - 8 = -1$
• For 9: $9 - 8 = 1$

Step 3: Square the Deviations

We square each of the deviations calculated in the previous step.

Squared Deviations ($(x_i - \bar{x})^2$):

• $(0)^2 = 0$
• $(-1)^2 = 1$
• $(1)^2 = 1$

Step 4: Sum the Squared Deviations

Add up all the squared deviations.

Sum of Squared Deviations: $0 + 1 + 1 = 2$

Step 5: Calculate the Sample Variance ($s^2$)

The sample variance is calculated by dividing the sum of squared deviations by ($n-1$).

Calculation: $ s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1} = \frac{2}{3-1} = \frac{2}{2} = 1 $

The sample variance ($s^2$) is 1.

Step 6: Calculate the Sample Standard Deviation ($s$)

The sample standard deviation is the square root of the sample variance.

Calculation: $ s = \sqrt{s^2} = \sqrt{1} $

Therefore, the sample standard deviation for the dataset {8, 7, 9} is $\sqrt{1}$.