Question

given the data set {4, 6, 8}, calculate the standard deviation. use the following formula to calculate standard deviation: ( s = sqrt{\frac{sum_{i=1}^{n}(x_i - \text{mean})^2}{n}} ), where ( x_i ) is each data point, and ( n ) is the number of data points. answer choices: 1.63, 1.5, 2.67, 2

Explanation:

Step1: Calculate the mean

The data set is {4, 6, 8}. The mean ($\bar{x}$) is calculated as the sum of the data points divided by the number of data points ($n$).
Sum of data points: $4 + 6 + 8 = 18$
Number of data points: $n = 3$
Mean: $\bar{x} = \frac{18}{3} = 6$

Step2: Calculate the squared differences from the mean

For each data point $x_i$:

• For $x_1 = 4$: $(4 - 6)^2 = (-2)^2 = 4$
• For $x_2 = 6$: $(6 - 6)^2 = 0^2 = 0$
• For $x_3 = 8$: $(8 - 6)^2 = 2^2 = 4$

Step3: Calculate the sum of squared differences

Sum of squared differences: $\sum_{i = 1}^{n}(x_i - \bar{x})^2 = 4 + 0 + 4 = 8$

Step4: Calculate the variance

Variance ($s^2$) is the sum of squared differences divided by $n$: $s^2 = \frac{8}{3} \approx 2.67$ (wait, no, wait for standard deviation. Wait, the formula for standard deviation is $s = \sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}}$? Wait, no, actually, for sample standard deviation, it's divided by $n - 1$, but the problem's formula is $s = \sqrt{\frac{\sum_{i = 1}^{n}(x_i - \text{mean})^2}{n}}$, so we follow the given formula. Wait, the problem's formula is $s = \sqrt{\frac{\sum_{i = 1}^{n}(x_i - \text{mean})^2}{n}}$. So:

Sum of squared differences is 8, $n = 3$. So $\frac{8}{3} \approx 2.67$, then square root of that is $\sqrt{\frac{8}{3}} \approx \sqrt{2.666...} \approx 1.63$

Wait, let's recalculate:

Sum of squared differences: $(4 - 6)^2 + (6 - 6)^2 + (8 - 6)^2 = 4 + 0 + 4 = 8$

Then divide by $n = 3$: $\frac{8}{3} \approx 2.6667$

Then take the square root: $\sqrt{\frac{8}{3}} \approx \sqrt{2.6667} \approx 1.63299... \approx 1.63$

Answer:

1.63