Question

What is the standard deviation of the dataset: 3, 7, 7, 8, 10, 15?

Answer 1

Step 1: Write down the data and find the mean.

Dataset: \( 3, 7, 7, 8, 10, 15 \) \[ \text{Mean} = \frac{3 + 7 + 7 + 8 + 10 + 15}{6} = \frac{50}{6} = 8.33 \]

Step 2: Find the deviation of each data point from the mean and square it.

Data (x)	\(x - \bar{x}\)	\((x - \bar{x})^2\)
3	\(-5.33\)	28.40
7	\(-1.33\)	1.77
7	\(-1.33\)	1.77
8	\(-0.33\)	0.11
10	\(+1.67\)	2.78
15	\(+6.67\)	44.49

Step 3: Find the variance.

Sum of squared deviations = \( 28.40 + 1.77 + 1.77 + 0.11 + 2.78 + 44.49 = 79.32 \) \[ \text{Variance} = \frac{79.32}{6} = 13.22 \]

Step 4: Take the square root to get the standard deviation.

\[ \text{Standard Deviation} = \sqrt{13.22} \approx 3.64 \]

Final Answer: \(\boxed{3.64}\)

Short Explanation (Why this works)

The standard deviation measures how spread out the data is from the mean. You find it by squaring the deviations, averaging them (to get the variance), and then taking the square root.