Question

we are going to calculate the standard deviation for the following set of sample data. 15,3,9,5,2

1. first, calculate the mean.

\\( \overline { x } = \\)

1. fill in the table below. fill in the differences of each data value from the mean, then the squared differences.
2. calculate the standard deviation.

standard deviation:
\\( s = \sqrt { \frac { \sum ( x - \overline { x } ) ^ { 2 } } { n - 1 } } = \\) round to two decimal places
question help: video

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 5\) and \(x_{1}=15,x_{2}=3,x_{3}=9,x_{4}=5,x_{5}=2\).
\(\bar{x}=\frac{15 + 3+9+5+2}{5}=\frac{34}{5}=6.8\)

Step2: Calculate \(x-\bar{x}\) and \((x - \bar{x})^{2}\)

• For \(x = 15\): \(x-\bar{x}=15 - 6.8=8.2\), \((x-\bar{x})^{2}=(8.2)^{2}=67.24\)
• For \(x = 3\): \(x-\bar{x}=3 - 6.8=- 3.8\), \((x-\bar{x})^{2}=(-3.8)^{2}=14.44\)
• For \(x = 9\): \(x-\bar{x}=9 - 6.8=2.2\), \((x-\bar{x})^{2}=(2.2)^{2}=4.84\)
• For \(x = 5\): \(x-\bar{x}=5 - 6.8=-1.8\), \((x-\bar{x})^{2}=(-1.8)^{2}=3.24\)
• For \(x = 2\): \(x-\bar{x}=2 - 6.8=-4.8\), \((x-\bar{x})^{2}=(-4.8)^{2}=23.04\)

The sum of \((x-\bar{x})^{2}\) is \(67.24+14.44 + 4.84+3.24+23.04=112.8\)

Step3: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum(x-\bar{x})^{2}}{n - 1}}\). Here \(n = 5\), \(\sum(x-\bar{x})^{2}=112.8\)
\(s=\sqrt{\frac{112.8}{5-1}}=\sqrt{\frac{112.8}{4}}=\sqrt{28.2}\approx5.31\)

Answer:

1. \(\bar{x}=6.8\)

2)

\(x\)	\(x-\bar{x}\)	\((x - \bar{x})^{2}\)
\(3\)	\(-3.8\)	\(14.44\)
\(9\)	\(2.2\)	\(4.84\)
\(5\)	\(-1.8\)	\(3.24\)
\(2\)	\(-4.8\)	\(23.04\)
	Total	\(112.8\)

1. \(s\approx5.31\)