Question

1. find the standard deviation of the data set (as a population) below. round your answer to the nearest hundredth if applicable. 4,5,3,6,2,1,5

Explanation:

Step1: Find the mean of the data set

The data set is \(4, 5, 3, 6, 2, 1, 5\). The formula for the population mean \(\mu\) is \(\mu=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n\) is the number of data points and \(x_{i}\) are the data values.
First, calculate the sum of the data: \(4 + 5+3 + 6+2 + 1+5=26\).
There are \(n = 7\) data points. So the mean \(\mu=\frac{26}{7}\approx3.7143\).

Step2: Calculate the squared differences from the mean

For each data point \(x_{i}\), calculate \((x_{i}-\mu)^{2}\):

• For \(x = 4\): \((4 - 3.7143)^{2}=(0.2857)^{2}\approx0.0816\)
• For \(x = 5\): \((5 - 3.7143)^{2}=(1.2857)^{2}\approx1.6531\)
• For \(x = 3\): \((3 - 3.7143)^{2}=(- 0.7143)^{2}\approx0.5102\)
• For \(x = 6\): \((6 - 3.7143)^{2}=(2.2857)^{2}\approx5.2245\)
• For \(x = 2\): \((2 - 3.7143)^{2}=(-1.7143)^{2}\approx2.9388\)
• For \(x = 1\): \((1 - 3.7143)^{2}=(-2.7143)^{2}\approx7.3673\)
• For \(x = 5\): \((5 - 3.7143)^{2}=(1.2857)^{2}\approx1.6531\)

Step3: Find the sum of the squared differences

Sum these squared differences: \(0.0816+1.6531 + 0.5102+5.2245+2.9388+7.3673+1.6531\)
\(=0.0816+(1.6531 + 1.6531)+0.5102+5.2245+2.9388+7.3673\)
\(=0.0816 + 3.3062+0.5102+5.2245+2.9388+7.3673\)
\(=(0.0816+3.3062)+0.5102+5.2245+2.9388+7.3673\)
\(=3.3878+0.5102+5.2245+2.9388+7.3673\)
\(=(3.3878 + 0.5102)+5.2245+2.9388+7.3673\)
\(=3.898+5.2245+2.9388+7.3673\)
\(=(3.898+5.2245)+2.9388+7.3673\)
\(=9.1225+2.9388+7.3673\)
\(=(9.1225 + 2.9388)+7.3673\)
\(=12.0613+7.3673\)
\(=19.4286\)

Step4: Calculate the population variance

The formula for population variance \(\sigma^{2}\) is \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}\).
We have \(\sum_{i = 1}^{n}(x_{i}-\mu)^{2}=19.4286\) and \(n = 7\), so \(\sigma^{2}=\frac{19.4286}{7}\approx2.7755\).

Step5: Calculate the population standard deviation

The population standard deviation \(\sigma\) is the square root of the variance, so \(\sigma=\sqrt{\sigma^{2}}=\sqrt{2.7755}\approx1.666\approx1.67\) (rounded to the nearest hundredth).

Answer:

\(1.67\)