Question

the standard deviation of the data set {7, 2, 5, 1, 12, 3, 7} is about _____. 3.773 3.599 2.989

Explanation:

Step1: Calculate the mean

First, find the sum of the data set: \(7 + 2 + 5 + 1 + 12 + 3 + 7 = 37\).
There are 7 data points, so the mean \(\bar{x}=\frac{37}{7}\approx5.2857\).

Step2: Calculate the squared differences

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\):

• For \(7\): \((7 - 5.2857)^2\approx(1.7143)^2\approx2.9388\)
• For \(2\): \((2 - 5.2857)^2\approx(-3.2857)^2\approx10.7959\)
• For \(5\): \((5 - 5.2857)^2\approx(-0.2857)^2\approx0.0816\)
• For \(1\): \((1 - 5.2857)^2\approx(-4.2857)^2\approx18.3673\)
• For \(12\): \((12 - 5.2857)^2\approx(6.7143)^2\approx45.0816\)
• For \(3\): \((3 - 5.2857)^2\approx(-2.2857)^2\approx5.2245\)
• For \(7\): \((7 - 5.2857)^2\approx(1.7143)^2\approx2.9388\)

Step3: Calculate the variance

Sum the squared differences: \(2.9388 + 10.7959 + 0.0816 + 18.3673 + 45.0816 + 5.2245 + 2.9388 = 85.4285\)
Variance \(s^2=\frac{85.4285}{7 - 1}=\frac{85.4285}{6}\approx14.2381\) (using sample variance, since it's a data set without population context).

Step4: Calculate the standard deviation

Standard deviation \(s = \sqrt{14.2381}\approx3.773\).

Answer:

3.773