Question

find the standard deviation for the group of data items.
14, 10, 14, 10, 14, 10, 14, 10
the standard deviation is
(round to two decimal places as needed.)

Explanation:

Step 1: Calculate the mean ($\mu$)

The data set is \(14, 10, 14, 10, 14, 10, 14, 10\). There are \(n = 8\) data points.
First, find the sum of the data: \(14 + 10 + 14 + 10 + 14 + 10 + 14 + 10 = (14\times4)+(10\times4)=56 + 40 = 96\)
The mean \(\mu=\frac{\sum x}{n}=\frac{96}{8} = 12\)

Step 2: Calculate the squared differences from the mean

For each data point \(x\), calculate \((x - \mu)^2\):

• For \(x = 14\): \((14 - 12)^2=2^2 = 4\) (there are 4 such points)
• For \(x = 10\): \((10 - 12)^2=(- 2)^2 = 4\) (there are 4 such points)

Step 3: Calculate the variance ($\sigma^2$)

Variance is the average of the squared differences.
Sum of squared differences: \(4\times4+4\times4 = 16 + 16 = 32\)
Variance \(\sigma^2=\frac{\sum(x - \mu)^2}{n}=\frac{32}{8}=4\)

Step 4: Calculate the standard deviation ($\sigma$)

Standard deviation is the square root of the variance.
\(\sigma=\sqrt{\sigma^2}=\sqrt{4} = 2.00\) (rounded to two decimal places)

Answer:

\(2.00\)