Question

In the data set below, what is the standard deviation?
7, 8, 1, 8 ,8
If the answer is a decimal, round it to the nearest tenth.

standard deviation (σ):

Answer 1

To calculate the standard deviation of the data set \(7, 8, 1, 8, 8\), we'll follow these steps:

1. **Calculate the mean (average)**:
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{7 + 8 + 1 + 8 + 8}{5} = \frac{32}{5} = 6.4
\]

2. **Calculate the squared deviations from the mean**:
\[
(7 - 6.4)^2 = (0.6)^2 = 0.36
\]
\[
(8 - 6.4)^2 = (1.6)^2 = 2.56
\]
\[
(1 - 6.4)^2 = (-5.4)^2 = 29.16
\]
\[
(8 - 6.4)^2 = (1.6)^2 = 2.56
\]
\[
(8 - 6.4)^2 = (1.6)^2 = 2.56
\]

3. **Sum of the squared deviations**:
\[
0.36 + 2.56 + 29.16 + 2.56 + 2.56 = 37.2
\]

4. **Calculate the variance**:
Since this is a sample standard deviation, we'll divide by \(n-1\) (4 in this case):
\[
\text{Variance} = \frac{37.2}{5 - 1} = \frac{37.2}{4} = 9.3
\]

5. **Calculate the standard deviation**:
\[
\text{Standard deviation} (σ) = \sqrt{9.3} \approx 3.0
\]

Thus, the standard deviation rounded to the nearest tenth is:
\[
\text{Standard deviation (σ)}: 3.0
\]