Question

Find the standard deviation of the data set

6.9, 8.7, 7.6, 4.8, 9.0

Answer 1

To find the standard deviation of the data set \( 6.9, 8.7, 7.6, 4.8, 9.0 \), follow the steps below:

1. **Calculate the mean (average)**:
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{6.9 + 8.7 + 7.6 + 4.8 + 9.0}{5}
\]
\[
\text{Mean} = \frac{37.0}{5} = 7.4
\]

2. **Find the squared differences from the mean**:
\[
(6.9 - 7.4)^2 = (-0.5)^2 = 0.25
\]
\[
(8.7 - 7.4)^2 = (1.3)^2 = 1.69
\]
\[
(7.6 - 7.4)^2 = (0.2)^2 = 0.04
\]
\[
(4.8 - 7.4)^2 = (-2.6)^2 = 6.76
\]
\[
(9.0 - 7.4)^2 = (1.6)^2 = 2.56
\]

3. **Calculate the variance**:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{\text{Number of data points}}
\]
\[
\text{Sum of squared differences} = 0.25 + 1.69 + 0.04 + 6.76 + 2.56 = 11.3
\]
\[
\text{Variance} = \frac{11.3}{5} = 2.26
\]

4. **Calculate the standard deviation**:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{2.26} \approx 1.504
\]

Thus, the standard deviation of the data set is approximately **1.504**.