To find the variance of the given data set: [tex]\(1, 3, 8, 10, 14, 24\)[/tex] with a mean ([tex]\(\bar{x}\)[/tex]) of 10, follow these steps:
1. Calculate the squared differences from the mean:
For each data point ([tex]\(x_i\)[/tex]), calculate [tex]\((x_i - \bar{x})^2\)[/tex]:
- For [tex]\(x_1 = 1\)[/tex]: [tex]\((1 - 10)^2 = (-9)^2 = 81\)[/tex]
- For [tex]\(x_2 = 3\)[/tex]: [tex]\((3 - 10)^2 = (-7)^2 = 49\)[/tex]
- For [tex]\(x_3 = 8\)[/tex]: [tex]\((8 - 10)^2 = (-2)^2 = 4\)[/tex]
- For [tex]\(x_4 = 10\)[/tex]: [tex]\((10 - 10)^2 = 0^2 = 0\)[/tex]
- For [tex]\(x_5 = 14\)[/tex]: [tex]\((14 - 10)^2 = 4^2 = 16\)[/tex]
- For [tex]\(x_6 = 24\)[/tex]: [tex]\((24 - 10)^2 = 14^2 = 196\)[/tex]
Therefore, the squared differences are:
[tex]\[
[81, 49, 4, 0, 16, 196]
\][/tex]
2. Sum the squared differences:
[tex]\[
81 + 49 + 4 + 0 + 16 + 196 = 346
\][/tex]
3. Calculate the variance:
The variance [tex]\(\sigma^2\)[/tex] is the average of these squared differences. Since there are 6 data points, divide the sum of the squared differences by 6:
[tex]\[
\sigma^2 = \frac{346}{6} = 57.666666666666664
\][/tex]
4. Round the variance to the nearest tenth:
[tex]\[
57.666666666666664 \approx 57.7
\][/tex]
Therefore, the variance ([tex]\(\sigma^2\)[/tex]) rounded to the nearest tenth is [tex]\(57.7\)[/tex].
