To find the variance of the given dataset, we will follow a step-by-step process.
1. Given Data:
[tex]\[
15, 21, 46, 49, 31, 24
\][/tex]
2. Mean of the Data ([tex]\(\bar{x}\)[/tex]):
[tex]\[
\bar{x} = 31
\][/tex]
3. Calculate Each Data Point's Deviation from the Mean, Square It:
- For 15:
[tex]\[
(15 - 31)^2 = (-16)^2 = 256
\][/tex]
- For 21:
[tex]\[
(21 - 31)^2 = (-10)^2 = 100
\][/tex]
- For 46:
[tex]\[
(46 - 31)^2 = 15^2 = 225
\][/tex]
- For 49:
[tex]\[
(49 - 31)^2 = 18^2 = 324
\][/tex]
- For 31:
[tex]\[
(31 - 31)^2 = 0^2 = 0
\][/tex]
- For 24:
[tex]\[
(24 - 31)^2 = (-7)^2 = 49
\][/tex]
4. Sum of the Squared Deviations:
[tex]\[
256 + 100 + 225 + 324 + 0 + 49 = 954
\][/tex]
5. Calculate the Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}
\][/tex]
where [tex]\(N\)[/tex] is the number of data points. Here, [tex]\(N = 6\)[/tex].
[tex]\[
\sigma^2 = \frac{954}{6} = 159.0
\][/tex]
Thus, the variance of the given dataset is:
[tex]\[
\sigma^2 = 159.0
\][/tex]
Variance [tex]\( \boxed{159.0} \)[/tex]
