To calculate the variance:
\[ \sigma^2 = \frac{\Sigma(x - \mu)^2}{N} \]
Where:
Distribution variance is a statistical measure that represents the spread or dispersion of a set of values in a dataset. It quantifies how much the values in the dataset deviate from the mean (average) value. A higher variance indicates that the values are more spread out from the mean, while a lower variance indicates that they are closer to the mean. Variance is an important concept in statistics and is used in various fields such as finance, engineering, and social sciences to analyze data variability and make informed decisions.
Consider the dataset: 5, 7, 8, 9, 10.
Calculate the mean: \(\mu = \frac{5 + 7 + 8 + 9 + 10}{5} = 7.8\)
Calculate the variance:
\[ \sigma^2 = \frac{(5 - 7.8)^2 + (7 - 7.8)^2 + (8 - 7.8)^2 + (9 - 7.8)^2 + (10 - 7.8)^2}{5} \]
\[ \sigma^2 = \frac{(-2.8)^2 + (-0.8)^2 + (0.2)^2 + (1.2)^2 + (2.2)^2}{5} = \frac{7.84 + 0.64 + 0.04 + 1.44 + 4.84}{5} = \frac{14.0}{5} = 2.96 \]
So, the variance is 2.80.
Consider the dataset: 2, 4, 6, 8, 10.
Calculate the mean: \(\mu = \frac{2 + 4 + 6 + 8 + 10}{5} = 6\)
Calculate the variance:
\[ \sigma^2 = \frac{(2 - 6)^2 + (4 - 6)^2 + (6 - 6)^2 + (8 - 6)^2 + (10 - 6)^2}{5} \]
\[ \sigma^2 = \frac{(-4)^2 + (-2)^2 + (0)^2 + (2)^2 + (4)^2}{5} = \frac{16 + 4 + 0 + 4 + 16}{5} = \frac{40}{5} = 8 \]
So, the variance is 8.
