Variance in statistics is a measure of how much a set of values differs from the mean (average) of the dataset. It quantifies the spread or dispersion of data points in a dataset. Mathematically, the variance is the average of the squared differences between each data point and the mean. A higher variance indicates that the data points are more spread out, while a lower variance suggests they are closer to the mean.

The formula for variance differs depending on whether you are calculating for a population or a sample:

1. Population Variance (denoted as $\sigma^2$): $\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2$ Where:

   • $N$ is the total number of data points in the population.
   • $x_i$ is each data point.
   • $\mu$ is the population mean.

2. Sample Variance (denoted as $s^2$): $s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$ Where:

   • $n$ is the number of data points in the sample.
   • $x_i$ is each data point in the sample.
   • $\bar{x}$ is the sample mean.

The key difference between the two is the denominator: $N$ is used for population variance, while $n-1$ is used for sample variance to account for the bias in using a sample to estimate a population parameter (this is called Bessel's correction).

Would you like further details or examples?

Related Questions:

1. What is the difference between variance and standard deviation?
2. How do you calculate the mean and variance of a dataset?
3. What is Bessel's correction, and why is it applied to sample variance?
4. Can variance be negative? Why or why not?
5. How is variance used in real-world applications like finance or science?

Tip: Variance is always non-negative, as it involves squaring the differences, which ensures all values are positive or zero.