You're right, they both look similar at first glance. You could interpret the sample variance of a random variable as the average sum of squared errors, where the error is the deviation of a sample from the sample mean of the given random variable. We have to be careful not to confuse an estimate of the variance with the true population variance though.

Given a true population mean $\mu$, aswell as all $N$ data samples from your distribution, we can calculate the variance as the average expected deviation of the samples from the population mean. 

$$
Var(X) = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2
$$

Whereas, if we interpreted the error as the deviation from the mean, the sum of squared errors would just be

$$
\sum_{i=1}^N (x_i - \mu)^2
$$

You can see, that the sum of squared errors is missing the correction factor $\frac{1}{N}$, in order to be equal to the population variance.
You're right, they both look similar at first glance. You could interpret the sample variance of a random variable as the average sum of squared errors, where the error is the deviation of a sample from the sample mean of the given random variable. We have to be careful not to confuse an estimate of the variance with the true population variance though.

Given a true population mean $\mu$, aswell as all $N$ data samples from your distribution, we can calculate the variance as the average expected deviation of the samples from the population mean. 

$$
Var(X) = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2
$$

Whereas, if we interpreted the error as the deviation from the mean, the sum of squared errors would just be

$$
\sum_{i=1}^N (x_i - \mu)^2
$$

You can see, that the sum of squared errors is missing the correction factor $\frac{1}{N}$, in order to be equal to the population variance.
user
The sample variance is an estimate of the population variance, and it is unbiased.

$$
\hat{Var}(X) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2
$$

Where $\bar{x}$ is the sample mean.

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is
assistant
You're right, they both look similar at first glance. You could interpret the sample variance of a random variable as the average sum of squared errors, where the error is the deviation of a sample from the sample mean of the given random variable. We have to be careful not to confuse an estimate of the variance with the true population variance though.

Given a true population mean $\mu$, aswell as all $N$ data samples from your distribution, we can calculate the variance as the average expected deviation of the samples from the population mean. 

$$
Var(X) = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2
$$

Whereas, if we interpreted the error as the deviation from the mean, the sum of squared errors would just be

$$
\sum_{i=1}^N (x_i - \mu)^2
$$

You can see, that the sum of squared errors is missing the correction factor $\frac{1}{N}$, in order to be equal to the population variance.
user
The sample variance is an estimate of the population variance, and is calculated as

$$
s^2 = \frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2
$$

Where $\bar{x}$ is the sample mean.

The sample variance is an unbiased estimator of the population variance, and is used to calculate the standard deviation of a sample.

The sample variance is also used to calculate the standard error of the mean, which is the standard deviation of the sampling distribution of the sample mean.

The sample variance is also used to calculate the confidence interval of the sample mean, which is the range of values within which the true population mean is likely to fall.

The sample variance is also used to calculate the t-statistic, which is used to test the null hypothesis that the population mean is equal to a specified value.

The sample variance is also used to calculate the F-statistic, which is used to test the null hypothesis that the population variances of two samples are equal.

The sample variance is also used to calculate the chi-squared statistic, which is used to test the null hypothesis that the population distribution is normal.

The sample variance is also used to calculate the correlation coefficient, which is used to measure the strength and direction of the linear relationship between two variables.

The sample variance is also used to calculate the coefficient of determination, which is used to measure the proportion of the variance in the dependent variable that is explained by the independent variable.

The sample variance is also used to calculate the multiple correlation coefficient, which is used to measure the strength and direction of the linear relationship between a dependent variable and two or more independent variables.

The sample variance is also used to calculate the multiple coefficient of determination, which is used to measure the proportion of the variance in the dependent variable that is explained by two or more independent variables.

The sample variance is also used to calculate the analysis of variance, which is used to test the null hypothesis that the population means of two or more samples are equal.

The sample variance is also used to calculate the regression analysis, which is used to estimate the relationship between a dependent variable and one or more independent variables.

The sample variance is also used to calculate the discriminant analysis, which is used to classify individuals into two or more groups based on their scores on two or more variables.

The sample variance is also used to calculate the factor analysis, which is used to identify the underlying factors that explain the variance in a set of variables.

The sample variance is also used to calculate the cluster analysis, which is used to group individuals into clusters based on their similarities.

The sample variance is also used to calculate the principal component analysis, which is used to reduce the dimensionality of a set of
assistant
You're right, they both look similar at first glance. You could interpret the sample variance of a random variable as the average sum of squared errors, where the error is the deviation of a sample from the sample mean of the given random variable. We have to be careful not to confuse an estimate of the variance with the true population variance though.

Given a true population mean $\mu$, aswell as all $N$ data samples from your distribution, we can calculate the variance as the average expected deviation of the samples from the population mean. 

$$
Var(X) = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2
$$

Whereas, if we interpreted the error as the deviation from the mean, the sum of squared errors would just be

$$
\sum_{i=1}^N (x_i - \mu)^2
$$

You can see, that the sum of squared errors is missing the correction factor $\frac{1}{N}$, in order to be equal to the population variance.
user
The sample variance is an estimate of the population variance, and it is unbiased.

$$
\hat{Var}(X) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2
$$

Where $\bar{x}$ is the sample mean.

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is an unbiased estimate of the population variance, because the correction factor $\frac{1}{N-1}$ is the expected value of the sample variance, given the population variance.

$$
E(\hat{Var}(X)) = \frac{1}{N-1}\sum_{i=1}^N (x_i - \mu)^2 = Var(X)
$$

The sample variance is
assistant