SDMs (squared deviations from the mean) are used in a variety of computations. The definition of variance in probability theory and statistics is either the anticipated value of the SDM (when examining a theoretical distribution) or its average value (for actual experimental data). In **both cases**, the result is called the "squared deviation from the mean".

- What is the square deviation?
- What is the variance and standard deviation of the rating?
- What is the standard deviation of a set?
- Why do we square the deviation scores when calculating the variance?
- Is the standard deviation a measure of variability?
- What is variance a measure of?
- What is the variance of the distribution?

The definition of variance is "the average of the squared departures from the mean." The square root of the variance is known as **the standard deviation**. The standard deviation and variance of a dataset indicate how far it deviates from the mean value. If the standard deviation is high, then the dataset is spread out across a wide range of values; if the standard deviation is low, then the dataset has mostly around mean value.

In statistics, the variance of a set of measurements is the expected value of the sum of the squares of the deviations of each measurement from the overall mean. In **other words**, it is the expected value of the total distance from the mean. For example, if we were to measure the heights of 10 people, and find that on average they were about 5 feet 10 inches tall, then the variance of their height would be 50 (or 0.5) since the average is 5 feet 10 inches and the standard deviation is 10 inches, so the distance between any two people will vary between 40 inches and 60 inches.

In mathematics, the variance of a set of numbers is the expected value of the sum of the products of **each number** in the set by its difference from the mean. That is, it is the expected value of the product of each number by its distance from the mean.

The standard deviation is defined as the square root of the variance. A standard deviation is one method for determining the dispersion of a collection of data. Variance. A measure of the data set's spread is equal to the mean of the squared variances of each data value from the data set's mean. The standard deviation is used because it gives an idea of how far the values are likely to be from their mean.

For example, if you take **a random sample** of sizes 5 and 7 from a population whose mean is 50 and whose variance is 20, then the test statistic (the ratio of the two samples' means) will be equal to 10, which is not close to its theoretical value of 5. Because the sample size of 7 is so much larger than the sample size of 5, some statistical methods would say that there is less uncertainty about the sample mean of 7 than about the sample mean of 5. Even though the sample mean of 5 is closer to **its population** mean than is the sample mean of 7, we can still say with certainty that the population mean $\mu$ is between 45 and 55. This is because the standard deviation of the sample mean, which is equal to the standard error of the mean, is only equal to 6.3 when $n = 7$.

The standard deviation is a statistic that uses the square root of the variance to determine how distant a bunch of values is from the mean. Because outliers are weighted more strongly than data closer to the mean, the variance calculation employs squares. The square root of the variance is then divided by the number of observations to obtain **the standard deviation**.

When the distribution is normal or nearly normal, the standard deviation is the square root of the variance and is a useful measure of variability (see below on the normality of distributions). When the distribution is not normal, the standard deviation does not necessarily correspond to how spread out the data are. For example, the standard deviation of -1 followed by 1 gives you 0, but this does not tell us whether the data are clustered around zero or spread out over 1.

We already know that variance is a measure of how dispersed **a data collection** is. The average squared deviation of **each value** from the mean of a data collection is used to compute it. For example, the mean of the integers 1, 2, and 3 is 2 while the variance is 0.667.

Varance is a measure of how spread out or scattered your data values are. It's used to describe the variability of a statistic derived from a sample of observations. For example, the standard deviation is a measure of variation around a mean; it tells us how much some number of observations vary about the mean. Variance is then used to describe the variability of a statistic derived from a sample of observations.

Another way to think about variance is that it's the average distance between every pair of values in the dataset. There are two ways to do this: calculate the average distance for all possible pairs of values, then divide by the number of pairs; or multiply each value by its distance from the mean (which is the same as saying that it contributes 1 to the sum for itself and minus 1 for every other value) and take the average of these results. Either way you get the same number.

The main use of variance is in statistics, where it's important to know how much one set of numbers varies from another.

The variance (s2) is defined as the total of the squared distances from the mean (m) of each term in the distribution divided by the number of terms in the distribution (N). Take the total of the squares of the distribution's terms and divide it by the number of terms in the distribution (N). The result is the variance.

There are several ways to calculate the variance. Here are two methods:

Variance = [x-m]^2/[N-1]

Var(X) = EX-E(X)^2

Using this formula, we can find the variance for any statistic. In practice, however, it is difficult to estimate these expectations analytically, so they are usually estimated using sample statistics. For example, if we have **a random sample** of size n from the distribution, then the sample variance is an estimate of the population variance.

Sample Variance

The sample variance is simply the sample average of the square of the statistic divided by the number of samples used.