What Is Variance in Statistics? Definition, Formula, and Example

Samples are taken to give an indication of the entire population data. Dividing by n-1 gives a sample variance or standard deviation that better reflects the population variance or standard deviation. Sample Variance – If the size of the population is too large then it is difficult to take each data point into consideration. In such a case, a select number of data points are picked up from the population to form the sample that can describe the entire group.

It shows the amount of variation that exists among the data points. Visually, the larger the variance, the “fatter” a probability distribution will be. In finance, if something like an investment has a greater variance, it may be interpreted as more risky or volatile. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. In sample variance and standard deviation, a denominator of n-1 is used to reduce bias in the estimation of the population.

There are two types of variance based on the type of data set being analyzed. Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared.

We first create a table to organise the data with the data listed in the first column as xi. A table is constructed with the data listed in the first column as xi. There are three measures of central tendency, namely, mean, median, and mode. Some of the properties of variance are given below that can help in solving both simple and complicated problem sums.

Sample Variance Formula

Thus, the sample variance can be defined as the average of the squared distances from the mean. The variance is always calculated with respect to the sample mean. While variance measures the spread of a single variable around its mean, covariance extends this concept to measure how two random variables change together.

Covariance tells us how the random variables are related to each other and it tells us how the change in one variable affects the change in other variables. In statistics, variance measures variability from the average or mean. The population variance is calculated when the data considered is that of an entire population. The sample variance is used when the data considered is a sample of a larger set of data. The population variance formula has a denominator of ‘N’, whereas the sample variance formula has a denominator of ‘n-1’.

For vector-valued random variables

Real-world observations such as the measurements of yesterday’s rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations.

Video Lesson: How to Calculate Variance

Sample Variance and Population Variance are the two types of variance. A general definition of variance is that it is the expected value of the squared differences from the mean. Similar to standard deviation, variance can be analyzed for ungrouped data (individual data points) and grouped data (data organized in intervals with frequencies).

Other tests of the equality of variances include the Box test, the Box–Anderson test and the Moses test.

The standard deviation and the expected absolute deviation can both be used as an indicator of the “spread” of a distribution. Sample variance is calculated when a sample of a larger set of data has been taken. The mean used is the sample mean, which is the mean of the data in the sample. Population variance is calculated whenever data concerning the whole population is known.

How to Calculate the Variance of a Data Set

Both variance and standard deviation indicate the dispersion of data points in a dataset by measuring their deviation from the mean. Thus, the population variance is 38.57, and the sample variance is 39.78. Thus, the population variance is 8, and the sample variance is 10. ‘Variance’ refers to the spread or dispersion of a dataset in relation to its mean value. A lower variance means the data set is close to its mean, whereas a greater variance indicates a larger dispersion.

Variance of Binomial Distribution

• Variance is defined as the square of the standard deviation, i.e., taking the square of the standard deviation for any group of data gives us the variance of that data set.
• The Standard Deviation is a measure of how spread out numbers are.
• Thus, the population variance is 8, and the sample variance is 10.
• There is a definite relationship between Variance and Standard Deviation for any given data set.
• The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n − 1.5 yields an almost unbiased estimator.

If the data is clustered near the mean then the variance will be variance interpretation lower. We can expect about 68% of values to be within plus-or-minus 1 standard deviation. Population variance is mainly used when the entire population’s data is available for analysis.

• There are three measures of central tendency, namely, mean, median, and mode.
• When we take the square of the standard deviation we get the variance of the given data.
• It tells us about how the population of a group varies with respect to the mean population.
• Both variance and standard deviation indicate the dispersion of data points in a dataset by measuring their deviation from the mean.

Sum of uncorrelated variables

There can be two types of variances in statistics, namely, sample variance and population variance. Variance is widely used in hypothesis testing, checking the goodness of fit, and Monte Carlo sampling. To check how widely individual data points vary with respect to the mean we use variance. In this article, we will take a look at the definition, examples, formulas, applications, and properties of variance. When the population data is very large, calculating the variance directly becomes difficult. In such cases, a sample is taken from the dataset, and the variance calculated from this sample is called the sample variance.

Population Variance – All the members of a group are known as the population. When we want to find how each data point in a given population varies or is spread out then we use the population variance. It is used to give the squared distance of each data point from the population mean. Variance is defined using the symbol σ2, whereas σ is used to define the Standard Deviation of the data set. Variance of the data set is expressed in squared units, while the standard deviation of the data set is expressed in a unit similar to the mean of the data set.