Variance in Statistics: Definition, Formula, and Examples

Variance is a statistical concept that is used to assess the spread of values in a set of data in relation to the data’s mean. We utilize variance to determine whether a distribution is pinched or stretched.

The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean.

Variance tells you the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean.

The variance determines the dispersion of data observations from the mean. The greater the variance, the greater the scatter from the mean. While a low variance number indicates that the data is less skewed from the mean.

What is a variance?

According to Wikipedia, a variance is the expectation of the squared deviation of a random variable for its population mean or sample. Variance is taken for two kinds of data, population data, and sample data.

If the variance is used in population data, then it is known as population variance. While if the variance is used for sample data, then it is known as sample variance. Variance can also be stated as the expected value of the squared differences of the number of observations and from the mean.

The main work of the variance is to measure the distance for each observation from the mean. The distance decides whether the number is closer to the mean or far. It is usually denoted by sigma square for the population variance written as Οƒ2. For sample variance, it is denoted by s2. We can also write variance such as var(x).

Formulas of the variance

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Main results


S^2=sample variance
x_i=the value of the one observation
bar{x}=the mean value of all observations
n=the number of observations

From the web for a population, the variance is calculated as ΟƒΒ² = ( Ξ£ (x-ΞΌ)Β² ) / N. Another equivalent formula is σ² = ( (Ξ£ xΒ²) / N ) – ΞΌΒ². If we need to calculate variance by hand, this alternate formula is easier to work with.

How to Calculate Variance

By using formulas, we can easily find the variance for the population data or sample data.

  1. Find the mean of the data set. Add all data values and divide by the sample size n.π‘₯⎯⎯⎯=βˆ‘π‘›π‘–=1π‘₯𝑖𝑛xΒ―=βˆ‘i=1nxin
  2. Find the squared difference from the mean for each data value. Subtract the mean from each data value and square the result.(π‘₯π‘–βˆ’π‘₯⎯⎯⎯)2(xiβˆ’xΒ―)^2
  3. Find the sum of all the squared differences. The sum of squares is all the squared differences added together.𝑆𝑆=βˆ‘π‘›π‘–=1(π‘₯π‘–βˆ’π‘₯⎯⎯⎯)2S S=βˆ‘i=1n(xiβˆ’xΒ―)^2
  4. Calculate the variance. Variance is the sum of squares divided by the number of data points.The formula for variance for a population is:Variance = πœŽ2=Ξ£(π‘₯π‘–βˆ’πœ‡)2/𝑛 Οƒ2=Ξ£(xiβˆ’ΞΌ)^2/n The formula for variance for a sample set of data is:Variance = π‘ 2=Ξ£(π‘₯π‘–βˆ’π‘₯⎯⎯⎯)^2/π‘›βˆ’1

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