Variance Calculator

Variance as a variability measure

One of the fundamental aspects of the statistical inference of a given data set is to measure a metric that characterizes the variability of data from their average. The most popular metrics measuring the variability are:

• Variance is the average of the squared deviations from the mean.
• Standard deviation - is the square root of the variance. Standard deviation is a commonly used metric to measure dispersion/variability.
• The coefficient of variation, which is also known as the relative standard deviation. The coefficient of variation is calculated as the ratio of the standard deviation σ to the mean μ or \$(C_v=\frac{σ}{μ})\$.

This calculator finds the variance of a given data set and displays the steps involved in the calculation.

The rules for using this calculator

The variance calculator accepts the input as a list of numbers separated by a delimiter. A few examples of possible input are shown in the table below.

row input	column input	column input	column input
44, 63, 72, 75, 80, 86, 87, 89	44	44,	44,63,72
44 63 72 75 80 86 87 89	63	63,	75,80
44,, 63,, 72, 75, 80, 86, 87, 89	72	72,	86,87
44 63 72 75, 80, 86, 87, 89	75	75,	89
44; 63; 72, 75,, 80, 86, 87, 89	80	80,
44,,, 63,, 72, 75, 80, 86, 87, 89	86	86,
44 63,, 72,,,, 75, 80, 86, 87, 89	87	87,
	89	89,

The numbers can be separated by a comma, a space, a line break, or a mix of more than one type of delimiter. You can use either the row or the column format. For all the formats shown in the above table, the calculator processes the input as 44, 63, 72, 75, 80, 86, 87, and 89.

Once entering the data, you may select whether it is sample data or population data. When you hit the calculate button, the calculator displays five statistical parameters of the dataset: count (number of observations), mean, sum of squared deviations, variance, and the standard deviation.

The calculator is designed to calculate the variance of a data set. It also provides an insight into the theory behind the calculation and shows all the steps involved.

When making inferences, it is preferable to use a large data set to obtain good statistics. But it is often difficult to obtain population data representing all possible observations. Therefore, as a rule, a "sample" is taken from the population. And conclusions about the population are usually drawn from the sample data.

Variance measures a data set's average dispersion in relation to the mean. It is often denoted by σ² for a population and by s² for a sample. A larger value of σ² or s² implies a larger dispersion of data points from the sample mean and vice versa.

Consider the following example data sets.

(Set I) 11, 3, 5, 21, 10, 15, 20, 25, 13, 26, 27,

(Set II) 12, 14, 14, 15, 15, 16, 16, 17, 18, 19, 20

Plugging Set I into the variance calculator yields:

n=11

x̄=16

SS=704

s²=70.4

s=8.39

for a sample, and

n=11

μ=16

SS=704

σ²=64

σ=8

for the population.

Similarly, plugging Set II into the calculator yields:

n=11

x̄=16

SS=56

s²=5.6

s=2.36

for a sample, and

n=11

μ=16

SS=56

σ²=5.09

σ=2.25

for the population.

• In Set I, the numbers deviated significantly from the sample mean

s²=70.4

σ²=64

• In Set II the variability is small

s²=5.6

σ²=5.09

The formula for variance: population variance vs. sample variance

Population variance

Population in statistics refers to all possible observations in an experiment. For N observations, the population variance is:

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

where

• σ² is population variance,
• Σ is summation,
• xᵢ is each observation,
• μ is the population mean,
• n is the number of observations in the population.

Sample variance

The sample variance is defined as

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

where

• s² is the sample variance,
• Σ is summation,
• xᵢ is each observation,
• x̄ is the sample mean,
• n is the number of observations in the sample.

Steps to calculate the variance

The following steps are involved in the calculation of variance.

Step 1: Calculate the sample/population mean. This is the sum of all data points divided by the number of data points (n for a sample and N for the population), i.e.,

Sample mean:

$$\bar{x}=\frac{\sum_{i=1}^{n} x_i}{n}$$

Population mean:

$$\mu=\frac{\sum_{i=1}^{N} x_i}{N}$$

Step 2: Calculate the deviations by subtracting the sample/population mean from each data point, i.e.,

Sample deviations:

$$(x_1-\bar{x}), (x_2-\bar{x}), (x_3-\bar{x}), \ldots, (x_n-\bar{x})$$

Population deviations:

$$(x_1-\mu), (x_2-\mu), (x_3-\mu), \ldots, (x_N-\mu)$$

Step 3: Calculate the squared deviations for each data point.

Sample squared deviations:

$$(x_1-\bar{x})^2, (x_2-\bar{x})^2, (x_3-\bar{x})^2, \ldots, (x_n-\bar{x})^2$$

Population squared deviations:

$$(x_1-\mu)^2, (x_2-\mu)^2, (x_3-\mu)^2, \ldots, (x_N-\mu)^2$$

Step 4: Calculate the sum of the squared deviations.

Sample sum of squared deviations:

$$SS=\sum_{i=1}^{n}(x_i-\bar{x})^2$$

Population sum of squared deviations:

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

Step 5: Divide the sum of the squared deviations by n-1 for a sample and N for the population to calculate the variance.

Sample variance:

$$s^2=\frac{SS}{n-1}$$

Population variance:

$$\sigma^2=\frac{SS}{N}$$

Example of Variance Calculation for a Sample

Let us consider the following data set: 1, 2, 4, 5, 6, and 12. To calculate the sample variance, we follow these steps:

Step 1: Compute the sample mean (average).

$$\bar{x}=\frac{1+2+4+5+6+12}{6}=\frac{30}{6}=5$$

Step 2: Compute the deviations from the mean for each data point.

x₁-x̄	x₂-x̄	x₃-x̄	x₄-x̄	x₅-x̄	x₆-x̄
1 - 5	2 - 5	4 - 5	5 - 5	6 - 5	12 - 5
-4	-3	-1	0	1	7

Step 3: Compute the squares of the deviations.

(x₁-x̄)²	(x₂-x̄)²	(x₃-x̄)²	(x₄-x̄)²	(x₅-x̄)²	(x₆-x̄)²
16	9	1	0	1	49

Step 4: Sum the squared deviations.

$$SS=\sum_{i=1}^{n}{(x_i-\bar{x})}^2=16+9+1+0+1+49=76$$

Step 5: Calculate the sample variance by dividing the sum of squared deviations by the degrees of freedom (n-1).

$$s^2=\frac{SS}{n-1}=\frac{76}{6-1}=\frac{76}{5}=15.2$$

For a population, we would divide by n (the total number of data points), rather than n-1, to calculate the population variance.

The significance of variance

Dispersion is used in investing. It helps asset managers improve the performance of their investments. Financial analysts can use variance to assess the individual performance of components of an investment portfolio.

Investors calculate variance when considering a new purchase to decide whether the investment is worth the risk. Dispersion helps analysts determine a measure of uncertainty, which is difficult to quantify without variance and standard deviation.

Uncertainty is not directly measurable. But the variance and standard deviation (the square root of the variance) help determine the perceived impact of a particular stock on a portfolio.

Scientists, statisticians, mathematicians, and data analysts can also use variance. It helps provide useful information about an experiment or sample population.

Scientists can look for differences between test groups to determine if they are similar enough to test a hypothesis successfully. The higher the variance of the data set, the more scattered the values in the data set. Data researchers can use this information to see how well the mean represents the data set.

The disadvantage of using variance is that large outliers in a set can lead to some distortion of the data. This is because the outliers can increase their weight even further once squared.

Many researchers prefer to work with the standard deviation, calculated as the variance's square root. The standard deviation is less affected by outliers, is a smaller figure, and is easier to interpret.