**Statistics Calculators** ▶ Variance Calculator

An online variance calculator will help you to determine the variance, sum of squares, and coefficient of variance for a specific data set. In addition, this calculator also displays mean, standard deviation through step-wise calculation. Read on to learn how to find variance and standard deviation using the sample variance formula.

The variance of a group or set of numbers is a number that represents the ‘spread’ of the set. Formally, this is the square of deviation in the set from the mean and the square of the standard deviation.

In other words, a small variance means that the data points tend to be close to the mean and very close to each other. A high variance indicates that the data points are far away from the mean and each other. The variance is the mean of the square of the distance from each point to the mean.

**Sample Variance: **the variance of the sample does not cover the entire possible sample (a random sample of people).

**Population Variance: **the variance which is measured from the whole population (for example, all people).

However, the online Standard Deviation Calculator allows you to determine the standard deviation (σ) and other statistical measurements of the given dataset.

The formula for variance (population) is:

Variance (denoted as σ2) is expressed as the root mean square deviation from the mean for all data points. We write:

$$ σ2 = ∑(xi – μ)^2 / N $$

where,

- σ2 is a variance;
- μ is the root mean square; and
- xᵢ represents the i-th data point among the N shared data points.

You can calculate it with a population variance calculator, otherwise, there are three steps to estimate the variance:

- To find the difference between the mean of a point, use the formula: xi – μ
- Now, take square the difference between the mean of each point: (xi – μ)^2
- Then, find the square mean of the difference from the mean: ∑(xi – μ)^2 / N.

This is the formula for population variance.

The sample variance equation has the following form:

s2 = ∑(xi – x̄)2 / (N – 1)

where,

s2is the variance estimate;

x̄ is the sample mean; and

xi is the i-th data point among the N shared data points.

To find the mean of the given data set. Substitute all values and divide by the sample size n.

ni = 1x in x = ∑ i = 1nx in

Now, find the root mean difference of data value, you need to subtract the mean of data value and square the result.

(xi − x)^2 (xi − x)^2

Then, calculate the quadratic differences, and the sum of squares of all the quadratic differences.

S= ∑ I = 1n (xi – x)^2

So, find the variance, the formula for the variance of the population is:

Variance = σ^2 = Σ (xi − μ)^2

The variance equation of the sample data set:

Variance = s^2 = Σ (xi − x)^{2n−1}

You don’t need to remember these formulas. To make it convenient for you, our sample variance calculator does all variance related calculations automatically by using them.

However, the Mean Median Mode Range Calculator helps you to calculate the mean median mode and range for the entered data set.

**Example calculation**

Let’s calculating the variance of five students’ exam scores: 50, 75, 89, 93, 93. Follow these steps:

**Find the mean**

To find the mean (x), divide the sum of all these values by the number of data points:

x̄ = (50 + 75 + 89 + 93 + 93) / 5

x̄ = 80

- Calculate the difference between the mean and the squared differences from the mean. Hence, the mean is 80, we use the formula to calculate the difference from the mean:

xi – x̄

The first point is 50, so the difference from the mean is 50 – 80 = -30

The squared deviation from the mean is the square of the previous step:

(xi – x̄)2

so, the square of the deviation is:

(50 – 80)2 = (-30)2 = 900

In the table below, the squared deviation calculated from the mean of all test results. The “Mean Deviation” column is the score minus 30, and the “Standard Deviation” column is the column before the square.

Score | Deviation from the mean | Squared deviation |

50 | -30 | 900 |

75 | -5 | 25 |

89 | 9 | 81 |

93 | 13 | 169 |

93 | 13 | 169 |

- Calculate the standard deviation and variance

Next, use the squared deviations from the mean:

σ2 = ∑(xi – x̄)2 / N

σ2 = (900 + 25 + 81 + 169 + 169) / 5

σ2 = 268.5

The exam scores’ variance was 268.8.

An online population variance calculator computes variance for given data sets. You can view the work done for the calculation from the dataset by following these instructions:

- First, enter data set values separated with a comma.
- Then, select variance for a sample or population set.
- Hit the calculate button for getting the results.

- The sample variance calculator displays variance, standard deviation, count, sum, mean, coefficient of variance, and the sum of squares.
- This calculator also provides step-by-step calculations for variance, coefficient of variance, and standard deviation.

The variance is the squared deviation of the mean, and the standard deviation is the square root of the number. Both indicators reflect the variability of the distribution, but their units are different: the standard deviation is determined in the same unit as the original value (for example, minutes or meters).

Low variance is associated with lower risk and lower return. High-variance stocks are generally beneficial to aggressive investors with lower risk aversion, while low-variance stocks are generally beneficial to conservative investors with lower risk tolerance.

The range is the difference between the high value and the low value. Since only extreme values are used because these values will greatly affect it. For finding the range of variance, take the maximum value and subtract the minimum value.

Use this online variance calculator which works for both sample and population datasets using population and sample variance formula. This is the best educational calculator that tells you how to calculate the variance of given datasets in a fraction of a second.

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