Variance Calculator

Computing the variance of a data set is not so easy. But there is no need to worry, because we are here with the simplest procedure to calculate the variance. So, keep on reading this section and follow these easy steps to solve variance quickly.

• First, get the sample or population data set.
• Find the sum of all the numbers in the data set.
• Divide the total value by the number of data values in the data set and note the result as mean.
• Compute the difference between data value and mean of the data set.
• Square the each and every output obtained after finding difference.
• Sum up all the squared values, divide it number of values in the data set.

Count:

The total number of elements in the data set are called count or size. It is defined as n.

Size = n = count(x_i)_{i = 1}ⁿ

Mean:

It is defined as the average of sample or population data set.

For sample data set

x_mean = (∑_{i = 1}ⁿx_i) / n

For Population data set

μ = (∑_{i = 1}ⁿx_i) / n = Sum / size

Sum of Squares:

It is the sum of squares of difference between data value and mean of the data set.

For a Population data set

SS = ∑_{i = 1}ⁿ (x_i - μ)²

For a Sample data set

SS = ∑_{i = 1}ⁿ (x_i -x_mean)²

Variance:

Variance formula for a a population is

σ² = [(∑_{i = 1}ⁿ (x_i - μ)²) / n]

For a sample data set is

s² = [(∑_{i = 1}ⁿ (x_i - x_mean)²) / (n - 1)]

Standard Deviation:

It is the square root of variance.

For a population

σ = √[(∑_{i = 1}ⁿ (x_i - μ)²) / n]

For a sample

s = √[(∑_{i = 1}ⁿ (x_i - x_mean)²) / (n - 1)]

Example

Question: Calculate variance, sum of squares, standard deviation, mean of the population data set {5, 6, 12, 89}?

Solution:

Given population data set is {5, 6, 12, 89}

count n = 4

Mean = μ = (∑_{i = 1}ⁿx_i) / n

= (5 + 6 + 12 + 89) / 4

= 112 / 4 = 28

Sum of Squares = ss = ∑_{i = 1}ⁿ (x_i -x_mean)²

= (5 - 28)² + (6 - 28)² + (12 - 28)² + (89 - 28)²

= (-23)² + (-22)² + (-16)² + (61)²

= 529 + 484 + 256 + 3721 = 4990

Variance = σ² = [(∑_{i = 1}ⁿ (x_i - μ)²) / n]

= [(5 - 28)² + (6 - 28)² + (12 - 28)² + (89 - 28)²] / 4

= [(-23)² + (-22)² + (-16)² + (61)²] / 4

= (529 + 484 + 256 + 3721) / 4 = 4990 / 4

= 1247.5

Standard Deviation = σ = √[(∑_{i = 1}ⁿ (x_i - μ)²) / n]

= √[[(5 - 28)² + (6 - 28)² + (12 - 28)² + (89 - 28)²] / 4]

= √[[(-23)² + (-22)² + (-16)² + (61)²] / 4]

= √[(529 + 484 + 256 + 3721) / 4]

= √[4990 / 4]

= √[1247.5]

= 35.31

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1. What is meant by variance?

In simply, variance is the average of the squared differences from the mean. It is a measure of how data points differ from the mean and it is represented by σ² for population data set , s² for the sample data set. Variance is the expected value of the squared variation of a random variable from its mean value.

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2. Why variance value is more than standard deviation value?

Standard deviation and variance are closely related topics. Standard deviation is more commonly used because it is more intuitive with respect to the measurement units. Variance is the square of standard deviation. So variance is more than the standard deviation.

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3. What is the variance symbol?

Variance symbol is σ², s².

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4. Compute the variance of sample data set {610, 160, 16}?

Mean = x_mean = (610 + 160 + 16) / 3 = 786/3 = 262

Variance s² = [(∑_{i = 1}ⁿ (x_i - x_mean)²) / (n - 1)]

= [((610 - 216)² + (160 - 216)² + (16 - 216)²) / (4 - 1)]

= (394² + (-56)² + (-200)²) / 3

= (155,236 + 3136 + 40000) / 3 = 198,372/3

= 66,124

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