 Variance Formula

In probability theory and statistics, the variance formula measures how far a set of numbers are spread out. It is a numerical value and is used to indicate how widely individuals in a group vary. If individual observations vary considerably from the group mean, the variance is big and vice versa.

A variance of zero indicates that all the values are identical. It should be noted that variance is always non-negative- a small variance indicates that the data points tend to be very close to the mean and hence to each other while a high variance indicates that the data points are very spread out around the mean and from each other.

Variance Formulas

Variance can be of either grouped or ungrouped data. To recall, a variance can of two types which are:

• Variance of a population
• Variance of a sample

The variance of a population is denoted by σ2 and the variance of a sample by s2.

Variance Formulas for Ungrouped Data

Formula For Population Variance

The variance of a population for ungrouped data is defined by the following formula:

• σ2 = ∑ (x − x̅)2 / n

Formula for Sample Variance

The variance of a sample for ungrouped data is defined by a slightly different formula:

• s2 = ∑ (x − x̅)2 / n − 1

Where,

σ2 = Variance

x = Item given in the data

x̅ = Mean of the data

n = Total number of items

s2 = Sample variance

Variance Formulas for Grouped Data

Formula for Population Variance

The variance of a population for grouped data is:

• σ2 = ∑ f (m − x̅)2 / n

Formula for Sample Variance

The variance of a sample for grouped data is:

• s2 = ∑ f (m − x̅)2 / n − 1

Where,

f = frequency of the class

m = midpoint of the class

Summary:

Variance Type For Ungrouped Data For Ungrouped Data
Population Variance Formula σ2 = ∑ (x − x̅)2 / n σ2 = ∑ f (m − x̅)2 / n
Sample Variance Formula s2 = ∑ (x − x̅)2 / n − 1 s2 = ∑ f (m − x̅)2 / n − 1

Also Check: Standard Deviation Formula

Variance Formula Example Question

Question: Find the variance for the following set of data representing trees heights in feet: 3, 21, 98, 203, 17, 9

Solution:

Step 1: Add up the numbers in your given data set.

3 + 21 + 98 + 203 + 17 + 9 = 351

351 × 351 = 123201

…and divide by the number of items. We have 6 items in our example so:

123201/6 = 20533.5

Step 3: Take your set of original numbers from Step 1, and square them individually this time:

3 × 3 + 21 × 21 + 98 × 98 + 203 × 203 + 17 × 17 + 9 × 9

Add the squares together:

9 + 441 + 9604 + 41209 + 289 + 81 = 51,633

Step 4: Subtract the amount in Step 2 from the amount in Step 3.

51633 – 20533.5 = 31,099.5

Set this number aside for a moment.

Step 5: Subtract 1 from the number of items in your data set. For our example:

6 – 1 = 5

Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the variance:

31099.5/5 = 6219.9

Step 7: Take the square root of your answer from Step 6. This gives you the standard deviation:

√6219.9 = 78.86634

The answer is 78.86.

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