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 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
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
f = frequency of the class
m = midpoint of the class
Try: Variance Calculator
|Variance Type||For Ungrouped Data||For Grouped 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
Step 1: Add up the numbers in your given data set.
3 + 21 + 98 + 203 + 17 + 9 = 351
Step 2: Square your answer:
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.