In statistics, the term variance refers to how spread out values are in a given dataset.
One common question students often have about variance is:
Can variance be negative?
The answer: No, variance cannot be negative. The lowest value it can take on is zero.
To find out why this is the case, we need to understand how variance is actually calculated.
How to Calculate Variance
The formula to find the variance of a sample (denoted as s2) is:
s2 = Σ (xi – x)2 / (n-1)
- x: The sample mean
- xi: The ith observation in the sample
- N: The sample size
- Σ: A Greek symbol that means “sum”
For example, suppose we have the following dataset with 10 values:
We can use the following steps to calculate the variance of this sample:
Step 1: Find the Mean
The mean is simply the average. This turns out to be 14.7.
Step 2: Find the Squared Deviations
Next, we can calculate the squared deviation of each individual value from the mean.
For example, the first squared deviation is calculated as (6-14.7)2 = 75.69.
Step 3: Find the Sum of Squared Deviations
Next, we can take the sum of all the squared deviations:
Step 4: Calculate the Sample Variance
Lastly, we can calculate the sample variance as the sum of squared deviations divided by (n-1):
s2 = 330.1 / (10-1) = 330.1 / 9 = 36.678
The sample variance turns out to be 36.678.
An Example of Zero Variance
The only way that a dataset can have a variance of zero is if all of the values in the dataset are the same.
For example, the following dataset has a sample variance of zero:
The mean of the dataset is 15 and none of the individual values deviate from the mean. Thus, the sum of the squared deviations will be zero and the sample variance will simply be zero.
Can Standard Deviation Be Negative?
A more common way to measure the spread of values in a dataset is to use the standard deviation, which is simply the square root of the variance.
For example, if the variance of a given sample is s2 = 36.678, then the standard deviation (written as s) is calculated as:
s = √ = √ = 6.056
Since we already know that variance is always zero or a positive number, then this means that the standard deviation can never be negative since the square root of zero or a positive number can’t be negative.