In statistics, the ax variance ad to how spread out values room in a offered dataset.

You are watching: Can the variance of a data set ever be negative

One usual question students regularly have around variance is:

Can variance be negative?

The answer: No, variance cannot be negative. The lowest worth it deserve to take ~ above is zero.

To uncover out why this is the case, we should understand just how variance is actually calculated.

### How to calculate Variance

The formula to discover the variance the a sample (denoted together s2) is:

s2= Σ (xi – x)2/ (n-1)

where:

x: The sample meanxi: The ith monitoring in the sampleN: The sample size Σ: A Greek symbol that way “sum”

For example, intend we have actually the adhering to dataset through 10 values: We deserve to use the adhering to steps to calculate the variance the this sample:

Step 1: uncover the Mean

The median is simply the average. This transforms out to be14.7.

Step 2: uncover the Squared Deviations

Next, we can calculate the squared deviation of each individual worth from the mean.

For example, the very first squared deviation is calculated together (6-14.7)2 = 75.69. Step 3: find the sum of Squared Deviations

Next, we deserve to take the sum of all the squared deviations: Step 4: calculation 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 be36.678.

### An instance of Zero Variance

The only means that a dataset deserve to have a variance the zero is if all the the values in the dataset room the same.

For example, the complying with dataset has a sample variance of zero: The typical of the dataset is 15 and none the the individual worths deviate native the mean. Thus, the sum of the squared deviations will certainly be zero and also the sample variance will simply be zero.

### Can traditional Deviation be Negative?

A an ext common method to measure up the spread out of values in a dataset is to use the traditional deviation, which is simply the square root of the variance.

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For example, if the variance of a offered sample is s2 = 36.678, climate the conventional deviation (written ass) is calculation as:

s = √s2= √36.678 =6.056

Since we already know the variance is always zero or a optimistic number, then this way that the conventional deviation deserve to never be an adverse since the square source of zero or a hopeful number can’t it is in negative.