What does a Standard Deviation of Zero Tell Us?

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In statistics, the standard deviation is used to measure the spread of values in a sample.

We can use the following formula to calculate the standard deviation of a given sample:

√Σ(x_i – x_bar)² / (n-1)

where:

• Σ: A symbol that means “sum”
• x_i: The i^th value in the sample
• x_bar: The mean of the sample
• n: The sample size

The higher the value for the standard deviation, the more spread out the values are in a .

The lower the value for the standard deviation, the more tightly packed together the values.

If the standard deviation of a sample is zero, this means that every value in the sample is the exact same.

In other words, there is zero spread in the values.

The following example shows how to interpret a standard deviation of zero in practice.

Example: How to Interpret a Standard Deviation of Zero

Suppose we collect a of 10 lizards and measure their lengths (in inches):

Lengths: 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

The mean length of lizards in the sample is 7 inches.

Knowing this, we can calculate the sample standard deviation (s) for this dataset:

• s = √Σ(x_i – x_bar)² / (n-1)
• s = √((7 – 7)² + (7 – 7)² + (7 – 7)² + … +  (7 – 7)²/ (10-1)
• s = √0² + 0² + 0² + … + 0²/ 9
• s = 0

The sample standard deviation turns out to be 0.

Will Standard Deviation Ever Be Zero in the Real World?

It’s entirely possible that a real-world dataset could have a standard deviation of zero, but it’s rare.

The most likely scenario where you could encounter a standard deviation of zero would be when collecting small samples for rare events.

For example, suppose you collect data on the number of traffic accidents during a one-week interval in a certain town.

It’s entirely possible that you could collect the following data:

In this scenario, the mean number of daily accidents would be zero and the standard deviation would also be zero.

Or perhaps you collect the following data on the number of monthly sales of some expensive product for some company during a 6-month time frame:

Since the product is so expensive, it just so happens that the company only sells exactly two each month.

In this scenario, the mean number of monthly products sold is two and the standard deviation of monthly products sold is zero.

Whenever you encounter a standard deviation of zero in a real-world dataset, just know that this means every value in the dataset is the exact same.

The following tutorials provide additional information about the standard deviation in statistics: