The Relationship Between Mean & Standard Deviation (With Example)


The mean represents the average value in a dataset.

It is calculated as:

Sample mean = Σxi / n

where:

  • Σ: A symbol that means “sum”
  • xi: The ith observation in a dataset
  • n: The total number of observations in the dataset

The standard deviation represents how spread out the values are in a dataset relative to the mean.

It is calculated as:

Sample standard deviation = √Σ(xi – xbar)2 / (n-1)

where:

  • Σ: A symbol that means “sum”
  • xi: The ith value in the sample
  • xbar: The mean of the sample
  • n: The sample size

Notice the relationship between the mean and standard deviation: The mean is used in the formula to calculate the standard deviation.

In fact, we can’t calculate the standard deviation of a sample unless we know the sample mean.

The following example shows how to calculate the sample mean and sample standard deviation for a dataset in practice.

Example: Calculating the Mean & Standard Deviation for a Dataset

Suppose we have the following dataset that shows the points scored by 10 different basketball players:

We can calculate the sample mean of points scored by using the following formula:

  • Sample mean = Σxi / n
  • Sample mean = (22+14+15+18+19+8+9+34+30+7) / 10
  • Sample mean = 17.6

The sample mean of points scored is 17.6. This represents the average number of points scored among all players.

Once we know the sample mean, we can the plug it into the formula to calculate the sample standard deviation:

  • Sample standard deviation = √Σ(xi – xbar)2 / (n-1)
  • Sample standard deviation = √((22-17.6)2 + (14-17.6)2 + (15-17.6)2 + (18-17.6)2 + (19-17.6)2 + (8-17.6)2 + (9-17.6)2 + (34-17.6)2 + (30-17.6)2 + (7-17.6)2) / (10-1)
  • Sample standard deviation = 9.08

The sample standard deviation is 9.08. This represents the average distance between each points value and the sample mean of points.

It’s helpful to know both the mean and the standard deviation of a dataset because each metric tells us something different.

The mean gives us an idea of where the “center” value of a dataset is located.

The standard deviation gives us an idea of how spread out the values are around the mean in a dataset. The higher the value for the standard deviation, the more spread out the values are in a sample.

By knowing both of these values, we can know a great deal about the distribution of values in a dataset.

Additional Resources

The following tutorials provide additional information about the mean and standard deviation:

Why is the Mean Important in Statistics?
Why is Standard Deviation Important in Statistics?
How to Calculate the Mean and Standard Deviation in Excel

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