How do you add standard deviations?

You can't directly add standard deviations. Instead, you need to add their variances first, and then take the square root of the sum to find the combined standard deviation.

Understanding Why You Can't Simply Add Standard Deviations

Standard deviation is a measure of the spread or dispersion of a set of data points around the mean. It tells you how much the values deviate from the average. Because it's rooted in these deviations, and not the raw data values themselves, directly adding standard deviations doesn't give a meaningful combined measure of spread.

The Correct Method: Adding Variances

Variance, on the other hand, can be combined, provided certain conditions are met. Variance is the square of the standard deviation.

Here's the process:

1. Calculate the Variance: If you have the standard deviation (SD), square it to find the variance (Variance = SD²).
2. Add the Variances: Add the variances of the individual datasets together. This combined variance represents the overall spread.
3. Calculate the Combined Standard Deviation: Take the square root of the sum of the variances to get the combined standard deviation. (Combined SD = √(Combined Variance))

Formula

If you have two datasets with standard deviations SD₁ and SD₂, and you know their variances are Variance₁ and Variance₂, then:

Combined Variance = Variance₁ + Variance₂
Combined SD = √(Variance₁ + Variance₂) = √(SD₁² + SD₂²)

Example

Let's say you have two sets of data:

• Dataset 1: Standard Deviation (SD₁) = 3
• Dataset 2: Standard Deviation (SD₂) = 4

1. Calculate the variances:
   • Variance₁ = 3² = 9
   • Variance₂ = 4² = 16
2. Add the variances:
   • Combined Variance = 9 + 16 = 25
3. Calculate the combined standard deviation:
   • Combined SD = √25 = 5

Therefore, the combined standard deviation is 5.

Important Considerations

• Independence: This method of adding variances assumes that the datasets are independent of each other. If the datasets are correlated, the formula becomes more complex and requires considering the covariance between the datasets.
• Sample Size: If you're working with sample standard deviations, you might need to apply Bessel's correction (using n-1 instead of n when calculating the variance) depending on the context of your analysis.
• Weighted Standard Deviations: If the datasets have different sizes or importances, you may need to use weighted variances and standard deviations.

In Summary

To combine standard deviations, you must first square them to obtain the variances, add the variances, and then take the square root of the result. This method provides a statistically sound way to understand the combined spread of data from multiple sources.