How to Calculate a Pooled Standard Error

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Statisticians often compare two or more groups when conducting research. Either because of participant dropout or funding reasons, the number of individuals in each group can vary. In order to make up for this variation, a special type of standard error is used which accounts for one group of participants contributing more weight to the standard deviation than another. This is known as a pooled standard error.

    Conduct an experiment and record the sample sizes and standard deviations of each group. For example, if you were interested in the pooled standard error of the daily caloric intake of teachers versus school children, you would record the sample size of 30 teachers (n1 = 30) and 65 students (n2 = 65) and their respective standard deviations (let’s say s1 = 120 and s2 = 45).

    Calculate the pooled standard deviation, represented by Sp. First, find the numerator of Sp²: (n1 – 1) x (s1)² + (n2 – 1) x (s2)². Using our example, you would have (30 – 1) x (120)² + (65 – 1) x (45)² = 547,200. Then find the denominator: (n1 + n2 – 2). In this case, the denominator would be 30 + 65 – 2 = 93. So if Sp² = numerator / denominator = 547,200 / 93 ? 5,884, then Sp = sqrt(Sp²) = sqrt(5,884) ? 76.7.

    Compute the pooled standard error, which is Sp x sqrt(1/n1 + 1/n2). From our example, you would get SEp = (76.7) x sqrt(1/30 + 1/65) ? 16.9. The reason you use these longer calculations is to account for the heavier weight of students affecting the standard deviation more and because we have unequal sample sizes. This is when you have to “pool” your data together to conclude more accurate results.


About the Author

Sky Smith has been writing on psychology, electronics, health and fitness since 2002 for various online publications. He graduated from the University of Florida with honors in 2005, earning a Bachelor of Science in psychology and statistics with a minor in math.

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