A pooled standard deviation is simply a weighted average of standard deviations from two or more independent groups.
In statistics it appears most often in the two sample t-test, which is used to test whether or not the means of two populations are equal.
The formula to calculate a pooled standard deviation for two groups is as follows:
Pooled standard deviation = √ (n1-1)s12 + (n2-1)s22 / (n1+n2-2)
where:
- n1, n2: Sample size for group 1 and group 2, respectively.
- s1, s2: Standard deviation for group 1 and group 2, respectively.
Note that the pooled standard deviation should only be used when the standard deviations between the two groups can be assumed to be roughly equal.
Also note that because the pooled standard deviation is a weighted average, it will give more “weight” to the group with the larger sample size.
Example: Calculating the Pooled Standard Deviation
Suppose we have two different groups with the following information:
Group 1:
- Sample size (n1): 15
- Sample standard deviation (s1): 6.4
Group 2:
- Sample size (n2): 19
- Sample standard deviation (s2): 8.2
We can calculated the pooled standard deviation for these two groups as:
Pooled standard deviation = √ (15-1)6.42 + (19-1)8.22 / (15+19-2) = 7.466
Notice how the value for the pooled standard deviation (7.466) is between the values for the standard deviation of group 1 (6.4) and group 2 (8.2).
This should make sense considering the pooled standard deviation is just a weighted average between the two groups.
Bonus: Pooled Standard Deviation Calculator
You can also use the Pooled Standard Deviation Calculator to quickly calculate the pooled standard deviation between two groups.
For example, we could plug in the values from the previous example to come up with the same pooled standard deviation that we calculated by hand:
Note that you can also use the “Enter raw data” option on the calculator to enter the raw data values for the two groups and calculate the pooled standard deviation in that manner.
Very good & clear site info thnks alot
Pair of brackets missing in formula for pooled st. dev.
Is this equation missing a pair of parentheses? I think you’re supposed to add the numbers in the numerator together before dividing, but I think the way it’s currently written does not make it clear that the numerator includes both of the terms. The way it’s written now means that you have to divide the second term ( (n2 – 1)s2^2 ) by the denominator and then add that number to the first term. If this were written like a fraction, with the numerator above the denominator, instead of like a division problem where all the terms are side-by-side, that would also make this order more clear.
Thanks!
You are missing brackets in your formula of pooled SD.
Pooled standard deviation = √ (15-1)6.4^2 + (19-1)8.2^2 / (15+19-2) as a linear formula following order of operations would compute (19-1)8.2^2 / (15+19-2) first followed by the addition of (15-1)6.4^2. The result of this, if plugged into a calculator in the way you have written the formula would be 24.800.
What you mean to write is:
Pooled standard deviation = √ ((15-1)6.4^2 + (19-1)8.2^2) / (15+19-2). This calculation would satisfy the order of operations to result in the given pooled standard deviation of 7.466.
Please update your website
From which source of literature do you have this formula?
Hi Julia…The formula for calculating **pooled standard deviation** is widely cited in statistical literature and resources. It is often derived and discussed in textbooks and papers on statistics, particularly in sections about hypothesis testing, analysis of variance (ANOVA), or when comparing means from two or more groups.
Here is the formula for pooled standard deviation:
\[
s_p = \sqrt{\frac{(n_1 – 1)s_1^2 + (n_2 – 1)s_2^2}{n_1 + n_2 – 2}}
\]
Where:
– \( s_p \) = Pooled standard deviation
– \( s_1, s_2 \) = Standard deviations of the two samples
– \( n_1, n_2 \) = Sample sizes of the two groups
### Sources of Literature
1. **Textbooks:**
– **”Statistics for the Behavioral Sciences” by Frederick J. Gravetter and Larry B. Wallnau**: This book explains pooled variance in the context of t-tests.
– **”Introduction to the Practice of Statistics” by David S. Moore et al.**: Covers pooled standard deviation in depth when discussing two-sample comparisons.
– **”Applied Statistics and Probability for Engineers” by Douglas C. Montgomery**: Includes pooled standard deviation in sections on hypothesis testing.
2. **Statistical Guides:**
– Online resources like **Khan Academy**, **Statistical Software Documentation (e.g., SPSS, R, Python’s SciPy)**, and university lecture notes often mention the formula.
3. **Research Papers:**
– Articles discussing **meta-analysis** or **t-tests for independent samples** frequently use pooled standard deviation for comparing group variability.