Sample Pooled/Combined/Composite/Overall Variance

• is a method for estimating the variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same

Sample Pooled/Combined/Composite/Overall Standard Deviation

• is just a weighted average of the standard deviation (variances) from two or more groups of data when they are assumed to come from populations with a common standard deviation

Sample Pooled Variance - Definition

• $s_p^2=\frac{(n_X-1)(s_X{)}^2+(n_Y-1)(s_Y{)}^2}{n_X+n_Y-2}$

where:

• 𝑠_𝑝² is the sample pooled variance
• 𝑛_𝑋is the sample size for 𝑋
• 𝑛_𝑌is the sample size for 𝑌
• (𝑠_𝑋)²is the sample variance for 𝑋
• (𝑠_𝑌)²is the sample variance for 𝑌

Sample Pooled Standard Deviation - Definition

• $s_p=\sqrt{\frac{(n_X-1)(s_X{)}^2+(n_Y-1)(s_Y{)}^2}{n_X+n_Y-2}}$

where:

• 𝑠_𝑝is the sample pooled standard deviation
• 𝑛_𝑋is the sample size for 𝑋
• 𝑛_𝑌is the sample size for 𝑌
• (𝑠_𝑋)²is the sample variance for 𝑋
• (𝑠_𝑌)²is the sample variance for 𝑌

Sample Pooled Standard Deviation - Expectation - When 𝜎_𝑋 = 𝜎_𝑌 = 𝜎

𝐄(𝑠_𝑝) = 𝜎

proof

• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( [ (𝑛_𝑋 - 1)(𝑠_𝑋)² + (𝑛_𝑌 - 1)(𝑠_𝑌)²]/[ 𝑛_𝑋 + 𝑛_𝑌 - 2 ] ))
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( [ (𝑛_𝑋 - 1)(𝜎_𝑋)² + (𝑛_𝑌 - 1)(𝜎_𝑌)² ] / [ 𝑛_𝑋 + 𝑛_𝑌 - 2 ] )) # expectation of sample standard deviation (𝑠𝑖) is population standard deviation (𝜎𝑖)
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( [ (𝑛_𝑋 - 1)𝜎² + (𝑛_𝑌 - 1)𝜎² ] / [ 𝑛_𝑋 + 𝑛_𝑌 - 2 ] )) # 𝜎𝑋 = 𝜎𝑌 = 𝜎
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( [ (𝑛_𝑋 - 1 + 𝑛_𝑌 - 1)𝜎² ] / [ 𝑛_𝑋 + 𝑛_𝑌 - 2 ] ))
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( [ (𝑛_𝑋 + 𝑛_𝑌 - 2)𝜎² ] / [ 𝑛_𝑋 + 𝑛_𝑌 - 2 ] ))
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( (𝑛_𝑋 + 𝑛_𝑌 - 2) 𝜎² / (𝑛_𝑋 + 𝑛_𝑌 - 2) ))
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡( 𝜎²(𝑛_𝑋 + 𝑛_𝑌 - 2)/(𝑛_𝑋 + 𝑛_𝑌 - 2) ))
• 𝐄(𝑠_𝑝) = 𝐄(𝑠𝑞𝑟𝑡(𝜎²))
• 𝐄(𝑠_𝑝) = 𝐄(𝜎)
• 𝐄(𝑠_𝑝) = 𝜎 # expected value of constant is itself