Learn more. Explanation for Additive Property of Variance? Ask Question. Asked 2 years, 9 months ago.
Active 2 years, 9 months ago. Viewed 2k times. Improve this question. Fudge Fudge 41 3 3 bronze badges. In my own uninitiated terms, what is the magical property of variance that standard deviation do not possess. More details here. Add a comment. Active Oldest Votes. Improve this answer.
Matthew Drury Matthew Drury 33k 2 2 gold badges silver badges bronze badges. ColorStatistics ColorStatistics 1, 8 8 silver badges 23 23 bronze badges. Sign up or log in Sign up using Google. How do I calculate the variance and standard deviation of , , , , ? What is the variance of 2, 9, 3, 2, 7, 7, 12? The sum of all X's is How can I calculate the variance? What Excel formula do I use to calculate the variance between two cells? How do you find the sample variance in 89, 57, , 73, 26, , 81?
How do you find the mean of the random variable x? He following table shows the probability distribution for a discrete random variable. What is the variance of X?
How do I find the variance in 40; 56; 59; 60; 60; 62; 65; 69; 75; 84 statistics? What is the difference between variance and variability? How do you find the variance of the data 2, 4, 6, 8, 10? Black Belts are taught that you add the variances.
Then you take the square root of the resulting variance to compute the standard deviation. Six Sigma instructors drill this idea into their students telling them time and time again that we add variances not standard deviations.
What you may not have learned is that this additive property relies on the linearity of the equation relating the distributions together. If the equation is quadratic, or some other non-linear combination, or if the coefficients of the transfer function are not equal to one, there is more to combining the variances than just summing them.
This discussion of the equations behind combining variances looks at three examples — two where the transfer function is linear and one where it is not. To discuss the equations governing the addition of variances, one must assume that the transfer function for a process can be obtained through statistically designed experiments or other means.
Now consider the basic equation in all Six Sigma projects:. Y Y is the dependent output variable of a process. It is used to monitor a process to see if it is out of control, or if symptoms are developing within a process. It is a function of the Xs that contribute to the process. Equation 1 states that the output, y, is a function of various inputs, or x values. In order to calculate the variance in the output based on the variance in all of the inputs, the equation is as follows:.
This equation is the part you may not have seen before. In words, it states that the total variance in the output is equal to the sum of the contributions from all of the inputs.
It further states that the contribution of each input is equal to the partial derivative of the function with respect to that input, which is then multiplied by the standard deviation of that input, and then that quantity is squared to put it in terms of variance.
Note that Equation 2 uses the first order terms of a Taylor Series expansion, and assumes that all inputs are normally, or at least symmetrically, distributed. Equation 2 also assumes that the inputs are not correlated, meaning that there is no covariance among them. Now Equation 2 will be applied to three different examples. Consider the case of a linear transfer function with all coefficients equal to one as shown in Equation 3. In this example, because the partial derivatives with respect to each of the inputs are equal to one, the variances of the input values can be added to arrive at the variance in the output.
Stated mathematically:. This case, where the variances in the inputs are simply added together to obtain the variance in the output, is the most basic example. However, be aware that combining variances this way requires that the transfer function is linear and that the coefficients of that transfer function all equal one. Now consider the case of a linear transfer function with coefficients not equal to one, such as Equation 5.
In this case, the partial derivative of the output with respect to the first input, x1, is equal to 3, while the partial derivative of the output with respect to the second input, x2, is 4.
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