Standard Errors and Estimation

On the previous page, we derived a formula for finding the Z-score when comparing two sample means.

The denominator of this formula is a general term for obtaining a standard error from any number of samples, each with their own SD.

SE = √(SD_a²/n_a + SD_b²/n_b + SD_c²/n_c)

We can use this formula in other ways. If we derived a value from an equation that involved adding or subtracting a number of different measurements, we can use the calculated SD of the individual measurements to yield an estimate of SE or confidence interval for the derived value. An example might be determining the total length of a road from adding measurements on a number of sub-sections. We could even do this for a single measurement of each variable; we would estimate the measuring accuracy to get an SD for each variable, sum the squares of the SDs (n = 1 in each case) and take the square root. Multiplying the SE by 1.96 as described before gives the confidence interval of the derived value.

You could do something similar for variable values plugged into more complex equations than simple addition or subtraction, but it would be disappointingly complex. For example the overall variance of the product of two variables is not their summation, but their product plus the mean of the first variable squared times the variance of the second variable plus the mean of the second variable squared times the variance of the first variable!

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