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How Population Standard Deviation Affects Standard Error

In statistics, an important component of standard error involves the amount of variability in the population (measured by standard deviation). In the standard error formula

you see the population standard deviation,

is in the numerator. That means as the population standard deviation increases, the standard error of the sample means also increases. Mathematically this makes sense; how about statistically?

Distributions of fish lengths a) in pond #1; b) in pond #2.
Distributions of fish lengths a) in pond #1; b) in pond #2.

Suppose you have two ponds full of fish (call them pond #1 and pond #2), and you’re interested in the length of the fish in each pond. Assume the fish lengths in each pond have a normal distribution. You’ve been told that the fish lengths in pond #1 have a mean of 20 inches and a standard deviation of 2 inches (see Figure (a)). Suppose the fish in pond #2 also average 20 inches but have a larger standard deviation of 5 inches (see Figure (b)).

Comparing Figures (a) and (b), you see the lengths for the two populations of fish have the same shape and mean, but the distribution in Figure (b) (for pond #2) has more spread, or variability, than the distribution shown in Figure (a) (for pond #1). This spread confirms that the fish in pond #2 vary more in length than those in pond #1.

Now suppose you take a random sample of 100 fish from pond #1, find the mean length of the fish, and repeat this process over and over. Then you do the same with pond #2. Because the lengths of individual fish in pond #2 have more variability than the lengths of individual fish in pond #1, you know the average lengths of samples from pond #2 will have more variability than the average lengths of samples from pond #1 as well. (In fact, you can calculate their standard errors using the above formula to be 0.20 and 0.50, respectively.)

Estimating the population average is harder when the population varies a lot to begin with — estimating the population average is much easier when the population values are more consistent. The bottom line is the standard error of the sample mean is larger when the population standard deviation is larger.

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