Solved – Why is the standard deviation of the sample mean less than the population SD

populationsamplingstandard deviation

The sample means do not vary as much as the individual values in the population. That the sample means are less variable than the individual values in the population follows directly from the fact that each sample mean averages together all the values in the sample. A population consists of individual outcomes that can take on a wide range of values, from extremely small to extremely large. However, if a sample contains an extreme value, although this value will have an effect on the sample mean, the effect is reduced because the value is averaged with all the other values in the sample. As the sample size increases, the effect of a single extreme value becomes smaller because it is averaged with more values.

(Excerpt from my stat book)

I find this contradictory because they said that a sample mean averages together all values in the sample, but a population mean also averages together all the values.
Also, they said that an extreme value's effect will be reduced when the sample size increases as it is averaged with more values. But in that case, since the population has the widest range doesn't the same thing apply even more?

So, why is the sample mean's standard deviation less than that of the population?

Best Answer

I find this contradictory because they said that a sample mean averages together all values in the sample, but a population mean also averages together all the values.

The excerpt never says anything about the population mean.

since the population has the widest range doesn't the same thing apply even more?

Absolutely. If you took the mean of the entire population then it would have even less variability. But that has nothing to do with what the excerpt is talking about.

So, why is the sample mean's standard deviation less than the population?

This is explained in the exerpt.

Related Solutions

Confidence Intervals – Estimating Mean Confidence Intervals for a Sample with Known Population Standard Deviation

Suppose you have 150 locations altogether, and you decide to base your confidence interval for the mean of the population (for some attribute) from a sample of size 10.

whole = rnorm(150, 50, 7)
x = sample(whole, 10)
summary(x);  length(x);  sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  37.86   43.03   45.24   47.92   52.61   59.93 
[1] 10         # sample size
[1] 7.470816   # sample standard deviation

t.test(x)$conf.int
[1] 42.57347 53.26207
attr(,"conf.level")
[1] 0.95

The mean for the whole company is 50; a 95% confidence interval for the mean is $(42.6, 53.3).$ I used the t.test procedure in R, but the 95% CI can be found from the formula $\bar X \pm t^* S/\sqrt{n},$ where $t^* = 2.262$ cuts probability 2.5% from the upper tail of Student's t distribution with $\nu = n-1 = 9$ degrees of freedom

qt(.975, 9)
[1] 2.262157

mean(x) + qt(c(.025,.975),9)*sd(x)/sqrt(10)
[1] 42.57347 53.26207

If you knew the population standard deviation $\sigma=7,$ then you could use $\bar X \pm 1.96(7/\sqrt{10}),$ which computes to $(42.6,53.3)).$ In general, this method has the potential to be a little more accurate, but there is no difference (to one place accuracy) from the CI above for this example.

mean(x) + qnorm(c(.025,.975))*7/sqrt(10)
[1] 43.57920 52.25633

Notes: (1) You are sampling from a finite population of size 150. As long as the sample size (here $n=10)$ is less than 10% of the population size, these formulas for sampling from essentially infinite populations should give useful results.

(2) These methods assume that the population values are approximately normally distributed. These methods would not work well if you had a few locations that are hugely different from any of the others.

(3) Your idea of doing some sort of stratified sampling so several provinces are represented or that some observations are from urban and some are from rural location might be useful. That would depend on whether there are large differences among provinces or between rural or urban locations. Stratified sampling would make it somewhat more difficult to make a confidence interval.

(4) Here, because I simulated the whole population, we can find the exact population mean and standard deviation and we know that the data are normal. In most actual applications this information would not necessarily be known.

(5) If you have some data for all 100+ scores, you might try the ttest` on a sample of a dozen or so locations to how well it workd in your application.

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