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For a population that has a standard deviation of 10, figure the standard deviation of the distribution of means for samples of size- (b) 3.

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Mar 20, 2011
Statistics - Distribution of Means
by: Staff


The question:

For a population that has a standard deviation of 10, figure the standard deviation of the distribution of means for samples of size- (b) 3.


The answer:

The Central Limit Theorem can be used to answer your question.


To illustrate:


Suppose you decided to compute the average (the mean) for a source population.

You take 10 separate samples. Each sample consists of 15 items (a sample size of 15). After taking the samples, you compute the average for each of the 10 samples.

The average computed for each of the 10 separate samples is not the same – close, but you are not entirely satisfied with the results.

You decide to take another 10 samples and compute the average for each. This time, you decide to take 10 samples of 20 items each (a sample size of 20).

As before, the averages for each of the 10 samples are not the same. However, the average values computed are much closer to one another because you used a larger sample size (20 items, in this example).

The Central Limit Theorem states that the larger the sample size (15, 20, . . .) used to compute the average, the more closely the various averages you compute will form a normal distribution. This is the normal distribution of the sample means (or, normal distribution of the sample averages).

The average (mean) of this normal distribution of means will be exactly the same as the average (mean) for your entire population.

The standard deviation of the sampling distribution of the means is:

standard deviation of the distribution of means = σ/sqrt(n)

σ = standard deviation of your population

n = sample size


your question: standard deviation of the distribution of means for samples of size 3.

σ = 10

n = 3

standard deviation of the distribution of means = 10/sqrt(3)

standard deviation of the distribution of means = 5.77

the final answer is: standard deviation of the distribution of means = ±5.77


Thanks for writing.


Staff
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