# Sampling Distributions

Suppose that we draw all possible samples of size *n* from a given
population. Suppose further that we compute a
statistic (e.g., a mean, proportion, standard deviation) for each
sample. The probability
distribution of this statistic is called a **sampling distribution**.

## Variability of a Sampling Distribution

The variability of a sampling distribution is measured by its variance or its standard deviation. The variability of a sampling distribution depends on three factors:

- N: The number of observations in the population.
- n: The number of observations in the sample.
- The way that the random sample is chosen.

If the population size is much larger than the sample size, then the sampling distribution has roughly the same sampling error, whether we sample with or without replacement. On the other hand, if the sample represents a significant fraction (say, 1/10) of the population size, the sampling error will be noticeably smaller, when we sample without replacement.

## Central Limit Theorem

The **central limit theorem** states that the
sampling distribution of any statistic will be normal or nearly normal,
if the sample size is large enough.

How large is "large enough"? As a rough rule of thumb, many statisticians say that a sample size of 30 is large enough. If you know something about the shape of the population distribution, you can refine that rule. The sample size is large enough if any of the following conditions apply.

- The population distribution is normal.
- The population distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less.
- The population distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40.
- The sample size is greater than 40, without outliers.

The exact shape of any normal curve is totally determined by its mean and standard deviation. Therefore, if we know the mean and standard deviation of a statistic, we can find the mean and standard deviation of the sampling distribution of the statistic (assuming that the statistic came from a "large" sample).

## Sampling Distribution of the Mean

Suppose we draw all possible samples of size *n* from a population of size *N*.
Suppose further that we compute a mean score for each sample. In this way, we
create a sampling distribution of the mean.

We know the following.
The mean of the population (μ) is equal
to the mean of the sampling distribution (μ_{x}).
And the standard error of the sampling distribution (σ_{x})
is determined by the standard deviation of the population (σ),
the population size, and the sample size. These relationships are shown in the
equations below:

μ_{x}
= μ and
σ_{x} = σ * sqrt( 1/n - 1/N )

Therefore, we can specify the sampling distribution of the mean whenever two conditions are met:

- The population is normally distributed, or the sample size is sufficiently large.
- The population standard deviation σ is known.

**Note:** When the population size is very large, the factor 1/N is approximately
equal to zero; and the standard deviation formula reduces to:
σ_{x} = σ / sqrt(n).
You often see this formula in introductory statistics texts.

## Sampling Distribution of the Proportion

In a population of size *N*, suppose that the probability of the occurrence
of an event (dubbed a "success") is P; and the probability of the event's
non-occurrence (dubbed a "failure") is Q. From this population, suppose that we
draw all possible samples of size *n*. And finally, within each sample,
suppose that we determine the proportion of successes *p* and failures *q*.
In this way, we create a sampling distribution of the proportion.

We find that the mean of the sampling distribution of the proportion (μ_{p})
is equal to the probability of success in the population (P). And the standard
error of the sampling distribution (σ_{p})
is determined by the standard deviation of the population (σ),
the population size, and the sample size. These relationships are shown in the
equations below:

μ_{p} = P
and σ_{p} = σ * sqrt( 1/n - 1/N ) =
sqrt[ PQ/n - PQ/N ]

where σ = sqrt[ PQ ].

**Note:** When the population size is very large, the factor PQ/N is approximately
equal to zero; and the standard deviation formula reduces to:
σ_{p} = sqrt( PQ/n ).
You often see this formula in intro statistics texts.

## Test Your Understanding

In this section, we offer two examples to illustrate how to apply the Central Limit Theorem to solve some commom statistical problems. Since the Central Limit Theorem makes use of the normal distribution, use the Normal Distribution Calculator to compute probabilities. The Calculator is free.

## Normal Distribution Calculator

The normal calculator solves common statistical problems, based on the normal distribution. The calculator computes cumulative probabilities, based on three simple inputs. Simple instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations. The calculator is free. It can be found under the Stat Tables tab, which appears in the header of every Stat Trek web page.

Normal Calculator |

**Example 1**

Assume that a school district has 10,000 6th graders. In this district, the
average weight of a 6th grader is 80 pounds, with a standard deviation of 20
pounds. Suppose you draw a random sample of 50 students. What is the
probability that the average weight of a sampled student will be less than 75
pounds?

*Solution:* To solve this problem, we need to define the sampling
distribution of the mean. Because our sample size is greater than
40, the Central Limit Theorem tells us that the sampling distribution will
approximate a normal distribution.

To define our normal distribution, we need to know both the mean of the sampling distribution and the standard deviation. Finding the mean of the sampling distribution is easy, since it is equal to the mean of the population. Thus, the mean of the sampling distribution is equal to 80.

The standard deviation of the sampling distribution can be computed using the following formula.

σ_{x}
= σ * sqrt( 1/n - 1/N )

σ_{x} = 20 * sqrt(
1/50 - 1/10000 ) = 20 * sqrt( 0.0199 ) = 20 * 0.141 = 2.82

Let's review what we know and what we want to know. We know that the sampling distribution of the mean is normally distributed with a mean of 80 and a standard deviation of 2.82. We want to know the probability that a sample mean is less than or equal to 75 pounds. To solve the problem, we plug these inputs into the Normal Probability Calculator: mean = 80, standard deviation = 2.82, and value = 75. The Calculator tells us that the probability that the average weight of a sampled student is less than 75 pounds is equal to 0.038.

**Example 2**

Find the probability that of the next 120 births, no more than 40% will be
boys. Assume equal probabilities for the births of boys and girls. Assume
also that the number of births in the population (N) is very large, essentially
infinite.

*Solution:* The Central Limit Theorem tells us that the proportion of boys
in 120 births will be approximately normally distributed.

The mean of the sampling distribution will be equal to the mean of the population distribution. In the population, half of the births result in boys; and half, in girls. Therefore, the probability of boy births in the population is 0.50. Thus, the mean proportion in the sampling distribution should also be 0.50.

The standard deviation of the sampling distribution can be computed using the following formula.

σ_{p} = sqrt[ PQ/n - PQ/N ]

σ_{p} = sqrt[ (0.5)(0.5)/120 ] = sqrt[
0.25/120 ] = 0.04564

In the above calculation, the term PQ/N was equal to zero, since the population size (N) was assumed to be infinite.

Let's review what we know and what we want to know. We know that the sampling distribution of the proportion is normally distributed with a mean of 0.50 and a standard deviation of 0.04564. We want to know the probability that no more than 40% of the sampled births are boys. To solve the problem, we plug these inputs into the Normal Probability Calculator: mean = .5, standard deviation = 0.04564, and value = .4. The Calculator tells us that the probability that no more than 40% of the sampled births are boys is equal to 0.014.

**Note:** This use of the Central Limit Theorem provides a good approximation
of the true probabilities. The exact probability, computed using a binomial
distribution, is 0.018 - very close to the approximation obtained with the
Central Limit Theorem. The accuracy of the approximation increases as sample
size increases.