Mean and Standard Deviation of the Sampling Distribution of x̄
Suppose that x̄ is the mean of an SRS of size n drawn from a large population with meanμand the standard deviation σ. Then:
- The mean of the sampling distribution of x̄ is µx̄ = μ
- The standard deviation of the sampling distribution of x̄ is σx = σ/√n) as long as the 10% condition is satisfied: n<(1/10)N
The behavior of x̄ in repeated samples is much like that of sample proportion p̂:
- The sample mean x̄ is an unbiased estimator of the population mean μ.
- The values of x̄ are less spread out for larger samples. Their standard deviation decreases at the rate √n), so you must take a sample four times as large to cut the standard deviation of the distribution of x̄ in half.
- You should use the formula σ/√n for the standard deviation of x̄ only when the population is at least 10 times as large as the sample (the 10% condition).
These facts about the mean and standard deviation of x̄ are true no matter what shape the population distribution has.
The shape of the distribution of x̄ depends on the shape of the population distribution, with one important case of note: if the population distribution is Normal, then so is the sampling distribution of x̄. This is true no matter what the sample size is.
Sampling Distribution of a Sample Mean from a Normal Population
Suppose that a population is Normally distributed with mean μ and standard deviation σ. Then the sampling distribution of x̄ has the Normal distribution with mean μ and standard deviation (provided the 10% condition is met) σ/√(n).
Students often forget to divide by the sample size when finding probabilities involving the sample mean.
Remember this from a previous unit? **Z-score = (x-mean)/standard deviation ** It is a remarkable fact that as the sample size increases, the sampling distribution of x̄ changes shape: it looks less like that of the population and more like a normal distribution.
When the sample size is large enough, the sampling distribution of x̄ is very close to normal. This is true no matter what shape the population distribution has, as long as the population has a finite standard deviation σ. This famous fact of probability theory is called the central limit theorem.
Draw an SRS of size n from any population with mean μ and finite standard deviation σ. The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean x̄ is approximately Normal.
How large a sample size n is needed for the sampling distribution of x̄ to be close to Normal depends on the population distribution. More observations are required if the shape of the population is far from normal. In that case the sampling distribution of x̄ will also be very non-Normal if the sample size is small.
Remember that the CLT is only about the shape of the sampling distribution of the sample mean, not the spread.
Normal/Large Sample Condition for Sample Means
If the population distribution is Normal, then so is the sampling distribution of x̄. This is true no matter what the sample size n is.
If the population distribution is not Normal, the CLT tells us that the sampling distribution of x̄ will be approximately Normal in most cases if n ≥ 30.