Mean of the sampling distribution of x̄

The mean of the sampling distribution of x̄ (also called mean of x̄) is the mean of all possible sample means.

In the lesson about sampling distribution, we ended up with the following sampling distribution of x̄.

 x̄  P(x̄)
83.33  0.10
85  0.20
85.66  0.20
86.66  0.10
87.33  0.20
89  0.20
  ∑P(x̄) = 1

What if we try to get the mean of all these sample means? Again, this mean is called mean of x̄ for short.

Mean =  Σx̄P(x̄) = 83.33 × 0.10 + 85 × 0.20 + 85.66 × 0.20 + 86.66 × 0.10 + 87.33 × 0.20 + 89 × 0.20

Mean  = 8.333 + 17 + 17.132 + 8.666 + 17.466 + 17.8

Mean = 86.397 and 86.397 rounded to the nearest tenth is 86.4

Recall though that we computed the population mean in the lesson about population distribution and we found that μ = 86.4.

As you can see, the mean of the sampling distribution of x̄ is equal to the population mean.

It is appropriate then to use the symbol μ for the mean of the sampling distribution of x̄ since it is equal to the population mean.

However, for the sake of clarity, we can use μ 

μ  = mean of the sampling distribution of x̄

We write  μ  = μ

The sample mean, x̄ , can also be called an estimator of the population mean,  μ

When  μ  = μ, we say that x̄ is an unbiased estimator.

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