The mean of the sampling distribution, often referred to as the expected value of the sample mean, can be calculated using the following formula:
[
mu_{bar{x}} = mu
]
Where:
– (mu_{bar{x}}) is the mean of the sampling distribution of the sample mean (also known as the expected value of the sample mean).
– (mu) is the population mean.
This principle is rooted in the Central Limit Theorem, which states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the sample size is sufficiently large (typically (n geq 30) is considered adequate).
Additional Information:
- Standard Deviation of the Sampling Distribution:
The standard deviation of the sampling distribution of the sample mean, also known as the standard error (SE), is given by:
[
sigma_{bar{x}} = frac{sigma}{sqrt{n}}
]
Where:
– (sigma_{bar{x}}) is the standard deviation of the sampling distribution (standard error).
– (sigma) is the population standard deviation.
– (n) is the sample size.
- Implications:
- As the sample size (n) increases, the standard error (sigma_{bar{x}}) decreases, which means the sample means will be more tightly clustered around the population mean (mu).
- This makes larger samples more reliable for estimating the population mean.
Example:
If a population has a mean (mu = 50) and a standard deviation (sigma = 10), and we take samples of size (n = 25), the mean of the sampling distribution would still be:
[
mu_{bar{x}} = 50
]
And the standard error would be:
[
sigma_{bar{x}} = frac{10}{sqrt{25}} = frac{10}{5} = 2
]
Thus, the sample means would be expected to cluster around 50 with a standard deviation of 2.