The sampling mean of the sample distribution is equal to the mean of the population from which the samples are drawn.
In simpler terms, if you take many different samples from a population and calculate the mean of each of those samples, and then you calculate the mean of all of those sample means, that final average will be very close to (and theoretically equal to) the actual mean of the entire population.
Understanding the Concepts
- Population: The entire group you are interested in studying.
- Sample: A subset of the population.
- Sampling Distribution of the Sample Mean: The distribution of sample means obtained from all possible samples of a specific size taken from a population.
- Sampling Mean (Mean of the Sample Distribution): The average of all the sample means in the sampling distribution. Denoted as μ¯¯¯x.
Why is this important?
This principle is a cornerstone of statistical inference. It allows us to estimate the population mean (μ) using sample data (¯¯¯x). Even though we may not be able to measure the entire population, we can use sample means to make inferences about it.
Formula
The sampling mean (μ¯¯¯x) is calculated as follows:
μ¯¯¯x = μ
Where:
- μ¯¯¯x is the mean of the sampling distribution of the sample means.
- μ is the population mean.
Standard Error of the Mean
It is also crucial to consider the standard deviation of the sampling distribution, also known as the standard error of the mean (σ¯¯¯x). This measures the variability of the sample means around the population mean. The formula for standard error is:
σ¯¯¯x = σ / √n
Where:
- σ¯¯¯x is the standard error of the mean.
- σ is the population standard deviation.
- n is the sample size.
A larger sample size (n) will result in a smaller standard error, meaning the sample means will be more closely clustered around the population mean, leading to a more precise estimation.
Example
Imagine you want to know the average height of all adults in a city. It's impossible to measure everyone. Instead, you take 100 random samples of 50 adults each and calculate the average height for each sample. The average of these 100 sample means will give you a very good estimate of the average height of all adults in the city.
