the standard deviation of the sampling distribution of the means is ____the standard deviation of the population under study.
SEm = SD/√n (?)
The standard deviation of the sampling distribution of the means is typically smaller than the standard deviation of the population under study. This phenomenon is known as the Central Limit Theorem.
To understand why this is the case, let's break down the process:
1. In a population, there is a certain standard deviation (σ) which measures the variability or spread of the data points. This is the standard deviation of the population.
2. When we take a sample from this population, we calculate the mean of the sample. By repeating this process multiple times, we can create a sampling distribution of the means.
3. According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the means approaches a normal distribution, regardless of the shape of the population distribution. As a result, the mean of the sampling distribution will be equal to the mean of the population.
4. The standard deviation of the sampling distribution, denoted as the standard error (SE), is related to the standard deviation of the population (σ) and the sample size (n) by the formula: SE = σ / √n.
From this formula, we can see that as the sample size (n) increases, the denominator becomes larger, thus decreasing the standard error (SE). A smaller standard error means that the values of the sample means are closer to the true population mean and less spread out.
In summary, the standard deviation of the sampling distribution of the means is smaller than the standard deviation of the population because of the effect of the Central Limit Theorem and the decrease in variability as the sample size increases.