(Basic Statistics for Citizen Data Scientist)
Basic Concepts of Sampling Distributions
Definition 1: Let x be a random variable with normal distribution N(μ, σ). Now consider a random sample {x1, x2,…, xn} from this population. The mean of the sample (called the sample mean) is
x̄ can be considered to be a number representing the mean of the actual sample taken, but it can also be considered to be a random variable representing the mean of any sample of size n from the population.
Observation: By Property 1 of Estimators, the mean of x̄ is μ (i.e. x̄ is an unbiased estimator of μ) even if the population being sampled is not normal. By Property 2 of Estimators, the variance of x̄ is σ2/n, and so the standard deviation of x̄ is
When the population is normal, we have the following stronger result.
Theorem 1: If x is a random variable with N(μ, σ) distribution and samples of size n are chosen, then the sample mean has normal distribution
Definition 2: The standard deviation of the sample mean is called the standard error of the mean.
Observation: As the sample size increases the standard error decreases, and so the precision of the sample mean as an estimator of the population mean improves.
Example 1: Test scores for a standardized test are distributed N(200, 40). If a random sample of 16 test papers is taken, what is the expected mean of the sample and what is the expected standard deviation of the sample around the mean (i.e. the standard error)? What if the sample has size 100?
The mean of the sample is expected to be 200 in either case. The standard error when n = 16 is 40/4 = 10, while the standard error when n = 100 is 40/10 = 4.
Statistics for Beginners in Excel – Sampling Distributions
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