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Mar 26, 2023 · The sample mean is a random variable; as such it is written \(\bar{X}\), and \(\bar{x}\) stands for individual values it takes. As a random variable the sample mean has a probability distribution, a mean \(μ_{\bar{X}}\), and a standard deviation \(σ_{\bar{X}}\).
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.
Let \(X_1,X_2,\ldots, X_n\) be a random sample of size \(n\) from a distribution (population) with mean \(\mu\) and variance \(\sigma^2\). What is the mean, that is, the expected value, of the sample mean \(\bar{X}\)?
Basically, you learn about populations working under the assumption that you know the mean/stdev, which is silly, as you say, but later you begin to drop these assumptions and learn to make inferences about populations based on your samples.
- 11 min
- Sal Khan
- There is a difference. Your "samples" (random selections of values "x") that are made up of "instances" (referred to as the variable "n") provide w...
- If a question talks about a "population proportion" then you are dealing with a binomial distribution, except that you divide by the sample size to...
- Yes, the sample sizes should be the same. The sample size is not considered to be a variable, it's considered to be a constant. The sampling distri...
- The sample distribution is what you get directly from taking a sample. You plot the value of each item in the sample to get the distribution of val...
- Properly, the sampling distribution APPROXIMATES a normal distribution for a sufficiently large sample (sometimes cited as n > 30). A coin flip is...
- Not every distribution goes to the Normal. the distribution of the sample mean does, but that's as the sample size increases. If you have smaller s...
- X-bar is the mean of a sample (as Sal says at 4:29 in https://www.khanacademy.org/math/probability/descriptive-statistics/central_tendency/v/statis...
The sample mean x−− is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. We will write X−− when the sample mean is thought of as a random variable, and write x−− for the values that it takes. The random variable X−− has a mean, denoted μX−−, and a standard deviation, denoted σX−−.
Solution. The sample mean is: x ¯ = 7 + 6 + 8 + 4 + 2 + 7 + 6 + 7 + 6 + 5 10 = 5.8. Sample Variance. The sample variance, denoted s 2 and read "s-squared," summarizes the "spread" or "variation" of the data: s 2 = ( x 1 − x ¯) 2 + ( x 2 − x ¯) 2 + ⋯ + ( x n − x ¯) 2 n − 1 = 1 n − 1 ∑ i = 1 n ( x i − x ¯) 2. Sample Standard Deviation.
