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- 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 ˉX when the sample mean is thought of as a random variable, and write x for the values that it takes. The random variable ˉX has a mean, denoted μˉX, and a standard deviation, denoted σˉX.
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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 \(σ_{\bar{X}}\).
- 5.5: Sample Mean - Statistics LibreTexts
The mean of SB girls depends on which sample we randomly...
- 6.2: The Sampling Distribution of the Sample Mean ...
Let \(\overline{X}\) be the mean of a random sample of size...
- 5.5: Sample Mean - Statistics LibreTexts
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 \(\bar{X}\)?
- Mean Or Expected Value: μ
- Variance: var
- Standard Deviation: Σ
When we know the probability p of every value xwe can calculate the Expected Value (Mean) of X: Note: Σ is Sigma Notation, and means to sum up. To calculate the Expected Value: 1. multiply each value by its probability 2. sum them up Note: this is a weighted mean: values with higher probability have higher contribution to the mean.
The Variance is: To calculate the Variance: 1. square each value and multiply by its probability 2. sum them up and we get Σx2p 3. then subtract the square of the Expected Value μ2
The Standard Deviation is the square root of the Variance: Let's have another example! (Note that we run the table downwards instead of along this time.) Let's try that again, but with a much higher probability for $50,000:
The sampling distribution of the sample mean can be thought of as "For a sample of size n, the sample mean will behave according to this distribution." Any random draw from that sampling distribution would be interpreted as the mean of a sample of n observations from the original population.
- 11 min
- Sal Khan
- Learning statistics can be a little strange. It almost seems like you're trying to lift yourself up by your own bootstraps. Basically, you learn ab...
- 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...
- 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−−.
