Sample Mean Value

A Sample Mean Value is an mean value that is a sample statistic value (for some sample population).

• AKA: Empirical Average Statistic.
• Context:
  • It is a point estimate of the population mean.
  • It is usually defined as [math]\displaystyle{ \bar{X}=\frac{1}{n}\sum_{i=1}^n X_i }[/math] where [math]\displaystyle{ n }[/math] is the sample size.
  • It approximates the population mean value when [math]\displaystyle{ n }[/math] (sample size) is large enough, i.e for [math]\displaystyle{ n \rightarrow \infty,\; \bar{X}=\mu }[/math].
  • It can (typically) be produced by a Sample Mean Estimator.
• Example(s):
  • 3.5 for sample {3,4} (drawn from some Population).
  • …
• Counter-Example(s):
  • a Population Mean.
  • a Sample Median.
  • a Sample Variance.
  • a Sample Covariance.
• See: Expected Value, Number Sequence.

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References

2017

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Mean
  • For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by [math]\displaystyle{ \bar{x} }[/math], pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted [math]\displaystyle{ \bar{x} }[/math]) to distinguish it from the population mean (denoted [math]\displaystyle{ \mu }[/math] or [math]\displaystyle{ \mu_x }[/math]).

    For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.

1996

• (Underbill & Bradfield, 1996) ⇒ Les Underhill, and Dave Bradfield (1996). Introstat. Juta and Company Ltd, page 181 ISBN:07021388X
  • QUOTE: We now have three concepts, each called a mean: the mean of a sample (chapter 1), the mean of a probability distribution (chapter 5) and now the mean of a population. The sample mean is used to estimate the population mean. When a probability distribution is chosen as a statistical model for a population, one of the criteria for determining the parameters of the probability distribution is that the mean of the probability distribution should be equal to the population mean. This paragraph so far is also true when we replace the word mean with the word variance. It is a universal convention to use the symbols [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \sigma^2 }[/math] for the population mean and population variance, respectively, and the fact that these symbols are also used for the mean and the variance of a probability distribution causes no confusion. Because these are important notions, we risk saying them again. The population mean and variance are quantities that belong to the population as a whole. If you could examine the entire population of interest then you could determine the one true value for the population mean and the one true value for the population variance. Usually, it is impracticable to do a census of every member of a population to determine the population mean. The standard procedure is to take a random sample from the population of interest and estimate [math]\displaystyle{ \mu }[/math], the population mean, by means of [math]\displaystyle{ \bar{x} }[/math], the sample mean.

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