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I am getting difficulty in understanding the definition of the statistic.

From wikipedia, I come to understand that statistic is any 'information' (for example, range, mean, variance) of any sample of any given population.

Whereas in my college the definition of statistic is given as:

Suppose $(X_1, X_2,... ,X_i,...,X_n)$ is a random sample of size $n$ from any given PDF or PMF. A function $T=t(X_1, X_2,... ,X_i,...,X_n)$ free from unknown parameter is called a statistic.

I cannot understand the definition given in my college. Are the two definitions related to each other? What is the need of finding the statistic?

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    $\begingroup$ In theory, a statistic is any kind of measurable quantity that is associated to a population (and by extension to a sample). In practice, the statistics most often used in inferential statistics are the mean, the variance, and the proportion. $\endgroup$ – Digio Oct 24 '17 at 14:08
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    $\begingroup$ The two definitions you quote are more or less the same. Wikipedia is giving you helpful examples and is aimed at the lay reader whereas your class text is coming from a mathematical perspective. $\endgroup$ – mdewey Oct 24 '17 at 14:28
  • $\begingroup$ This is purely a matter of notation and terminology: simply read "$t$" as "information" and "pdf or pmf" as "population." The two descriptions are then identical. The phrase "free from unknown parameter" is superfluous (do you see any unknown parameters in the arguments to $t$??). $\endgroup$ – whuber Oct 24 '17 at 15:50
  • $\begingroup$ I think the main point of both definitions is that if you've got $X_1, \dots, X_n \sim \mathcal N(\mu, 1)$ it's not useful to allow your "statistic" to be $t(X_1, \dots, X_n) = \mu$. The whole point is that we need to be able to actually calculate these things, hence they need to be a function only of observable quantities. $\endgroup$ – jld Oct 24 '17 at 16:41
  • $\begingroup$ A statistic is a function of a sample, a definition you will find in almost any textbook $\endgroup$ – Repmat Oct 24 '17 at 19:53
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A statistic is a function of your data.

That's all it is. In different context, you may be interested in different statistics. Maybe T = number of observations. That's a valid statistic. Or T = max value observed. T = seventh observation. T = sixth largest observation. I guess it's valid to say T=1 too, just a constant, although it's quite meaningless. Most commonly as an introduction, we look at T = sample mean, and use it to infer the population parameter.

Maybe your samples aren't event real numbers! For example, maybe your X1, X2, ... are students first names. A valid statistic could be T= most common name. Or T = total number of characters in every name.

To reiterate, a statistic is just a function of your data. In introductory courses, you often work with sample means, sample variances, and play with some algebra to make a mathematical statement about a population's parameter.

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Asking questions in class is never a bad idea. A statistic is an estimate of a population parameter based on a sample. So if mu is the population mean, the sample mean x-bar is the statistic.

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    $\begingroup$ Well, a statistic is not necessarily an estimator ... $\endgroup$ – kjetil b halvorsen Sep 23 at 15:08
  • $\begingroup$ A statistic is an estimator of a parameter. Otherwise it’s just a number. $\endgroup$ – HEITZ Oct 25 at 4:55
  • $\begingroup$ Well, a statistic can also be a test statistic, such as a chi-square, which does not estimate any parameter. $\endgroup$ – kjetil b halvorsen Oct 25 at 12:55

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