statistic
A statistic, or sample statistic, is simply a function, usually real-valued, of a set of (sample) data or observations : . More formally, let be the sample space of the data , then is a function from to some set , usually a subset of . The data is usually considered as a vector of iid random variables .
Examples
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1.
100 light bulbs out of 1,000,000 are tested for their functionality. Then the number , of defective light bulbs in the 100 samples is a statistic. To see this, define, for each from 1 to 100,
Then , a function of the data. Similarly, the number of operating light bulbs is also a statistic if we switch the 1 and 0 in the above definitions for the ’s. If we make all , then is just the count of the observations, one of the simplest forms of sample statistics. If we make all , then is a statistic that is not at all useful.
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2.
Let be the weights of 20 students from a particular college. Then the average weight defined by
is a statistic. It is commonly called the sample mean. It is often used to estimate , the expectation of a particular random variable, which, in this case, is the weight of a student in the college. Of course, other averages, such as medians, mode, trimmed mean, are also examples of (sample) statistics.
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3.
Using the same example as in (2), we can define
This is also a statistic, for, after some substitution and rewriting,
which is a function explicitly in terms of the ’s. This statistic is known as the sample variance, which is a common estimate of , the variance of the random variable . Again, in this example, the is the weight of a student in the college.
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4.
Again, borrowing from the same example above, we can simply order the weights of the 20 students in an ascending order, so we get a vector of 20 real numbers . This is also a statistic, called an order statistic. It is not real-valued and its range is a subset of .
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5.
Given a set of numeric observations , without knowing the distribution of these observations, one can define what is known as the empirical distribution function , which is a real-valued function, based on the observations. This is an example of a statistic whose range is a function space.
Remarks.
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Any function of a statistic is again a statistic.
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Since the underlying data is assumed to be random, a statistic is necessarily a random variable.
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Although mostly real-valued, a statistic can be vector-valued, or even function-valued as we have seen in earlier examples.
Title | statistic |
---|---|
Canonical name | Statistic |
Date of creation | 2013-03-22 14:46:18 |
Last modified on | 2013-03-22 14:46:18 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 62A01 |
Defines | sample mean |
Defines | sample variance |