What is a Statistic?

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A statistic is a single measure of some attribute of a sample e.g. the mean. It is calculated by applying a function to the set of data.

More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample’s distribution; that is, the function can be stated before the data is taken. The term statistic is used both for the function and for the value of the function on a given sample.

A statistic is distinct from a statistical parameter – the mean or standard deviation for example – which is not computable because often the population is much too large to examine and measure all its items, or because the population is transitory – for example to find the average age of the UK population, you could find the mean of the whole population, but people are dying and being born all the time, so the average age is not a constant number but is itself fluctuating. The average age just described is in fact a random variable.

A statistic, when used to estimate a population parameter, is called an estimator. For instance, the sample mean is a statistic which estimates the population mean, which is a parameter. In practice, all parameters are estimated by some statistic. Estimators of parameters are often denoted with a hat, so an estimator ofmay be written

Example: Suppose we have a sample of observations

We can find an estimate of the meanThis is a statistic because it is computed from the data, even though not all the data are used.

Example: Using the same sample above, we can find an estimator for the variance.

The function above is a statistic because it is calculated using only the data, but

is not a statistic because it is calculated using the parameterwhich is not calculated from the data.

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