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Article 5. An introduction to estimation—2: from z to t
  1. P Driscoll,
  2. F Lecky
  1. Accident and Emergency Department, Hope Hospital, Salford M6 8HD, UK
  1. Correspondence to: Mr Driscoll, Consultant in Accident and Emergency Medicine (pdriscoll{at}

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  • Comparing a large sample with a population with unknown standard deviation

  • Using a large sample to estimate a population's probability value

  • Comparing a small sample with a population with unknown standard deviation

In covering these objectives we will introduce the following terms:

  • Estimated standard error of the mean

  • Degrees of freedom

  • t statistic


In the previous article we found that it was possible to estimate the probability of getting an element greater than or equal to a particular value (X) in a population with the known parameters, mean (μ) and standard deviation (σ).1 In these cases the z statistic is calculated to locate the position of X in a standard normal distribution where:Math

A similar process can be used when dealing with sample means. If a sufficient number of samples have been taken, and their means plotted, then they begin to take up a normal distribution. It can be shown mathematically that the mean of this distribution (μx) is the same as the population mean (μ). Furthermore, the standard deviation of the distribution is equal to σ/√n, where n is the number of cases in the sample. This is known as the standard error of the mean (SEM). To estimate the probability of getting a value greater than or equal to a particular sample mean (x), in a population with a known mean (μ) and standard deviation (σ), we again calculate the z statistic. However, as we are dealing with the distribution of the means, we use the SEM rather than the population's standard deviation:Math

You will have noticed that both of these calculations are dependent upon knowing the population's mean and standard deviation. In clinical and experimental practice this is rarely the case. However, we know that the best single estimate we have for the parameter μ …

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