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Article 4. An introduction to estimation—1. Starting from Z
1. P Driscoll,
2. F Lecky,
3. M Crosby
1. Accident and Emergency Department, Hope Hospital, Salford M6 8HD, UK
1. Correspondence to: Mr Driscoll, Consultant in A&E Medicine (pdriscoll{at}hope.srht.nwest.nhs.uk)

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## Objectives

• Estimating an element from a population with known standard deviation

• Estimating a statistic from a population with known standard deviation

In covering these objectives we will introduce the following terms:

• Standard normal distribution

• z statistic

• Confidence intervals

In the previous article the term inferential statistic was introduced.1 This form of numeric manipulation is often used to estimate a population's parameter from a sample's statistic. For example, inferential statistics would be used to estimate a population's mean from a sample's mean. It can also be used to do the opposite—that is, estimate a sample statistic from a population's parameter. This is not commonly done because it requires the population's mean and standard deviation to be known and this is rarely the case.

In both calculations the values obtained are only estimations because of the normal variation that occurs. We can however work out the probability of a particular value based upon information from either the sample or population. Central to this is converting the original data to a standard normal distribution so that these estimations can be made.

## Standard normal distribution

A standard normal distribution is a particular type of normal distribution that has the following properties:

• Symmetrical bell shaped curve

• Mean equal to zero

• Standard deviation equal to 1

• Total area under the curve equal to 1

It is possible to convert any normal distribution to a standard normal distribution by adjusting it such that the population mean becomes zero and the standard deviation is equal to 1. To do this, each of the data points (elements) is modified by:

• Subtracting the population mean

• Dividing the result by the standard deviation of the population

The final value is known as the z statistic. The z statistic is therefore describing the size of the difference between the element (X) and population's mean (μ) …

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