Australian Bureau of Statistics

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Relative Standard Error
 

Measuring variability due to sampling and randomly adjustment

There are two sources of uncertainty or variability associated with survey estimates that are released by Survey TableBuilder. The first source of variability is due to sampling and the second is due to random adjustment of cell counts.

Variability due to Sampling

Since the estimates from surveys are based on information obtained from a sub-sample of usual residents of a sample of dwellings, they are subject to sampling variability; that is, they may differ from those that would have been produced if all usual residents of all dwellings had been included in the survey. This component of variability is measured by the group Jacknife method.

Variability due to Random Adjustment

The random adjustment of totals and subtotals introduces another source of variability into the estimates. As these adjustments are generated in a predictable way the impact they have on estimates can be measured directly.

Standard Errors

The variability due to sampling and random adjustment are combined into a single measure called the Standard Error (SE). The SE indicates the extent to which an estimate might have varied by chance, because only a sample of dwellings was included, and by random adjustment.

There are about two chances in three that a sample estimate will differ by less than one SE from the number that would have been obtained if all dwellings had been included and there was no random adjustment. There is about 19 chances in 20 that the difference will be less than two SEs. Another measure of the likely difference is the relative standard error (RSE), which is obtained by expressing the SE as a percentage of the estimate.

RSE%( x ) =SE(x) / x * 100

SEs of proportions and percentages

Proportions and percentages formed from the ratio of two estimates are also subject to sampling errors. The size of the error depends of the accuracy of both the numerator and denominator. A formula to approximate the RSE of a proportion is given below:

RSE%( x/y ) =sqrt[RSE%(x)]2 - [RSE%(y)]2

Note - this formula only holds when the x is a subset of y. It should not be used if this is not the case i.e. estimates of 'rates' as opposed to proportions.

Using this formula, the RSE of the estimated proportion or percentage will be lower than the RSE estimate of the numerator. Therefore an approximation for SEs of proportions or percentages may be derived by neglecting the RSE of the denominator i.e. obtaining the RSE of the number of persons corresponding to the numerator of the proportion or percentage and then applying this figure to the estimated proportion or percentage. This approach was adopted for the purposes of assigning the * or ** to indicate a 25% or 50% RSE threshold in publications from the NHS and NHSI.

SEs may also be used to calculate SEs for the difference between two survey estimates (numbers or percentages). The sampling error of the difference between the two estimates depends on their individual SEs and the relationship (correlation) between them. An approximate SE of the difference between two estimates (x-y) may be calculated by the following formula:
SE(x-y) =sqrt[SE(x)]2 +[SE(y)]2

While this formula will only be exact for differences between separate and uncorrelated characteristics of subpopulations, it is expected to provide a reasonable approximation for most differences likely to be of interest in relation to this survey.

The imprecision due to sampling variability, which is measured by the SE, should not be confused with inaccuracies that may occur because of imperfections in reporting by respondents and recording by interviewers, and errors made in coding and processing data. Inaccuracies of this kind are referred to as non-sampling error, and they may occur in any enumeration, whether it be a full count or a sample. Every effort is made to reduce non-sampling error to a minimum by careful design of questionnaires, intensive training and supervision of interviewers, and efficient operating procedures.

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