
TECHNICAL NOTE  SAMPLING VARIABILITY
MEASURING SAMPLING VARIABILITY
1. Since the estimates in this publication are based on information obtained from occupants 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 dwellings had been included in the survey. One measure of the likely difference is given by the standard error (SE), which indicates the extent to which an estimate might have varied by chance because only a sample of dwellings was included. 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 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.
2. A table of SEs and RSEs for estimates of numbers of persons included in this publication appears at the end of this Technical note. These values do not give a precise measure of the SE or RSE for a particular estimate but will provide an indication of its magnitude.
CALCULATION OF STANDARD ERRORS
3. An example of the calculation and the use of SEs in relation to estimates of persons is as follows. Consider the estimate for Australia of preretired persons with no superannuation, which is 2,836,200. Since this estimate is between 2,000,000 and 5,000,000 in the SE table, the SE for the estimate will be between 30,580 and 45,310 and can be approximated by interpolation as 34,700 (rounded to the nearest 100). Therefore, there are about two chances in three that the value that would have been produced if all dwellings had been included in the survey will fall within 34,700 persons of the survey estimate, i.e. in the range 2,801,500 to 2,870,900, and about 19 chances in 20 that the value will fall within 69,400 persons of the survey estimate, i.e. in the range 2,766,800 to 2,905,600. This example is illustrated in the diagram below.
4. As can be seen from the SE table at the end of this note, the smaller the estimate the higher the RSE. Very small estimates are subject to very high SEs (relative to the size of the estimate). This detracts significantly from their value for most reasonable uses.
5. In the tables in this publication, only estimates with RSEs of less than 25%, and percentages based on such estimates, are considered sufficiently reliable for most purposes. However, estimates with larger RSEs have been included. Estimates with RSEs between 25% and 50% are preceded by an asterisk (*) to indicate they are subject to high SEs and should be used with caution. Estimates with RSEs greater than 50% are preceded by a double asterisk (**) to indicate that they are considered too unreliable for general use.
6. The standard error can be calculated from the relative standard error and the estimate using the following formula.
SE =RSE x Estimate
PROPORTIONS AND PERCENTAGES
7. Proportions and percentages formed from the ratio of two estimates are also subject to sampling errors. The size of the error depends on the accuracy of both the numerator and the denominator. A formula to approximate the RSE of a proportion is given below:
8. Consider the example above of the number of preretired persons with no superannuation (2,836,200). Of these, 1,304,400 or 46.0% were estimated to be male. The SE of 2,836,200 is approximately 34,700 so the RSE is 1.2%. The RSE for 1,304,400 is 1.9%. Applying the formula above, the RSE of the proportion is 1.5% giving a SE for the proportion (46.0%) of 0.68 percentage points. Therefore there are about two chances in three that the proportion of preretired persons with no superannuation who were male is between 45.3% and 46.7% and 19 chances in 20 the proportion is within the range 44.6% and 47.4%
9. Published estimates may also be used to calculate the difference between two survey estimates (numbers or percentages). Such an estimate is subject to sampling error. The sampling error of the difference between the two estimates depends on their SEs and the relationship (correlation) between them. An approximate SE of the difference between two estimates (xy) may be calculated by the following formula:
10. While this formula will be exact only for differences between separate and uncorrelated characteristics of subpopulations, it is expected to provide a good approximation for all differences likely to be of interest in this publication.
11. 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 interviewers and respondents and errors made in coding and processing data. Inaccuracies of this kind are referred to as nonsampling error and they may occur in any enumeration, whether it be a full count or a sample.
STANDARD ERRORS OF ESTIMATES

 Standard Error
 Australia

Size of estimate (persons)  NSW  Vic.  Qld  SA  WA  Tas.  NT  ACT  Standard error  Relative standard error 
 no.  no.  no.  no.  no.  no.  no.  no.  no.  % 

400  0  0  0  0  0  0  0  230  460  115 
500  0  0  0  0  0  320  0  260  520  104 
600  0  0  0  0  0  350  400  290  570  95 
700  0  0  0  0  0  370  430  310  620  89 
800  0  0  0  0  0  400  460  330  670  84 
900  0  0  0  0  0  420  480  350  710  79 
1,000  0  0  0  0  0  440  510  370  750  75 
1,100  0  0  0  0  0  460  530  390  790  72 
1,200  0  0  0  650  700  480  550  410  830  69 
1,300  0  0  0  680  730  500  570  420  870  67 
1,400  0  0  0  710  760  520  590  440  900  64 
1,500  0  0  0  730  780  530  610  460  940  63 
1,600  0  0  950  760  810  550  620  470  970  61 
1,700  0  0  980  780  840  570  640  480  1,000  59 
1,800  0  1,080  1,010  800  860  580  660  500  1,030  57 
1,900  0  1,110  1,030  830  890  600  670  510  1,060  56 
2,000  0  1,140  1,060  850  910  610  690  520  1,090  55 
2,100  0  1,170  1,090  870  930  630  700  530  1,120  53 
2,200  0  1,200  1,120  890  960  640  720  550  1,150  52 
2,300  0  1,230  1,140  910  980  660  730  560  1,170  51 
2,400  0  1,250  1,170  930  1,000  670  750  570  1,200  50 
2,500  1,440  1,280  1,190  950  1,020  680  760  580  1,230  49 
3,000  1,580  1,410  1,310  1,040  1,120  740  830  630  1,350  45 
3,500  1,710  1,530  1,420  1,120  1,210  800  880  680  1,460  42 
4,000  1,830  1,640  1,520  1,200  1,300  850  940  720  1,570  39 
4,500  1,940  1,740  1,620  1,270  1,370  900  990  760  1,670  37 
5,000  2,050  1,840  1,710  1,340  1,450  950  1,040  800  1,760  35 
6,000  2,240  2,020  1,870  1,460  1,590  1,040  1,120  860  1,940  32 
8,000  2,590  2,330  2,160  1,680  1,830  1,190  1,280  980  2,240  28 
10,000  2,890  2,600  2,420  1,860  2,040  1,320  1,410  1,080  2,510  25 
20,000  4,050  3,650  3,400  2,560  2,860  1,840  1,920  1,430  3,560  18 
30,000  4,930  4,420  4,140  3,070  3,470  2,230  2,300  1,680  4,360  15 
40,000  5,650  5,050  4,760  3,480  3,980  2,560  2,620  1,880  5,020  13 
50,000  6,280  5,590  5,290  3,840  4,420  2,850  2,890  2,040  5,590  11 
100,000  8,700  7,630  7,320  5,130  6,080  3,970  3,940  2,610  7,810  8 
200,000  11,970  10,300  10,070  6,790  8,310  5,530  5,360  3,300  10,820  5 
300,000  14,400  12,200  12,090  7,960  9,950  6,710  0  3,750  13,070  4 
400,000  16,390  13,740  13,740  8,880  11,280  7,700  0  0  14,910  4 
500,000  18,120  15,040  15,160  9,660  12,430  0  0  0  16,510  3 
1,000,000  24,620  19,760  20,490  12,440  16,700  0  0  0  22,540  2 
2,000,000  33,280  25,680  27,490  15,830  22,280  0  0  0  30,580  2 
5,000,000  49,130  35,680  40,100  0  0  0  0  0  45,310  1 
10,000,000  0  0  0  0  0  0  0  0  60,540  1 
15,000,000  0  0  0  0  0  0  0  0  71,500  0 

ESTIMATES WITH RELATIVE STANDARD ERRORS OF 25% AND 50%

Size of estimate (persons)  NSW  Vic.  Qld  SA  WA  Tas.  NT  ACT  Aust. 
 no.  no.  no.  no.  no.  no.  no.  no.  no. 

Estimates with RSEs of 25%  13277  10825  9356  5705  6718  2956  3566  2175  10115 
Estimates with RSEs of 50%  3348  2638  2272  1429  1643  784  1022  551  2405 

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