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6360.0 - Superannuation: Coverage and Financial Characteristics, Australia, Jun 2000  
Latest ISSUE Released at 11:30 AM (CANBERRA TIME) 17/09/2001   
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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 pre-retired 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.

Equation - published estimate



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:

Equation

8. Consider the example above of the number of pre-retired 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 pre-retired 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 (x-y) may be calculated by the following formula:

Equation

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 non-sampling 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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