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RELIABILITY OF ESTIMATES

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 Space does not allow for the separate indication of the SEs of all estimates in this publication. Tables of SEs and RSEs for estimates of numbers of persons appear 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. SEs and RSEs for estimates of median personal income per week and separate SEs and RSEs for persons living in cared accommodation have not been included in this publication, but are available on request.

3 The size of the SE increases with the level of the estimate, so that the larger the estimate the larger is the SE. However, the larger the sample estimate the smaller the SE will be in percentage terms (that is, the RSE). Thus, larger estimates will be relatively more reliable than smaller estimates. In the tables in this publication, only estimates with RSEs of 25% or less, and percentages and medians based on such estimates, are considered sufficiently reliable for most purposes. However, estimates, percentages and medians with RSEs between 25% and 50% have been included and are preceded by an asterisk (e.g. *3.4) to indicate that they are subject to high SEs and should be used with caution. Estimates with RSEs greater than 50% are also included and are preceded by a double asterisk (e.g. **0.1). Such estimates are considered too unreliable for general use.

CALCULATION OF STANDARD ERRORS

Standard error of an estimate

4 An example of the calculation and use of SEs is given below. Table 8 in this publication shows that the estimated number of males aged 15-64 years living in households with a moderate core-activity limitation in 2003 was 203,300. The SE for this size of estimate is calculated as follows: the estimate lies between 200,000 and 300,000. The corresponding SEs for these two numbers in the table are 11,750 and 14,250. The SE for 203,300 is calculated by interpolation using the following formula:

5 Therefore, there are about two chances in three that the actual number of males aged 15-64 years living in households with a moderate core-activity limitation was within the range 191,100 to 215,500 and about 19 chances in 20 that it was within the range 178,900 to 227,700.

Standard error of a proportion

6 Proportions and percentages formed from the ratio of two estimates are also subject to sampling error. The size of the error depends on the accuracy of both the numerator and the denominator. The formula for the RSE of a proportion or percentage is :

7 In using the formula, the numerator and the denominator will be estimates over subsets of the population. The formula is only valid when the set for the numerator is a subset of the set for the denominator.

8 The SE of an estimated percentage or rate, computed by using sample data for both numerator and denominator, depends on both the size of the numerator and the size of the denominator. However, the RSE of the estimated percentage or rate will generally be lower than the RSE of the estimate of the numerator.

9 An example from Table 8 is the unemployment rate for females aged 15-64 years with a disability living in households, 8.3%.

10 In this equation:the numerator, the number of unemployed females aged 15-64 years with a disability living in households, is 42,300the denominator, the number of females in the labour force aged 15-64 years with a disability living in households, is 511,700SE for 42,300 = 5,437 or 12.9% RSESE for 511,700 = 18,215 or 3.6% RSEThe difference of the RSE squares = 153.45The square root of the difference is 12.4%, the RSE of the proportion.

Standard error of a difference

11 The difference between two survey estimates is itself an estimate and is therefore subject to sampling variability. The SE of the difference between two survey estimates depends on the SEs of the original estimates and on the relationship (correlation) between the two original estimates. An approximate SE of the difference between two estimates (x-y) may be calculated using the following formula:

12 While this formula will only be exact for differences between separate and uncorrelated (unrelated) characteristics or sub-populations, it is expected to provide a reasonable approximation for all of the differences likely to be of interest.

Significance testing

13 Statistical significance testing has been undertaken for the comparison of estimates between 1998 and 2003 in Tables 3 and 4. The statistical significance test for these comparisons was performed to determine whether it is likely that there is a difference between the corresponding population characteristics. The standard error of the difference between two corresponding estimates (x and y) can be calculated using the formula in the paragraph above. This standard error is then used to calculate the following test statistic:

14 If the value of this test statistic is greater than 1.96 then there are 19 chances in 20 that there is a real difference in the two populations with respect to that characteristic. Otherwise, it cannot be stated with confidence that there is a real difference between the populations.

15 Tables 3 and 4 are annotated to indicate whether or not the estimates which have been compared are statistically significantly different from each other with respect to the test statistic. In all other tables which do not show the results of significance testing, users should take account of RSEs when comparing estimates for different populations.

Non-sampling error

16 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 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.

Age standardisation

17 For this publication the direct age standardisation method was used. The standard population used was the 2003 SDAC survey population. Estimates of age-standardised rates were calculated using the following formula: where:C_direct = the age-standardised rate for the population of interest a = the age categories that have been used in the age standardisationC_a = the estimated rate for the population being standardised in age category a P_sa = the proportion of the standard population in age category a.

18 The age categories used in the standardisation for this publication were 0-4 years, 5-14 years, 15-24 years, 25-34 years, 35-44 years, 45-54 years, 55-64 years, then five-year groups to 90 years and over.

19

T1 Number of persons, Estimates with relative standard errors of 25% and 50%
	NSW	Vic	Qld	SA	WA	Tas	ACT	Australia(a)
Size of estimate
RSE of 25%	14,668	11,833	10,468	6,288	7,540	3,677	3,675	10,350
RSE of 50%	2,949	2,949	2,172	1,200	1,471	772	786	2,139
(a) Includes Northern Territory.

T2 Standard errors of person estimates
Size of estimate	NSW	Vic	Qld	SA	WA	Tas	ACT	Australia(a)
Standard error (number)
500	490	400	420	340	370	300	300	450
1,000	760	650	660	540	580	450	450	690
1,500	980	850	850	690	740	570	570	870
2,000	1,170	1,030	1,020	820	890	670	670	1,030
2,500	1,340	1,190	1,160	930	1,010	750	750	1,170
3,000	1,490	1,330	1,290	1,040	1,120	830	830	1,300
3,500	1,630	1,460	1,420	1,130	1,230	900	900	1,420
4,000	1,760	1,590	1,530	1,220	1,330	960	960	1,530
4,500	1,890	1,700	1,640	1,310	1,420	1,020	1,010	1,630
5,000	2,010	1,810	1,740	1,390	1,500	1,070	1,070	1,730
6,000	2,230	2,020	1,930	1,530	1,660	1,170	1,160	1,920
8,000	2,620	2,380	2,260	1,790	1,950	1,350	1,330	2,250
10,000	2,970	2,700	2,550	2,020	2,190	1,490	1,460	2,540
20,000	4,330	3,910	3,670	2,870	3,130	2,020	1,940	3,680
30,000	5,360	4,800	4,500	3,500	3,820	2,380	2,250	4,560
40,000	6,210	5,520	5,180	4,010	4,380	2,660	2,490	5,290
50,000	6,950	6,130	5,760	4,440	4,860	2,890	2,680	5,930
100,000	9,760	8,370	7,910	6,030	6,620	3,670	3,290	8,390
200,000	13,500	11,130	10,640	8,010	8,830	4,530	3,910	11,750
300,000	16,200	12,990	12,540	9,380	10,360	5,060	4,270	14,250
400,000	18,380	14,410	14,040	10,430	11,540	5,450	4,500	16,290
500,000	20,230	15,580	15,280	11,310	12,530	5,760	4,680	18,060
1,000,000	26,960	19,490	19,630	14,330	15,940	-	-	24,690
2,000,000	35,380	23,760	24,700	-	-	-	-	33,420
5,000,000	49,450	-	-	-	-	-	-	49,050
10,000,000	-	-	-	-	-	-	-	64,800
20,000,000	-	-	-	-	-	-	-	84,600
Relative standard error (%)
500	97.6	80.4	84.2	68.7	74.3	59.2	59.4	90.3
1,000	76.1	65.2	66.1	53.6	57.9	45.0	45.3	68.5
1,500	65.3	57.0	56.8	45.9	49.6	37.9	38.1	58.0
2,000	58.4	51.5	50.8	40.9	44.3	33.3	33.5	51.4
2,500	53.4	47.5	46.5	37.3	40.4	30.1	30.2	46.8
3,000	49.7	44.3	43.1	34.6	37.5	27.6	27.6	43.3
3,500	46.6	41.8	40.5	32.4	35.1	25.6	25.6	40.5
4,000	44.1	39.7	38.3	30.6	33.2	24.0	24.0	38.2
4,500	42.0	37.8	36.4	29.0	31.5	22.6	22.6	36.3
5,000	40.2	36.3	34.8	27.7	30.1	21.5	21.4	34.7
6,000	37.2	33.6	32.1	25.5	27.7	19.6	19.4	32.0
8,000	32.8	29.7	28.3	22.4	24.3	16.8	16.6	28.1
10,000	29.7	27.0	25.5	20.2	21.9	14.9	14.6	25.4
20,000	21.7	19.5	18.4	14.4	15.7	10.1	9.7	18.4
30,000	17.9	16.0	15.0	11.7	12.7	7.9	7.5	15.2
40,000	15.5	13.8	12.9	10.0	11.0	6.6	6.2	13.2
50,000	13.9	12.3	11.5	8.9	9.7	5.8	5.4	11.9
100,000	9.8	8.4	7.9	6.0	6.6	3.7	3.3	8.4
200,000	6.8	5.6	5.3	4.0	4.4	2.3	2.0	5.9
300,000	5.4	4.3	4.2	3.1	3.5	1.7	1.4	4.7
400,000	4.6	3.6	3.5	2.6	2.9	1.4	1.1	4.1
500,000	4.0	3.1	3.1	2.3	2.5	1.2	0.9	3.6
1,000,000	2.7	1.9	2.0	1.4	1.6	-	-	2.5
2,000,000	1.8	1.2	1.2	-	-	-	-	1.7
5,000,000	1.0	-	-	-	-	-	-	1.0
10,000,000	-	-	-	-	-	-	-	0.6
20,000,000	-	-	-	-	-	-	-	-
- nil or rounded to zero (including null cells)
(a) Includes the Northern Territory.