6265.0 - Underemployed Workers, Australia, Sep 2004  
ARCHIVED ISSUE Released at 11:30 AM (CANBERRA TIME) 15/03/2005   
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TECHNICAL NOTE DATA QUALITY


INTRODUCTION

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 estimates 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 (67%) 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 (95%) 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 Due to space limitations, it is impractical to print the SE of each estimate in the publication. Instead, a table of SEs is provided to enable readers to determine the SE for an estimate from the size of that estimate (see table T1). The SE table is derived from a mathematical model, referred to as the 'SE model', which is created using data from a number of past Labour Force Surveys. It should be noted that the SE model only gives an approximate value for the SE for any particular estimate, since there is some minor variation between SEs for different estimates of the same size.



CALCULATION OF STANDARD ERROR

3 An example of the calculation and the use of SEs in relation to estimates of persons is as follows. Table 5 shows the estimated number of female part-time workers who want more hours was 382,600. Since this estimate is between 300,000 and 500,000, table T1 shows that the SE for Australia will lie between 7,250 and 8,800 and can be approximated by interpolation using the following general formula:

Equation: Calculation of standard errors


4 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 the range 374,700 to 390,500 and about 19 chances in 20 that the value will fall within the range 366,800 to 398,400. This example is illustrated in the following diagram.

Diagram: Confidence intervals of estimates


5 In general, the size of the SE increases as the size of the estimate increases. Conversely, the RSE decreases as the size of the estimate increases. Very small estimates are thus subject to such high RSEs that their value for most practical purposes is unreliable. In the tables in this publication, only estimates with RSEs of 25% or less are considered reliable for most purposes. Estimates with RSEs greater than 25% but less than or equal to 50% are preceded by an asterisk (e.g.*3.2) to indicate they are subject to high SEs and should be used with caution. Estimates with RSEs of greater than 50%, preceded by a double asterisk (e.g.**0.3), are considered too unreliable for general use and should only be used to aggregate with other estimates to provide derived estimates with RSEs of less than 25%.



MEANS AND MEDIANS

6 The RSEs of estimates of mean duration of insufficient work, median duration of insufficient work and mean preferred number of extra hours are obtained by first finding the RSE of the estimate of the total number of persons contributing to the mean or median (see table T1) and then multiplying the resulting number by the following factors:

  • mean duration of insufficient work: 1.7
  • median duration of insufficient work: 2.1
  • mean preferred number of extra hours: 0.8.

7 The following is an example of the calculation of SEs where the use of a factor is required. Table 5 shows that the estimated number of male part-time workers who want more hours was 230,300 with a median duration of insufficient work of 26 weeks. The SE of 230,300 can be calculated from table T1 (by interpolation) as 6,600. To convert this to a RSE we express the SE as a percentage of the estimate or 6,600/230,300 =2.9%.


8 The RSE of the estimate of median duration of insufficient work is calculated by multiplying this number (2.9%) by the appropriate factor shown in paragraph 6 (in this case 2.1): 2.9 x 2.1 = 6.1%. The SE of this estimate of median duration of insufficient work is therefore 6.1% of 26, i.e. about 2 (rounded to the nearest whole week). Therefore, there are two chances in three that the median duration of insufficient work for males that would have been obtained if all dwellings had been included in the survey would have been within the range 24-28 weeks, and about 19 chances in 20 that it would have been within the range 22-30 weeks.



PROPORTIONS AND PERCENTAGES

9 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. This formula is only valid when x is a subset of y.

Equation: Calculation of relative standard errors of proportions and percentages


10 Considering the example from paragraph 3, of the 382,600 females who usually work part time and want more hours, 143,400 or 37.5% had insufficient work for 52 weeks or more. The SE of 143,400 may be calculated by interpolation as 5,500. To convert this to an RSE we express the SE as a percentage of the estimate, or 5,500/143,400 = 3.8%. The SE for 382,600 was calculated previously as 7,900, which converted to an RSE is 7,900/382,600 = 2.1%. Applying the above formula, the RSE of the proportion is:

Equation: Example calculation of relative standard errors of proportions


11 Therefore, the SE for the proportion of females who have a current period of insufficient work of 52 weeks or more is 1.2 percentage points (=(37.5/100)x3.2). Therefore, there are about two chances in three that the proportion of females who have a current period of insufficient work of 52 weeks or more was between 36.3% and 38.7% and 19 chances in 20 that the proportion is within the range 35.1% to 39.9%.



DIFFERENCES

12 Published estimates may also be used to calculate the difference between two survey estimates (of numbers or percentages). Such an estimate is subject to sampling error. The sampling error of the difference between 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: Calculation of differences between estimates


13 While this formula will only be exact for differences between separate and uncorrelated characteristics or subpopulations, it is expected to provide a good approximation for all differences likely to be of interest in this publication.

T1 STANDARD ERRORS OF ESTIMATES

AUST.
NSW
Vic.
Qld
SA
WA
Tas.
NT
ACT
SE
RSE
Size of estimates (persons)
no.
no.
no.
no.
no.
no.
no.
no.
no.
%

100
90
100
180
160
160
110
110
130
80
80.0
200
160
170
260
220
220
140
150
160
140
70.0
300
220
230
310
260
260
170
180
180
190
63.3
500
330
320
390
320
340
210
220
220
270
54.0
700
420
400
460
370
390
240
250
240
350
50.0
1,000
530
500
540
420
460
280
290
270
440
44.0
1,500
690
630
650
500
550
330
340
310
580
38.7
2,000
820
750
740
570
620
370
380
350
700
35.0
2,500
950
850
800
600
700
400
400
400
800
32.0
3,000
1,050
950
900
650
750
450
450
400
900
30.0
3,500
1,150
1,000
950
700
800
450
450
450
1,000
28.6
4,000
1,250
1,100
1,000
750
850
500
500
450
1,050
26.3
5,000
1,400
1,200
1,100
850
900
550
550
500
1,200
24.0
7,000
1,650
1,400
1,300
950
1,050
600
600
550
1,450
20.7
10,000
1,950
1,700
1,500
1,100
1,200
700
700
650
1,750
17.5
15,000
2,350
2,000
1,800
1,300
1,450
800
800
750
2,150
14.3
20,000
2,700
2,250
2,050
1,450
1,600
900
900
850
2,450
12.3
30,000
3,150
2,650
2,450
1,700
1,850
1,050
1,050
1,000
2,950
9.8
40,000
3,500
2,900
2,750
1,900
2,100
1,200
1,150
1,100
3,350
8.4
50,000
3,800
3,150
3,000
2,100
2,250
1,300
1,250
1,250
3,700
7.4
100,000
4,750
4,000
4,000
2,750
2,900
1,700
1,600
1,650
4,850
4.9
150,000
5,350
4,600
4,750
3,250
3,350
1,950
1,850
2,000
5,600
3.7
200,000
5,900
5,150
5,300
3,650
3,750
2,150
2,050
2,300
6,250
3.1
300,000
6,900
6,100
6,250
4,300
4,300
2,500
. .
2,750
7,250
2.4
500,000
8,550
7,700
7,650
5,250
5,050
3,050
. .
. .
8,800
1.8
1,000,000
11,950
10,800
10,050
6,850
6,350
. .
. .
. .
11,550
1.2
2,000,000
17,600
15,650
13,100
9,000
7,800
. .
. .
. .
15,250
0.8
5,000,000
31,550
26,900
18,450
. .
. .
. .
. .
. .
23,400
0.5
10,000,000
. .
. .
. .
. .
. .
. .
. .
. .
40,950
0.4

. . not applicable

T2 LEVELS AT WHICH ESTIMATES HAVE RELATIVE STANDARD ERRORS OF 25% AND 50%(a)

NSW
Vic.
Qld
SA
WA
Tas.
NT
ACT
Aust.
no.
no.
no.
no.
no.
no.
no.
no.
no.

RSE OF 25%

Mean duration of current period of insufficient work
11,800
10,800
8,300
4,600
5,600
2,000
1,300
2,100
12,000
Median duration of current period of insufficient work
18,900
14,300
12,200
6,700
8,000
3,200
2,800
2,900
16,300
Mean preferred number of extra hours
5,300
4,400
4,200
2,300
2,700
900
800
1,100
4,100
All other estimates
6,200
4,700
4,100
2,500
2,900
1,200
1,000
1,100
4,600

RSE OF 50%

Mean duration of current period of insufficient work
2,800
2,800
2,400
1,400
1,700
600
400
700
2,500
Median duration of current period of insufficient work
5,000
3,900
3,500
2,100
2,400
1,000
900
1,000
3,700
Mean preferred number of extra hours
900
900
1,200
700
800
300
300
400
600
All other estimates
1,200
1,000
1,200
800
900
400
300
400
700

(a) Refers to the number of persons contributing to the estimate.