TECHNICAL NOTE SAMPLING ERROR FROM THE 2012 FORMS OF EMPLOYMENT SURVEY
INTRODUCTION
1 Since the estimates in Section One of 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.
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 ERRORS
3 An example of the calculation and the use of SEs in relation to estimates of people is as follows. Diagram 1 in Appendix 1 shows that the estimated number of persons in Australia who were other business operators was 1,036,900. Since this estimate is between 1,000,000 and 2,000,000, table T1 shows the SE for Australia will be between 11,750 and 17,050 and can be approximated by interpolation using the following general formula:
4 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 in the range 1,025,000 to 1,048,800, and about 19 chances in 20 that the value will fall within the range 1,013,100 to 1,060,700. This example is illustrated in the diagram below:
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. *3.4) to indicate that 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 25% or less.
PROPORTIONS AND PERCENTAGES
6 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:
7 Considering the example above, of the 1,036,900 persons who were other business operators, 407,800 or 39.3% were female. The SE of 407,800, may be calculated by interpolation as 7,500. To convert this to an RSE we express the SE as a percentage of the estimate, or 7,500/407,800 = 1.8%. The SE for 1,036,900 was calculated previously as 11,900, which converted to an RSE is 11,900/1,036,900 = 1.1%. Applying the above formulae, the RSE of the proportion is:
8 The SE for the proportion of females who were other business operators, is 0.6 percentage points, calculated as (39.3/100)x1.4. There are about two chances in three that the proportion of female business operators is between 38.7% and 39.9% and 19 chances in 20 that the proportion is within the range 38.1% to 40.4%.
9 All other estimates produced from population estimates smaller than the values in T2 have RSEs larger than 25% and should be used with caution. T2 also indicates the size of the population estimates that would produce all other estimates with RSEs greater than 50% are considered too unreliable for general use.
DIFFERENCES
10 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 appropriate SE of the difference between two estimates (xy) may be calculated by the following formula:
11 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.
STANDARD ERRORS
T1 STANDARD ERRORS OF ESTIMATES 

         Aust. 
 NSW  Vic.  Qld.  SA  WA  Tas.  NT  ACT  SE  RSE 
Size of estimate  no.  no.  no.  no.  no.  no.  no.  no.  no.  % 

100  290  290  220  180  220  110  80  100  110  110.0 
200  400  380  320  240  290  160  120  170  190  95.0 
300  470  440  390  280  340  190  150  210  260  86.7 
500  580  540  500  340  420  240  190  270  380  76.0 
700  660  620  580  390  480  270  230  300  480  68.6 
1 000  760  710  680  450  550  310  260  330  610  61.0 
1 500  900  830  810  530  640  360  310  360  780  52.0 
2 000  1 010  930  910  590  710  390  340  390  920  46.0 
2 500  1 100  1 000  1 000  650  800  400  350  400  1 050  42.0 
3 000  1 200  1 100  1 050  700  850  450  400  450  1 150  38.3 
3 500  1 250  1 150  1 100  700  900  450  400  450  1 250  35.7 
4 000  1 300  1 200  1 200  750  900  500  450  450  1 350  33.8 
5 000  1 450  1 300  1 250  800  1 000  500  450  500  1 500  30.0 
7 000  1 650  1 500  1 450  900  1 150  600  550  600  1 700  24.3 
10 000  1 850  1 700  1 600  1 050  1 300  700  700  700  2 000  20.0 
15 000  2 150  1 950  1 800  1 200  1 500  850  1 000  850  2 350  15.7 
20 000  2 400  2 200  1 950  1 350  1 650  1 000  1 250  1 000  2 550  12.8 
30 000  2 800  2 550  2 250  1 550  1 900  1 250  1 750  1 250  2 900  9.7 
40 000  3 100  2 800  2 500  1 800  2 100  1 500  2 250  1 500  3 150  7.9 
50 000  3 350  3 050  2 750  2 000  2 300  1 700  2 650  1 650  3 400  6.8 
100 000  4 250  4 000  3 750  3 000  3 400  2 400  4 650  2 250  4 300  4.3 
150 000  5 000  4 850  4 600  3 850  4 450  2 850  6 350  2 500  5 000  3.3 
200 000  5 750  5 650  5 400  4 550  5 350  3 200  7 950  2 650  5 600  2.8 
300 000  7 250  7 250  6 850  5 550  6 750  3 700  10 850  2 800  6 650  2.2 
500 000  10 150  10 050  9 250  7 000  8 600  4 250  . .  2 800  8 350  1.7 
1 000 000  15 100  15 250  13 200  8 900  10 950  4 850  . .  . .  11 750  1.2 
2 000 000  20 350  22 550  17 700  10 600  12 700  . .  . .  . .  17 050  0.9 
5 000 000  25 900  36 100  23 900  11 900  13 250  . .  . .  . .  28 450  0.6 
10 000 000  27 750  49 750  27 950  . .  . .  . .  . .  . .  37 950  0.4 
15 000 000  . .  . .  . .  . .  . .  . .  . .  . .  42 850  0.3 

. . not applicable 
T2 Population levels at which estimates have rses of 25% and 50% 

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

Relative Standard Error (RSE) of 25%  6 300  5 400  5 100  2 600  3 500  1 400  1 100  1 400  6 800 
Relative Standard Error (RSE) of 50%  2 000  1 800  1 700  800  1 200  500  300  600  1 600 

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