Allocate a stratified sample
For a proportional stratified sample, calculate each subgroup's proportional amount, round to the nearest whole number with halves rounded up, then correct the total using eligible groups
- Find the population total Add all subgroup sizes before forming any subgroup fraction \(\displaystyle N=44+53+103=200\)
- Multiply each subgroup fraction Divide a subgroup size by the population total, then multiply by the target sample size \(\displaystyle a_1=\frac{44}{200}\times20=4.4\)
- Round each amount Round each proportional amount to the nearest whole number, rounding halves up \(\displaystyle \operatorname{round}(4.4)=4,\quad \operatorname{round}(5.3)=5,\quad \operatorname{round}(10.3)=10\)
- Correct the total For a shortfall, add one to groups rounded down, starting with the largest fractional parts. For a surplus, subtract one from groups rounded up, starting with the smallest fractional parts. Adjust each selected group once; break equal-fraction ties by adjusting the earlier table row \(\displaystyle 4+5+10=19<20\;\Rightarrow\;5+5+10=20\)
Worked example
A survey record set is divided into three groups with population sizes 42, 57 and 81. A stratified random sample of 24 records is required. Allocate the sample in proportion to the group sizes. Round to the nearest whole number, rounding halves up. For a shortfall, add one to groups rounded down, starting with the largest fractional parts. For a surplus, subtract one from groups rounded up, starting with the smallest fractional parts. Adjust each selected group once; break equal-fraction ties by adjusting the earlier table row
- \(\displaystyle N=42+57+81=180\)Add the three subgroup sizes to obtain the population total
- \(\displaystyle a_1=\frac{42}{180}\times24=5.6,\quad a_2=\frac{57}{180}\times24=7.6,\quad a_3=\frac{81}{180}\times24=10.8\)Multiply each subgroup fraction by the target sample size to obtain the unrounded amounts
- \(\displaystyle r_1=\operatorname{round}(5.6)=6,\quad r_2=\operatorname{round}(7.6)=8,\quad r_3=\operatorname{round}(10.8)=11,\quad 6+8+11=25\)Round each amount and compare the rounded total with the target of 24
- \(\displaystyle f_1=0.6,\quad f_2=0.6,\quad f_3=0.8,\quad s_1=6-1=5,\quad s_2=8,\quad s_3=11,\quad 5+8+11=24\)The total is one too large. Among groups rounded up, subtract one from a group with the smallest fractional part. Groups 1 and 2 tie; adjust the earlier row, Group 1
Common mistakes
Difficulty levels
Questions about this topic
Why must the subgroup fraction use the population total?+
The subgroup fraction describes the subgroup's share of the whole population, so its denominator is the population total. The target sample size is then used to scale that share to the required sample
Why might the rounded allocations need an adjustment?+
Individual proportional amounts are usually not whole numbers. Rounding them separately can make their sum one or more away from the target, so a smallest whole-number correction is required
How is a group chosen for the rounding adjustment?+
For a shortfall, add one to groups rounded down, starting with the largest fractional parts. For a surplus, subtract one from groups rounded up, starting with the smallest fractional parts. Adjust each selected group once; break equal-fraction ties by adjusting the earlier table row
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 14: Statistical inference