# Subgroups

Household Income (monthly)Population shareProportional design100 respondents per subgroupCell Weights
I: up to $2,5000.280100 (25%)0.8 II:$2,500+ to 5,0000.4160100 (25%)1.6
III: $5,000+ to 7,5000.3120100 (25%)1.2 IV: above$7,5000.140100 (25%)0.4
TOTAL1400400 (100%)_

#### Exhibit 34.5   Proportional and disproportional sample designs for subgroups.

In many quantitative research studies, it is common to report results for relevant subgroups. These subgroups can include demographics (such as income, gender, and age), buying behaviours (such as brand/category usage or usage rate), and geographical differences.

When the sample size for each subgroup is proportional to its population size, it is possible that some subgroup samples end up being too small to analyse effectively. For example, in Exhibit 34.5, subgroups IV and I have sample sizes of less than 100.

To address these inadequacies, each subgroup’s sample size should meet specific requirements. In the cited example, a subgroup sample size of 100 limits the sampling error to ±10%. In disproportionate designs like this, each subgroup must be weighted to account for the disproportions.

The cell weight, as shown in Exhibit 34.5, is calculated by dividing the population share by the sample share. For instance, for the first group, this would be 20% divided by 25%, resulting in a weight of 0.8. If the number of “yes” or “success” respondents is 20 for subgroup I, 30 for II, 40 for III and 50 for IV, the weighted average for the entire sample would be 20 × 0.8 + 30 × 1.6 + 40 × 1.2 + 50 × 0.4 = 132 or 33% (132/400). If the population is 1 million, the estimated number of “successes” is 330 thousand (33% × 1 million, or 132 × 1 million/400).

For major subgroups, research agencies typically recommend sample sizes of 100 or more (with a margin of error of ±10%). For minor subgroups, the sample size should be at least 50 (with a margin of error of ±14%).

Disproportional designs are commonly used in the majority of quantitative research studies. However, if information about the proportions is not available, disproportional designs should not be employed.

The sampling error for disproportional samples is slightly greater compared to proportional sample designs. In the aforementioned example, the confidence interval would be ±5.4% for a confidence level of 95% (with Z = 1.96), as opposed to ±4.9% for proportional sampling.

To compute the margin of error, we need to estimate the variance, which in the case of subgroup (or stratified) sampling is computed as follows: $$σ^2= \sum_{k=1}^K(N_k/N)^2×(N_k-n_k)/N_k×p_k(1-p_k)/n_k$$

Where:

• σ2: population variance.
• K: the number of subgroups.
• N: total universe population
• Nk: population of subgroup k.
• nk: sample size for subgroup k.
• pk: probability of “success”.

Conservatively setting pk = 0.5, and assuming Nk >> nk, the formula reduces to:

$$σ^2=0.25\sum_{k=1}^K(N_k/N)^2×1/n_k$$

For the example in Exhibit 34.5:

$$σ^2=0.25/100(0.2^2+0.4^2+0.3^2+0.1^2)=0.0025×0.3=0.00075$$ $$σ=0.0274$$

The margin of error, e = Zσ=1.96 × 0.027=0.054 or 5.4%.

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