# 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 33.5   Proportional and disproportional sample designs for subgroups.

Most quantitative research studies report results by relevant subgroups; for instance analyses within demographics (income, gender, age etc.), buying behaviours (use of brand/category, usage rate) and geography.

If each subgroup sample size is in proportion to the subgroup’s population, you may find that some subgroup samples are too small to analyse. The example in Exhibit 33.5, for instance, yields samples of less than 100 for subgroups IV and I.

These inadequacies may be resolved if each subgroup meets the sample size requirement. In the above cited example, the subgroup sample size of 100 restricts the sampling error to ±10%. For such disproportionate designs, each subgroup must then be weighted to account for the disproportions.

The cell weight (refer Exhibit 33.5) is equal to the population share divided by the sample share (e.g. for the first group, this is 20%/25% = 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 is 20 × 0.8 + 30 × 1.6 + 40 × 1.2 + 50 × 0.4 = 132 or 33%. 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 recommend sample sizes of 100 (margin of error ±10%) or more. For minor subgroups, the sample size should be at least 50 (±14%).

Disproportional design is the norm for the majority of quantitative research studies. If however, information on the proportions is not available, disproportional designs should not be used.

The sampling error for the disproportional samples is a little greater than that for the proportional sample design. For instance in the above cited example, the confidence interval is ±5.4% for confidence level of 95% (Z=1.96), compared to ±4.9% as would be the case for proportionate sampling.

To compute the margin of error, we need to estimate the variance, which in the case of subgroup (or stratified) sampling is obtained 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 33.5:

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

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

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