Survey Sampling & Experimental Design
Tschuprow-Neyman Optimal Stratified Allocation Formula
Calculate Cost-Adjusted Optimal Stratified Sample Sizes n_h* proportional to W_h * S_h / SQRT(c_h), Minimum Estimator Variance under fixed budget B_tot, and Variance Reduction vs Proportional Allocation.
Oversample noisy strata and undersample expensive strata by SQRT(cost)!
In stratified survey sampling (Tschuprow 1923; Neyman 1934; Cochran 1977), Proportional Allocation n_h = n * W_h ignores both within-stratum heterogeneity S_h and per-interview field cost c_h! Minimizing stratified mean variance Var(y_st) = SUM(W_h^2 * S_h^2 / n_h) subject to Variable Field Budget SUM(c_h * n_h) = B_tot yields the Optimal Cost-Weighted Allocation rule: sample Stratum h in proportion to W_h * S_h / SQRT(c_h)! Notice the square-root damping on cost: if rural in-person interviews cost 2.25x as much as urban web surveys ($9 vs $4), you divide its sampling weight by SQRT(2.25) = 1.5—not 2.25—while multiplying directly by its standard deviation S_h!
Verify on Stratum_1_Weight W_1 = 0.80 (B2 -> Stratum_2_Weight W_2 = 1 - W_1 = 0.20), Stratum_1_SD S_1 = 10 (B3), Stratum_1_Unit_Cost c_1 = $4.00 (B4), Stratum_2_SD S_2 = 30 (B5, 3x noisier enterprise/rural tail!), Stratum_2_Unit_Cost c_2 = $9.00 (B6), and Variable_Survey_Budget B_tot = $1,700 (B7): Allocation scores are u_1 = 0.80 * 10 / SQRT(4) = 4.0 and u_2 = 0.20 * 30 / SQRT(9) = 2.0, giving Sample Fractions f_1* = 2/3 and f_2* = 1/3, Average Cost (2/3)*4 + (1/3)*9 = $5.6667, Total Sample n_opt = 300 (n_1* = 200 in E2, n_2* = 100 in E3; check cost: 200*$4 + 100*$9 = $1,700 exact!), and Minimum Estimator Variance V_opt = (0.80*10*2 + 0.20*30*3)^2 / 1700 = 34^2 / 1700 = 0.6800 (vs V_prop = 0.7647 under Proportional Allocation of n_prop = 340 interviews)!
| Stratified Survey Metric (W1=0.8, S1=10, c1=$4; W2=0.2, S2=30, c2=$9; B=$1,700) | Formula | Verified Output |
|---|---|---|
| Optimal Stratum 1 Sample Size n_1* | B_tot * (W1*S1/SQRT(c1)) / (c1*u1 + c2*u2) | 200 interviews ($800) |
| Optimal Stratum 2 Sample Size n_2* | B_tot * (W2*S2/SQRT(c2)) / (c1*u1 + c2*u2) | 100 interviews ($900) |
| Optimal Stratified Variance V_opt | (W1*S1*SQRT(c1) + W2*S2*SQRT(c2))^2 / B_tot | 0.6800 |
| Proportional Allocation Variance V_prop | (W1*S1^2 + W2*S2^2)*(W1*c1 + W2*c2) / B_tot | 0.7647 (+12.5% Higher Var) |
Optimal Cost-Constrained Stratum Sample Size & Minimum Variance Formula
Note that when per-unit interview costs are equal (c_1 = c_2), SQRT(c_h) cancels out and the formula reduces cleanly to classic Neyman Allocation n_h proportional to W_h * S_h!
=LET(w_s1, B2, sd_1, B3, cs_1, B4, sd_2, B5, cs_2, B6, bg_t, B7, IF(OR(w_s1 <= 0, w_s1 >= 1, sd_1 <= 0, cs_1 <= 0, sd_2 <= 0, cs_2 <= 0, bg_t <= 0), "Invalid Survey Design", LET(w_s2, 1 - w_s1, ROUND((w_s1 * sd_1 * SQRT(cs_1) + w_s2 * sd_2 * SQRT(cs_2)) ^ 2 / bg_t, 4))))Proportional-Allocation Benchmark Variance Formula
Under Proportional Allocation n_h = n_prop * W_h, total affordable interviews equal n_prop = B_tot / (W_1 * c_1 + W_2 * c_2), yielding V_prop = (W_1 * S_1^2 + W_2 * S_2^2) / n_prop!
=LET(w_s1, B2, w_s2, 1 - B2, IF(ISTEXT(E4), E4, ROUND((w_s1 * B3 ^ 2 + w_s2 * B5 ^ 2) * (w_s1 * B4 + w_s2 * B6) / B7, 4)))Try the example
Use the original values or change them to test your understanding.