Introduction to cochranSize

What is Cochran’s formula?

Cochran’s formula estimates the minimum sample size needed for a survey to achieve a given margin of error at a given confidence level:

\[n_0 = \frac{z^2 \, p (1-p)}{e^2}\]

Where:

z is the z-score for the desired confidence level (e.g. 1.96 for 95%)
p is the estimated proportion of the population with the attribute of interest (use 0.5 if unknown — the most conservative assumption)
e is the desired margin of error (e.g. 0.05 for +/-5%)

If the population size N is known and relatively small, a finite population correction is applied:

\[n = \frac{n_0}{1 + \frac{n_0 - 1}{N}}\]

Basic usage

By default, cochran_sample_size() uses a 95% confidence level and a 5% margin of error — but these are only defaults, not fixed assumptions. Every parameter can be set explicitly:

cochran_sample_size(e = 0.05, conf.level = 0.95)
#> Cochran's Formula - Sample Size Calculation
#> --------------------------------------------
#> Confidence level:      95%
#> Margin of error (e):    0.050
#> Expected proportion (p): 0.500
#> 
#> Unadjusted sample size (n0): 385
#> Population size (N):         not supplied (treated as infinite)

Known finite population

cochran_sample_size(N = 2000, e = 0.05, conf.level = 0.95)
#> Cochran's Formula - Sample Size Calculation
#> --------------------------------------------
#> Confidence level:      95%
#> Margin of error (e):    0.050
#> Expected proportion (p): 0.500
#> 
#> Unadjusted sample size (n0): 385
#> Population size (N):         2000
#> Adjusted sample size (n):    323

Custom confidence level, margin of error, and expected proportion

cochran_sample_size(N = 500, p = 0.3, e = 0.03, conf.level = 0.99)
#> Warning in cochran_n_adj(n0, N): `n0` exceeds `N`; the corrected sample size
#> will approach the full population size.
#> Cochran's Formula - Sample Size Calculation
#> --------------------------------------------
#> Confidence level:      99%
#> Margin of error (e):    0.030
#> Expected proportion (p): 0.300
#> 
#> Unadjusted sample size (n0): 1549
#> Population size (N):         500
#> Adjusted sample size (n):    379

Using the underlying functions directly

n0 <- cochran_n(p = 0.5, e = 0.05, conf.level = 0.95)
n0
#> [1] 384.1459
cochran_n_adj(n0, N = 1000)
#> [1] 277.7335

Comparing confidence levels and margins of error

settings <- expand.grid(
  conf.level = c(0.90, 0.95, 0.99),
  e = c(0.05, 0.03, 0.01)
)
settings$n0 <- mapply(cochran_n, e = settings$e, conf.level = settings$conf.level)
settings$n0 <- ceiling(settings$n0)
settings[order(settings$conf.level, -settings$e), ]
#>   conf.level    e    n0
#> 1       0.90 0.05   271
#> 4       0.90 0.03   752
#> 7       0.90 0.01  6764
#> 2       0.95 0.05   385
#> 5       0.95 0.03  1068
#> 8       0.95 0.01  9604
#> 3       0.99 0.05   664
#> 6       0.99 0.03  1844
#> 9       0.99 0.01 16588

As the table shows, tightening the margin of error or raising the confidence level both increase the required sample size — margin of error has the larger effect, since it is squared in the denominator.