Calculating variance and DEFF for stratified sampling

I think you want a confidence interval around a point estimate.

Here is a simple approach ...

You data:

  Strata     N    n    P
  ------ ----- --- -----
       1  9870 225 0.813
       2 33599 219 0.548
  ------ ----- --- -----
       3 14130 212 0.311

  w = N / sum(N)

  Strata     N   n     P     w
  ------ ----- --- ----- -----
       1  9870 225 0.813 0.171
       2 33599 219 0.548 0.583
       3 14130 212 0.311 0.245
  ------ ----- --- ----- -----

  sum(P * w) = 0.813 * 0.171 + 0.548 * 0.583 + 0.311 * 0.245 = 0.535

  sum((w^2 * p * (1 - p)) / n)

  Strata     N   n     p     w  x$w^2*x$p*(1-x$p))/x$n
  ------ ----- --- ----- ----- -----------------------
       1  9870 225 0.813 0.171            1.975795e-05
       2 33599 219 0.548 0.583            3.844253e-04
       3 14130 212 0.311 0.245            6.067027e-05
  ------ ----- --- ----- ----- -----------------------
                                      SUM = 0.00046485
                               -----------------------
                                SQRT(SUM) = 0.02156046
                               -----------------------

  Lower 95% CL = 0.535 - 1.96 * 0.02156046 = 0.4927 (49.27%)
  Upper 95% CL = 0.535 + 1.96 * 0.02156046 = 0.5773 (57.73%)

BTW : The design effect is the ratio of the variance (calculated above) and the variance calculated from a simple random sample. With your data:

c = round(225 * 0.813 + 219 * 0.548 + 212 * 0.245)
  = 355

n = 225 + 219 + 212
  = 656

p = 355 / 656
  = 0.541
  
var = (p * (1 - p)) / (n - 1)
    = (0.541 * (1 - 0.541)) / (656 - 1)
    = 0.000379113

DEFF = 0.00046485 / 0.000379113
     = 1.23

Avoid rounding early (as I have done above).

I hope this helps.

You should check my arithmetic.

Just to be clear ... rounding errors can accumulate and end up becoming quite large over a series of calculations ... I have been lazy above ... keep all numbers to full precision throughout and round only the final results. If you do calculations in a spreadsheet with raw numbers then you should be OK as the full precision is often retained "behind the scenes".