It's a normal approximation of the expected variance (in terms of standard deviation) in the estimate of the mean of the sum of 200 Bernoulli random variables. Each nurses' response is considered an observation of a Bernoulli distributed random variable, and we trying to determine the rate of that variable.

You are incorrect that "n is 1" since, by that logic one survey talking to 100,000 nurses would be the same as one talking to 3.

If you would like an alternate, more Bayesian formulation we can use the Beta distribution which is parameterized by alpha (numbers of 'yes') and beta (number of 'no').

This approach is a bit more intuitive than the Frequentist method since it answers the question "what do we believe to be the expected rate of nurses answering 'yes'"

In this case alpha=180 and beta=20, we'll include uniform prior of alpha_prior = 1, and beta_prior = 1

For Betas the posterior is defined quite nicely as:

Beta(alpha_posterior, beta_posterior) = Beta(alpha_likelihood + alpha_prior, beta_likelihood + beta_prior)

In general for Beta distributions we can compute the expectation as:

E[Beta(alpha,beta)] = alpha/(alpha + beta)

In this case: 181/202 = ~0.9

And the variance of a Beta distributed random variable is:

Var[Beta(alpha, beta)] = (alpha*beta)/((alpha+beta)^2 * (alpha + beta + 1))

Which for our case is:

0.00046

and the standard deviation of this is just it's square root:

0.021

Which gives us the same answer as we get with the normal approximation.