I was thinking as follows. The width of the confidence interval of quantile q for a confidence level alpha isF(q2 = q + z*(q*(1 - q)/n)^0.5) - F(q1 = q—z*(q*(1 - q)/n)^0.5), where P(z ⇐ X | X ~ N(0, 1)) = 1 - (1 - alpha)/2. A greater variation across estimates does not change q1 nor q2, but it increases the width F(q2) - F(q1).
That being said, I have to concede that what we care about is the mass in the tails, not the tails of the median. So one should care about the difference between e.g. the 97.5th and 2.5th percentile, not F(q2) - F(q1).
Thanks for following up!
I was thinking as follows. The width of the confidence interval of quantile q for a confidence level alpha is F(q2 = q + z*(q*(1 - q)/n)^0.5) - F(q1 = q—z*(q*(1 - q)/n)^0.5), where P(z ⇐ X | X ~ N(0, 1)) = 1 - (1 - alpha)/2. A greater variation across estimates does not change q1 nor q2, but it increases the width F(q2) - F(q1).
That being said, I have to concede that what we care about is the mass in the tails, not the tails of the median. So one should care about the difference between e.g. the 97.5th and 2.5th percentile, not F(q2) - F(q1).