A quantity is measured 100 times, and, after calculating the mean, the standard deviation, and the standard error on the mean, and assuming the errors on the individual measurements are all random, you come to the conclusion that your best value, and standard error on the mean, are X = 21.0 +/- 0.4
This is fine, but you'd really like to have a final accuracy of 0.1 cm. So you take more measurements. Assuming that in the second bunch of measurements, any given measurement has about as much random error as any given measurement in your original 100 measurements, how many measurements should you make, in total, so that the standard error on the average of all your measurements is 0.1 cm