A theoretical development is presented which results in a relationship between the expectation value of the standard deviation of the frequency fluctuations for any finite number of data

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The results of a joint experiment aimed primarily at the determination of the frequency of the H1 hyperfine transition (F = 1, mF =O) cf (F = 0, mF 0) is reported. In terms of the frequency of the Cs133 hyperfine transition (F = 4, mF = 0) c-* (F = 3, m F O), defined as 9192 631 770 Hz, fortheunperturbedhydrogen transition frequency the value VH = 1420 405 751.768 HZ is obtained. This result is the mean of two independent evaluations against the same cesium reference, which differ by 2 X 10-3 Hz. We estimate the one-sigma uncertainty of the value vH also to be 2 X 10-2 Hz. One evaluation is based on wall-shift experiments at Harvard University; the other is a result of a new wall-shift measurement using many storage bulbs of dflerent sizes at the National Bureau of Standards. The experimental procedures and the applied corrections are described. Results for the wall shit and for the frequency of hydrogen are compared with previously published values, and error limits of the experiments are discussed.