. Elementary biophysics: selected topics. Biophysics. MEASUREMENTS AND THEIR VARIATION 7 a deviation of —2s or more (in the negative direction) will occur once in 40 measurements. This table will provide the basis for much of the analysis we will present. With its presentation, we have answered one of the two ques- tions we set out to answer about the standard deviation: what is its significance? The other question concerns the utility of the averaging procedure employed in obtaining the arithmetic average. We already know that the clustering of measurements around this value is greatest, but
. Elementary biophysics: selected topics. Biophysics. MEASUREMENTS AND THEIR VARIATION 7 a deviation of —2s or more (in the negative direction) will occur once in 40 measurements. This table will provide the basis for much of the analysis we will present. With its presentation, we have answered one of the two ques- tions we set out to answer about the standard deviation: what is its significance? The other question concerns the utility of the averaging procedure employed in obtaining the arithmetic average. We already know that the clustering of measurements around this value is greatest, but still more can be said. If we ask the mathematicians to compute the cluster- ing of the averages themselves, we are asking what would be the distri- bution of averages if we made a set of n measurements many, many times. Another way of asking the same question is to ask for the standard deviation of the average itself. The mathematician's reply to the latter question is V or n V V = — n That is, the average variation, or variance, of the average measurement is only one-nth of the variance of the individual measurements. This is the result we want. It says that the reason for taking the average is that the average is roughly -\Jn times more likely to be the true value than is any individual measurement. sa is called the standard error by statis- X or a Fig. 4. A schematic representation of a set of individual measurements, x, and of a set of averages, a, obtained by making many repeated sets of measure- ments. The relative narrowness of the curve for a shows that the average ob- tained in any single set of measurements is very likely to be close to the "true" value. Still another way of indicating the same information is shown in the schematic drawing in Fig. 4, which shows the distribution of individual measurements and also the distribution of averages. The latter is so narrow that it is very unlikely that we will ever get a much different result, even
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