What is the origin of the formula for repeatablity?

I am often asked what is the origin of the repeatability formula:

r = 2 root(2) sw

where sw is the standard deviation within a subject.

It is an estimate of the maximum difference which we might get between two measurements made at random on the same subject. To be more precise, it is the limit within which 95% of differences will lie.

The difference between two measurements on the same subject has variance given by the sum of the two variances, i.e. sw2 + sw2 = 2 sw2. The standard deviation is the square root of this: root(2)sw.

The differences may be expected to have an approximately Normal distribution, because we are subtracting one error from another. Hence there will be 95% of differences within 1.96 standard deviations from the mean. They will have mean zero, because there is no reason why the first or second observation should be the larger. Hence 95% of differences will be between -1.96 root(2)sw and +1.96 root(2)sw.

If we ignore the sign, we can say that 95% of differences will be less than 1.96 root(2)sw in magnitude.

Some people prefer to approximate this 1.96 as 2.0, rounding to one decimal place. This gives

r = 2 root(2)sw.

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Last updated: 18 February, 2005.

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