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

*r* = 2 root(2) *s _{w}*

where *s _{w}* 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.
*s _{w}*

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)*s _{w}* and
+1.96 root(2)

If we ignore the sign, we can say that 95% of differences will be less than
1.96 root(2)*s _{w}* in magnitude.

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

*r* = 2 root(2)*s _{w}*.

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Last updated: 18 February, 2005.