The following transformation was made on 79 values of a variable W. U=W-54.5/6. Statistics computed from U were U bar = 6.8, variance of u = 1.8. Find W bar and variance of W

Question

U =W-54.5/6. Statistics computed Mean of U as 6.8 and Variance as 1.8. Find mean of w and variance of w

Answer 1

INTERPRETATION 1

If U=W-(54.5/6) then W=U+(54.5/6).

Assuming a one-for-one transformation, the mean for W is found by adding all the W data together then dividing by 79.

That is, Wbar=(w1+w2+w3+...+w79)/79.

But w1=u1+54.5/6, w2=u2+54.5/6, etc.

So Wbar=(u1+u2+...+u79)/79+(79×54.5/6)/79=Ubar+54.5/6=6.8+54.5/6=15.8833 approx.

The variance is unaffected by the transformation since variance is simply the spread of the data which doesn’t change.

Wbar=Ubar=1.8.

(But see note at bottom of page)

INTERPRETATION 2

If U=(W-54.5)/6 then W=6U+54.5. This changes the mean to 6×6.8+54.5=95.3.

The variance is also affected: Wbar=6Ubar=10.8 because the spread of the data is amplified by a factor of 6.

Note that the sizes of the datasets are not relevant, but it has been assumed that W and U are the same size and that the means and variances were calculated using the same data sizes. If the stats for U are population stats, and 79 is the sample size (population>79) then the variance for the sample may need taking into account so that the variance of U becomes 1.8/79 instead of 1.8. This changes the estimation of the variance of W to 1.8/79=0.0228 (interpretation 1) and 0.1367 (interpretation 2).