Better a Dead Clock Than a Slow One?

I remember my grandfather asking me which was better: a clock that was always 10-minutes behind, or one that had stopped completely. I confidently told him that the slow clock would be better but, tongue-in-cheek, he argued that at the least the clock that had stopped told the right time twice a day!

My grandfather’s joke isn’t that far removed from the statistical anecdote of the carpenter seeking an apprentice. To him, the most important rule of carpentry is the ability to measure. He interviews two young lads and asks them each to measure the same piece of wood three times. One of the lads records three slightly different measurements, whilst the other one records the same measurement all three times. The carpenter immediately rejects the first lad and hires the second. What’s wrong with his approach?

The ability to measure well certainly requires precision, but also accuracy. The carpenter sees that the first lad’s measurements are imprecise in that he obtained three different values and assumes that he’s no good at measuring. The second lad was consistent in that he got the same reading three times, but there’s nothing to suggest that he was accurate.

Precision and accuracy are two very important concepts in statistics and a good statistician will always assess them both when determining what constitutes the best estimate for an unknown value of interest. For any estimation procedure there is usually a trade-off between the bias (measures accuracy) and the variance (measures precision) determined by what’s known as the mean-squared error (MSE) which is defined as the variance plus the square of the bias. The estimator with the smallest MSE is the best.

The MSE in action can be seen by returning to the carpenter. Suppose the piece of wood above was 100cm long and the first lad’s readings were 99, 100 and 101cm, whilst the second lad’s readings were all 103cm. The variance of the first lad’s measurements is 2/3 but with zero bias because the average of his readings is correct. The second lad has zero variance, but a bias of 3, leading to an MSE of 9 vs the first lad’s MSE of 2/3. Clearly the first lad was better though, of course, if he could improve his precision he’d be better still.

And what of my grandfather’s clock? Well another tool in the statistician’s armoury is the ability to estimate bias and therefore remove it. In this case adding 10 minutes to the slow clock gives you the correct time and you need never be late again.