Statistics

Sampling

In French: Échantillonnage

Sampling consists of measuring a part to conclude about the whole. Its validity depends on the way the units are drawn, far more than on their number.

Representativeness before size

A sample is not a small set: it is a set in which every unit of the population had a known chance of being retained. That condition is what authorises generalising; without it, no size makes up for the flaw.

A draw made from what is accessible — the parts on top of the crate, the files still open, the customers who reply — faithfully describes that subset and nothing else. The error does not show in the calculations, which work perfectly on biased data.

That is why a very large, badly drawn sample is more dangerous than a small, correct one: its size inspires confidence and tightens the displayed uncertainty around a wrong value.

The ways of drawing

Simple random. Every unit has the same chance of being retained, which requires a complete list and a genuinely random draw — a draw "at random" done in one’s head is not.

Stratified. The population is split into homogeneous groups — machines, shifts, periods — and units are drawn from each. It is the most useful choice in industry, because it guarantees that every source of variation is represented and makes it possible to compare it with the others.

Systematic. One unit is taken every n. Simple to implement, this mode becomes misleading if production follows a cycle with the same period as the sampling interval.

Cluster. Whole sets are drawn rather than isolated units, for lack of access to each unit. The procedure is economical and gives lower precision at equal sample size, which must be explicitly accepted.

How many to draw

Size is not chosen by habit: it is derived from what one wants to be able to assert. The smaller the gap to bring out, and the more dispersed the measurements, the more units are needed to distinguish it from chance. That sizing precedes collection — deciding it while looking at the first results amounts to choosing one’s conclusion.

Precision improves as the sample size increases, but more and more slowly: quadrupling the number of measurements only doubles the precision. It is that diminishing return that in practice sets the reasonable limit of a campaign.

The size of the population, for its part, hardly comes into it as long as it is large compared with the sample. A draw of two hundred units tells you as much about a batch of ten thousand as about a batch of a hundred thousand — a counter-intuitive result that avoids many oversized campaigns.

The common mistakes

Drawing what is convenient. The easily accessible units almost always differ from the others — by their position, their moment of production or the person who handled them.

Deciding the size after seeing the first results. It is the most discreet way of manufacturing the desired conclusion, and it leaves no trace in the data.

Treating a heterogeneous set as a single population. Mixing two machines in the same draw produces a mean that describes neither of them and hides precisely the difference that was being sought.

On the shop floor — Food processing

Weight checks were done on the first units of each pallet, for ease of access. Stratified sampling across the whole production showed a drift at the end of the run, invisible until then: the units checked were systematically produced in the first ten minutes of the batch.

The other tools illustrated by this trade: food and beverage.

Frequently asked questions

What sample size is needed?

It is calculated from the gap one wishes to detect, the expected spread and the accepted risk of error — there is no number valid in general, and rules of the "thirty is enough" kind are dangerous shortcuts. The calculation is done before collection, which forces one to state precisely what one is trying to establish.

Is a larger sample always better?

It tightens the uncertainty, but only if the draw is correct: a bias is not diluted by increasing the sample size, it is confirmed. A large, badly constituted sample therefore gives a wrong answer with great assurance, which is the least desirable situation of all.

How do you draw at random in a workshop?

By relying on a draw external to judgement: serial numbers drawn by a tool, sampling times set in advance, positions decided before going on site. The instruction "take a few at random" invariably produces a sample influenced by what is visible, reachable and seemingly representative to the person drawing.

Should the size of the population be taken into account?

Rarely: as long as the population is large compared with the sample, its size has practically no influence on the precision obtained. The correction becomes necessary only if a large fraction of the whole is drawn, say beyond a tenth, in which case the precision is better than the usual formula indicates.