Is the Mean Better?

One or Two Measurements?
To increase accuracy you must try to make measurements unbiased. You must also try to reduce variability. In the rifle shooting Ann's shots were less varied than Barry's. In measurement you want less variability. One way is to use the mean. This section shows why using the mean reduces variability. However, the mean is of no use if your results are biased.

Diana is measuring lengths. Equally often she is 1 mm under, 1 mm over and exactly right. This is illustrated in Figure 5.

Figure 5 - Errors with one measurement.

Diana decides to take pairs of measurements and find the mean.

There are nine different outcomes, shown in Table 1. e.g. a first measurement with error -1, followed by, a second measurement with error -1, gives a total error of -2, which is a mean error of -1.

Table 1 - Total error in two measurements.

Table 2 on page R1 shows the mean error for some of these figures.

From Table 2 you can see that one out of the nine outcomes gives a mean error of -1. As a proportion this is ¹/₉
Two out of the nine outcomes give a mean error of -¹/₂. As a proportion this is ²/₉.

Taking Four Measurements
Suppose Diaina took the mean of four measurements. It is difficult to work out proportions in the same way. We can see what happens by, throwing cubes or dice.

You will need four cubes (or dice).

On each of the four cubes mark two faces -1, two faces 0 and two faces +1. When you throw the cubes, the faces show the error Diaina made so that -1, -1, 0, 1 gives a total error of -1, which is a mean error of -¹/₄.

(If you have dice use the following code: one and two become -1, three and four become 0, five and six become + 1).

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