Gravitational center. This exercise will demonstrate how the mean is the balancing point of a set of numbers.

(a) Distribution A. The values 1 and 5 are marked as “Xs” on the following number line. Calculate the mean and mark its location on the number line.

(b) Distribution B. The values 1, 5, and 5 are shown on the number line. Calculate the mean and show its location on the number line. Notice how the extra 5 pulls the mean to the right.

TABLE 4.1 The mean calculated from frequencies. Data are frequencies of fatal horse kicks in the Prussian Army; a classical example from Bortkiewicz (1898).

= Σpixi

= (0.545· 0)+(0.325·1)+(0.110·2)+(0.015·3)+(0.005·4) = 0.61 Data from Bortkiewicz, L. V. (1898). Das Gesetz Der Kleinen Zahlen. Liepzig:

Tuebner.

(c) Distribution C. Calculate and show the location of the mean for the data points 2.75, 3.00, and 3.25 on the following number line:

(d) Distribution D. Calculate and show the mean of these three points:
Exercise 3.1 should convince you that the mean tells you nothing about the spread or shape of a distribution. All four distributions are different, yet distributions A and C have the same means ( = 3), as do distributions B and D ( = 3.67).

Reliance solely on the mean would have missed the full picture. Consider:

• Describing the central location of a pendulum tells you little of its motion.

• If you have your head in the freezer and your feet in the oven, your average body temperature can still be normal.

• You can drown in a deep area of a lake that has an average depth of just a few inches.

Sole reliance on a mean often misses the true picture.