2.3.3. (arithmetic) Mean

What is it?

The mean is often defined by how it is calculated: the sum of all values divided by the number of items.

It is strange that for most measures of central tendency the definition often explains conceptually what it is, but for the mean the formula is often the definition. Some authors though use a conceptual definition for the mean as well (Watier, Lamontagne, & Chartier, 2011):

• It is the amount or value that each group member would receive if the sum would be divided equally among all members.
• It is the fulcrum (balancing point) for the distribution.

The fulcrum is the triangle in Figure 21.

Figure 21. Visualisation of the mean

A proof that the formula often seen for the arithmetic mean (sum up all items, divide by number of items) is indeed the fulcrum, can be found in Appendix D . An alternative method based on the fulcrum idea is known as using an ‘assumed mean’. In the side notes this is discussed in more details.

There are two symbols for the mean that are often used. The Greek letter mu (μ) is used to indicate a mean from a population, while a letter with a horizontal line (bar) is used for a mean from a sample (e.g. x̄ ).

Examples

List of values (raw data)
The mean from 2, 4, 8, 8, 10 is calculated by adding all the values and dividing it by the number of values. In this case: 2+4+8+8+10 = 32, and since there are 5 values the mean is 32/5 = 6.4.

Frequency table
If we convert the list of values in the previous example into a frequency table as shown in Table 11, it can be used to illustrate that the calculation for the mean in a frequency table is actually the same as for a list of values.

Table 11
Example frequency table for mean

Value (x)	Frequency (f)
2	1
4	1
8	2
10	1

To determine the mean we multiply each value by its frequency, sum these up and then divide by the number of values (the total frequency). In this case 1*2 + 1*4 + 2*8 + 1*10 = 32 and divide this by 1+1+2+1 = 5 to obtain again 32/5 = 6.4.

Grouped table 
The problem with a grouped table is that we do not have a single value (as was the case in the frequency table), but a range of values (classes), hence it is impossible to multiply the frequency by the class (i.e. how would you calculate 3 * (5<10)?). To overcome this problem we use the class midpoint to represent the entire class. We can then use these class midpoints as the value for that class and use the same technique as in the frequency table example to get an estimate for the mean. The grouped frequency table from the previous example might look like Table 12.

Table 12
Example for mean calculation from a grouped table

Class	Midpoint (x)	Frequency (f)
0<6	3	2
5<9	7	2
9<11	10	1

The estimated mean now becomes (2*3 + 2*7 + 1*10) / (2 + 2 + 1) = (6 + 14 + 10) / 5 = 30 / 5 = 6. Note that this is different than the 6.4. By using the midpoints we assume the values in that class are distributed evenly (which might not be the case). The calculated mean from a grouped table is therefore always only an estimate of the mean.

Why (not) use the mean?

The mean can only be determined for numerical values, so should not be computed for measurements on nominal and ordinal level. Some argue though to also calculate the mean for ordinal data. An interesting article with references to both sides (those who favor using the mean for ordinal data also and those who don't) is from Knapp (1990).

The mean takes not only the order of the values into consideration, but also their 'weight'. It uses most information possible from the values and is therefore the preferred measure of central tendency for interval and ratio measurements (Stevens, 1946).

The mean is (because it takes the weight of the values into consideration) quickly influenced by extreme values (or outliers). In those cases the median might give a better idea of the center.

History

In Pythagoras time the Greek studied the arithmetic mean, but from a geometrical perspective. Aristotle (384-322 BC) (1850, p. 43) refers to the arithmetic mean. Hald (2003, p. 147) mentions that Tycho Brahe uses the arithmetic mean to average out his astronomical observations between 1582 and 1588. The original might be in ‘Meteorologiske dagbog’ (Brahe, 1876), probably somewhere after page 283 (sammendrag), but I don’t read Danish.

The use of a bar for the mean comes from applied mathematics where any average was represented by a bar (e.g. v̄ for velocity mean from Maxwell (1867, p. 64) and the centre of inertia from Kelvin & Trait (1879)). The first found use of μ for the mean (actually as a moment, not specifically the mean) can be found in the work from Pearson (1895, p. 347).

>>Next section: Using technology to determine measures of central tendency

References
Aristotle. (1850). The Nicomachean Ethics of Aristotle. (R. W. Browne, Trans.). London: Henry G. Bohn.
Brahe, T. (1876). Meteorologiske dagbog. Kjøbenhavn H.H. Thiele. Retrieved from http://archive.org/details/meteorologiskeda00brahuoft 
Hald, A. (2003). A history of probability and statistics and their applications before 1750. Hoboken, N.J.: Wiley.
Kelvin, W. T., & Tait, P. G. (1879). Treatise on Natural Philosophy. London: University Press. Retrieved from http://archive.org/details/treatiseonnatur01darwgoog
Knapp, T. R. (1990). Treating Ordinal Scales as Interval Scales: An Attempt To Resolve the Controversy. Nursing Research, 39(2), 121–123. doi:10.1097/00006199-199003000-00019
Maxwell, J. C. (1867). On the Dynamical Theory of Gases. Philosophical Transactions of the Royal Society of London, 157, 49–88. 
Pearson, K. (1895). Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material. Philosophical Transactions of the Royal Society of London. (A.), 186, 343–414. doi:10.1098/rsta.1895.0010
Stevens, S. S. (1946). On the Theory of Scales of Measurement. Science, 103(2684), 677–680. doi:10.1126/science.103.2684.677 
Watier, N. N., Lamontagne, C., & Chartier, S. (2011). What does the mean mean? Journal of Statistics Education, 19(2), 1–20.