Raising the bar

Chapter 3. Describing Data: Numerical

Unit 3.2. Measures of central tendency

Our plan is to detail Figure 3.1.

Figure 3.1. Numerical characteristics of samples.

Mean. The sample mean (or simply mean or average) is the usual arithmetic mean
$latex \bar x = \frac{{{x_1} + ... + {x_n}}}{n}. \qquad (\textrm{Eq. 3.3})$

Does it change if instead of the raw representation one uses the ordered representation?

Median - intuitive definition. The median is such a point that half of the observations lie to the left of it and another half to the right.

It is useful to consider two samples (using the ordered representation is better).

Sample 1: $latex {y_1} = 0,\,{y_2} = 1,\,{y_3} = 5$.

Sample 2: $latex {y_1} = 0,\,{y_2} = 1,\,{y_3} = 3,\,{y_4} = 5$.

In Sample 1, where $latex n$ is odd, only the value 1 satisfies the intuitive definition of the median. In Sample 2, where $latex n$ is even, any point between 1 and 3 can serve as a median. To achieve uniqueness, it is customary to take as the median the number 2, the middle of the interval $latex (1,3)$.

Before generalizing upon these examples, we have to think about a general form of writing even and odd numbers. You can verify that

(a) any number in the sequence 2, 4, 6, ... can be written in the form $latex 2k$ where $latex k$ runs over naturals, and that
(b) any number in the sequence 1, 3, 5, ... can be written in the form $latex 2k + 1$ where $latex k$ runs over nonnegative integers.

Median - formal (and working) definition.

Step 1. Arrange the observations in an ascending order.

Step 2. (a) If the number of observations is even, $latex n = 2k$, then the observations $latex {y_k}$ and $latex {y_{k + 1}}$ will be the equivalents of the observations $latex {y_2}$ and $latex {y_3}$ in Sample 2 and by definition

$latex \textrm{median} = \frac{1}{2}({y_k} + {y_{k + 1}}).$

(b) If the number of observations is odd, $latex n = 2k + 1$, then the observation $latex {y_{k + 1}}$ will be the equivalent of the middle observation $latex {y_2}$ in Sample 1 and we put

$latex \textrm{median} = {y_{k + 1}}.$

Mode. The mode is the most frequent observation. Obviously, in this case the frequency representation must be used.

It is possible for a frequency distribution to have more than one most frequent observation. In that case all of them will be modes. In the literature you can see names like bimodal and trimodal distributions.

Which of the measures of central tendency to use depends on the context and data, see Table 3.2.

Table 3.2. Exemplary comparison of measures of central tendency.

Don't try to memorize this table. Think about the examples in NCT. Try to feel the definitions. Don't think about observations as something frozen. Try to move them around. For example, you can fix the median, and move the points to the left of it back and forth. As you do that, do the sample mean and mode change? Or you can think about dependence on outliers, which, by definition, are points which lie far away from the majority of the observations. If you move them further away, which of the measures of central tendency may change and which do not?

Previous: Three representations of data: raw, ordered and frequency | Next: Shape of the distribution

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