Measures of Central Tendency
David M. Lane
- Compute mean
- Compute median
- Compute mode
In the previous section we saw that there are
several ways to define central tendency. This section defines
the three most common measures of central tendency: the mean,
the median, and the mode. The relationships among these measures
of central tendency and the definitions given in the previous
section will probably not be obvious to you. Rather than just
tell you these relationships, we will allow you to discover
them in the simulations in the sections that follow.
This section gives only the basic definitions of
the mean, median and mode. A further discussion of the relative
merits and proper applications of these statistics is presented
in a later section.
The arithmetic mean is the most common measure
of central tendency. It is simply the sum of the numbers divided
by the number of numbers. The symbol "μ" is used for the
mean of a population. The symbol "M" is used for the mean of
a sample. The formula for μ is shown below:
μ = ΣX/N
where ΣX is the sum of all the numbers in the population
N is the number of numbers in the population.
The formula for M is essentially identical:
M = ΣX/N
where ΣX is the sum of all the numbers in the sample and
N is the number of numbers in the sample.
As an example, the mean of the numbers 1, 2, 3,
6, 8 is 20/5 = 4 regardless of whether the numbers constitute
the entire population or just a sample from the population.
Table 1 shows the number of touchdown (TD) passes
thrown by each of the 31 teams in the National Football League
in the 2000 season. The mean number of touchdown passes thrown
is 20.4516 as shown below.
μ = ΣX/N
Table 1. Number of touchdown passes.
|37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19
19 18 18 18 18 16 15 14 14 14 12 12 9 6
Although the arithmetic mean is not the only "mean"
(there is also a geometric mean), it is by far the most commonly
used. Therefore, if the term "mean" is used without
specifying whether it is the arithmetic mean, the geometric mean,
or some other mean, it is assumed to refer to the arithmetic mean.
The median is also
a frequently used measure of central tendency. The median is the
midpoint of a distribution: the same number of scores is above
the median as below it. For the data in Table 1, there are 31
scores. The 16th highest score (which equals 20) is the median
because there are 15 scores below the 16th score and 15 scores
above the 16th score. The median can also be thought of as the
Computation of the Median
When there is an odd number of numbers, the median
is simply the middle number. For example, the median of 2, 4,
and 7 is 4. When there is an even number of numbers, the median
is the mean of the two middle numbers. Thus, the median of the
numbers 2, 4, 7, 12 is (4+7)/2 = 5.5. When there are numbers with the same values, then the formula for the third definition of the 50th percentile should be used.
The mode is the most frequently occurring value.
For the data in Table 1, the mode is 18 since more teams (4) had
18 touchdown passes than any other number of touchdown passes.
With continuous data such as response time measured to many decimals,
the frequency of each value is one since no two scores will be
exactly the same (see discussion of continuous
variables). Therefore the mode of continuous data is normally
computed from a grouped
frequency distribution. Table 2 shows a grouped frequency
distribution for the target response time data. Since the interval
with the highest frequency is 600-700, the mode is the middle
of that interval (650).
Table 2. Grouped frequency distribution.
td=c(37,33,33,32,29,28,28,23,22,22 ,22, 21,21,21, 20, 20, 19,19,18,18,18,18,16,15,14,14,14,12,12,9,6)
quantile(td, probs = c(.5), type = 6)
z=c(2, 4, 7, 12)
Please answer the questions: