CONCEPT OF AVERAGES
Biological phenomena vary around the average. The average represents what is normal by being the point
of equilibrium. The average is a representative summary of the data using one value. Three averages are commonly used: the
mean, the mode, and the median.
There are 3 types of means: the arithmetic mean, the geometric mean, and the harmonic mean. The most
popular is the arithmetic mean. The arithmetic mean is considered the most useful measure of central tendency in data analysis.
The geometric and harmonic means are not usually used in public health. The median is gaining popularity. It is the basis
of some non-parametric tests as will be discussed later. The mode has very little public health importance.
2.0 THE ARITHMETIC MEAN
The arithmetic mean is the sum of the observations' values divided by the total number of observations
and reflects the impact of all observations. The robust arithmetic mean is the mean of the remaining observations when a fixed
percentage of the smallest and largest observations are eliminated. The mid-range is the arithmetic mean of the values of
the smallest and the largest observations. The weighted arithmetic mean is
used when there is a need to place extra emphasis on some values by using different weights. The indexed arithmetic
mean is stated with reference with an index mean. The consumer price index (CPI) is an example of an indexed mean.
The arithmetic mean enjoys 2 desirable statistical advantages. It is the best single summary statistic.
It has a rigorous mathematical definition. Its disadvantage is that it is affected by extreme values.
3.0 THE MODE
The mode is the value of the most frequent observation. It is rarely used in science. It is intuitive,
easy to compute, and is the only average suitable for nominal data. It is useless for small samples because it is unstable
due to sampling fluctuation. It cannot be manipulated mathematically. It is not a unique average, one data set can have more
than 1 mode.
4.0 THE MEDIAN
The median is value of the middle observation in a series ordered by magnitude. It is intuitive and
is best used for erratically spaced or heavily skewed data. The median can be computed even if the extreme values are unknown
in open-ended distributions. It is less stable to sampling fluctuation than the arithmetic mean.
5.0 RELATIONSHIPS AMONG AVERAGES
Mean = mode = median for symmetrical data.
Mean > median for right skewed data.
Mean < median for left skewed data..
In general, mode-median = 2(median-mean).
The mean is best used to summarize symmetrical data. The median is used to summarize skewed
data. For some data sets it is best to show all the 3 types of averages.