An average or a measure of central tendency refers to a single value that can be used to summarize a large set of numerical data. They are also known as measures of location since the average gives us some idea about where the “central” value of the data lies.
Some of the common averages in use are the mean, median, and mode. We now list some of the properties that an ideal measure of central tendency must possess.
1. Rigidly Defined / Well Defined
This means that it should be clear to everyone how to calculate the particular average for any given set of data. The value of the average should not depend upon any choice made by the investigator.
Let us look at an example of an average that is not well-defined. Consider the following sequence of data values: 1, 4, 8, 8, 8, 9, 9, 9, 13, 13, 18, 20. The mode is defined as the most frequently occurring observation but in the above data set both the numbers 8 and 9 occur the highest number of times.
Here we have two possible choices of mode and which choice to make is left at the discretion of the investigator. Thus two different investigators might reach a different value for the mode. Thus, the mode is not rigidly defined in such situations.
2. Easy to Understand and Calculate
Since the averages are often used to communicate information about the data to the general public, it is important that the averages are simple to understand and interpret. The arithmetic involved in the calculation must also not be too cumbersome.
3. Based on all Observations
Since the average is meant to represent the entire data set, it is reasonable to expect that all data values are used to obtain the value of the average.
For example, the arithmetic means is based on all observations since it involves adding all data values and dividing by the total number of values.
On the other hand, the median is not based on all observations since it only depends on the middlemost values.
4. Resistant to Outliers
The average should represent the majority of the data values and not be affected too much by one or two extreme observations. The average should be resistant to the effect of outliers on the data.
It is easy to see that the arithmetic mean is very much affected even by the presence of a single value that is too large.
On the other hand, since the median is calculated only on the basis of the middlemost values, it is much more resistant to the presence of outliers.
5. Not Affected by Sampling Fluctuations
By this, we mean that if we take independent random samples of the same size from a given population and compute the average for each of these samples then, for an ideal average, the values so obtained from different samples should not vary much from one another.
The difference in the values of the average for different samples is attributed
to the so-called fluctuations of sampling. This property is also explained by saying that an ideal average should possess sampling stability.
This is important because the average value for the sample can then be used to estimate the value of the average for the entire population from which the sample is drawn.
6. Capable of Further Mathematical Treatment
It should be easy to apply further mathematical/statistical operations on the basis of the average calculated.
For example, given the arithmetic mean of two groups, it is possible to find the combined arithmetic mean of both groups taken together.
Also given the arithmetic mean of a sample we can conduct testing of hypotheses such as the Z test, T-test, etc. in order to test the value of the population means.