Measuring Averages: Mean, Median and Mode
Whilst selling the attributes of a house to a potential buyer, the real estate agent describes the neighbourhood as comprising many beautiful homes with the average value being in the vicinity of $800,000. Yet, when quoting the average value of homes in this same area for the purposes of ascertaining council rates, it is recorded as $325,000. Which average is correct? Can they both be right or is there some deception going on?
The answer is that both estimates can be correct and in fact, are. Both figures quoted are averages, but the important question to ask is “which type of average is it?”
There are three measures of average: the mean, median and mode. All perform the function of estimating the value that represents where the majority of observations fall in a set of data. However, depending on what is being measured and indeed, how it is distributed, each of these three measures may produce very different estimates of the average.
Where a data set has a few very large outliers, its distribution is positively skewed and the mean is pulled to the right. In some cases, this can result in the majority of observations being below average. Similarly, with negatively skewed distributions, the majority of values may be above average. In both such cases, quoting the arithmetic mean becomes not only meaningless, but often misleading.
A better estimate of the average in such cases of skewness is provided by the median, where the size of the outliers has less impact.
Understanding the difference between a mean and a median is what explains the apparent contradiction in our opening example relating house prices. For illustrative purposes, let’s assume that there are ten houses in the said neighbourhood, which is described as an ideal place to retire. The first eight houses, belonging to retired pensioners, begin at $280,000 and rise in increments of $10,000. The last two homes, however, were built as weekenders by multi-millionaires and are valued at $2.5 and $3.0 million respectively. To exaggerate the average value of the homes, the arithmetic mean of $802,000 would be quoted. Downplaying the value of the homes would be better served by reporting the median of only $325,000.
Whilst both approaches report a correctly calculated average, in each case the method of calculating the average was well chosen to suit the purpose.
It should be noted that for some kinds of information, there is very little difference across the averages reported by the mean, median or mode. This is true where the frequency distribution of the data is approximately bell-shaped. Also known as normal distributed, it is here that the mean, median and mode fall at the same value.
The warning remains – where the data of interest is likely to be skewed (i.e. income, house prices, satisfaction etc.), be sure to question how the average was calculated.
