The purpose of this article is to compare indices applied in analyses of the residential market. Using a more precise statistical analysis of price distribution, I will attempt to present differences between the following measures of central tendency: the mode, the median and the mean, which will allow to answer the question, which of these should be considered in order to provide the most accurate analysis of market changes and market situation.

Measures of central tendency define the location of „average values of the distribution”. They can be further divided into measures of position and classic measures. Measure of position is a value occurring in a specific statistical unit. Examples of such measures are e.g. the median (the value of the middle unit) or the mode (the value of the most frequent unit). On the other hand, the classic measure is calculated based on all researched units occurring in the set, e.g. the arithmetic mean or the geometric mean.

When analyzing prices occurring in the residential market, REAS uses the three measures of central tendency defined below regarding the prices of residential units calculated per one square meter of usable floor area.

The median – the price of the middle unit in the set. In order to understand this measure better, one should imagine that all residential units have been arranged according to their price per one square meter from the cheapest to the most expensive unit. Perceived in this manner, the median is the price of the middle dwelling in this line.

The mode – the price occurring most frequently in the market, characteristic of the highest number of residential units.

The mean – the average price in the market is calculated as the arithmetic mean of prices of all residential units in all projects, considering that in such calculations every dwelling– regardless of its floor area – is represented by one number.

The distribution of prices in the residential market is most frequently unimodal and is positively skewed (it has one distinct peak and its elongated tail extends to the right part of the distribution), which means that usually we can observe the following inequality:

mode < median < mean
The three described measures have been illustrated on the graph on a factual distribution of prices in the residential market in Warsaw. (Graph 1)
Therefore, the mode is the peak of the graph, while the median is the place where the area under the graph in the left equals the area under the graph in the right. As opposed to the mean value, both these measures are expressed by the price of a specific residential unit exposed in the market, whereas the mean, as the classic measure, does not have to reflect the price of any particular residential unit exposed in the market (thought it is possible.) Drawing conclusions based on statistical measures
As part of the dispute focused on inadequacy of the mean as the index characterizing the residential market and significant supremacy of the median in this respect, it is worth to consider the advantages and disadvantages of both measures, as well as their possible applications. The mean is a classic measure, therefore during calculations, one must consider all residential units exposed in the market in a given moment, which intuitively suggests its high representativeness, i.e. a great advantage. Unfortunately, at the same time it is a drawback of that measure, since it makes it sensitive to extreme values. This shortcoming manifests itself in the rapid increase of the mean value when a new investment project with uncharacteristically high asking prices is introduced to the market (for instance, Sky Tower in Wrocław or the U Scheiblera lofts in Łódź). This type of problems does not occur in the case of measures of position, which „disregard” extreme values. As a result, introducing a new investment project with an extreme price level to the market will not have significant influence on the median and will probably not impact the mode. However, this advantage has also negative consequences, as explained in the below examples. Let’s imagine a situation, where the developers decide to reduce prices in investment projects located „to the left from the median value”, i.e. cheapest residential units in the market (e.g. due to the „Family’s Own Home” programme). It is very probable that the median value would not reflect the change (the price of the middle unit will remain unchanged). We may also imagine a situation, where a market with the median value of PLN 7,500 per sq.m. experiences the launch of two investment projects, whose median prices amount to PLN 6,500 per sq.m. Intuitively, we conclude that such change ought to cause a decrease of the median value in the coming research, however the example presented below shows that the market effect may be completely different. For the purposes of the example, let us assume that the two investments, where respectively 160 and 150 residential units are planned for construction, are characterized by extreme levels of asking prices at PLN 6,500 per sq.m. in the first and PLN 18,000 per sq.m. in the second one. As a result, in a market where no sales occurred, the median value will generally remain unaltered. However, when we witness even minimal change in the offer priced below the median value (e.g. when 20 residential units with prices below the level of PLN 7,500 per sq.m. are sold), as a result the median value may increase. This means that by analyzing solely the fluctuation of the median value, it is difficult to draw correct conclusion about the market. In practice, in long term analyses and price change studies, the mean is more accurate than the median, since it takes into consideration the whole statistical population. On the other hand, the median value allows a more precise analysis of a single situation and is (like the mode) a very good index supporting deduction. Therefore, when analyzing the market, one should use all of the mentioned indices, since each is characterized by a different informative value. Taking into account the most frequent characteristics of price distribution in the residential markets, (unimodality, right-skewness), which have been confirmed by many years of REAS observations and analyses, we can assume that in a residential market characterized by a given mean price, most residential units are priced lower. One ought to bear it in mind when applying the index of the mean (average) price. It is also worth to add that in the vast majority of international studies and analyses concerning long-term trends in the real estate markets, the mean price is the index selected when one can present only one measure. Adam Kulpa is REAS analyst, graduate of the Warsaw School of Economics in the faculty of Quantitative Methods and Information Systems in Economics. In REAS he is responsible for analyzing data obtained during the monitoring of residential markets. Adam Kulpa
Junior Consultant