Limitations of measures of centre and spread

We are often interested in the centre of a set of data in the sense that we would like to know where on the number line the data are mostly located. Together with the location of the centre, we may want to know how widely spread the data points are away from the centre.

The usual measures of centre are the arithmetic mean, the median and the mode. In applying these, we first examine the data, bearing in mind what our investigation is intended to show.  Often, we do this with the help of a visual display of some sort, to decide whether any of the standard measures really makes sense for the given data set and to decide which of them would be the most appropriate to use.

Measures of centre can fail to make sense and fail to give useful information in various ways. 

Consider the maternity wing of a hospital. In a given month, $25$25 boys were born and $21$21 girls. The mean number of births in the two genders would not be a very meaningful number. A more useful statistic would be the proportions of each in the total number. In general, it usually makes little sense to find the average of the number of subjects that fall into a small number of categories.

In general, we need data that can take a more nearly continuous range of values in order for an average to give useful information. If, for example, the birth weights of the boys and the girls born in the hospital were recorded, it would then be of interest to know the location of the centre of the spread of the weights.

Consider a set of data about sale prices of houses in a particular city over some span of time. An examination of the data is likely to reveal that a few houses sold for exceptionally high prices while most sold for prices nearer to the middle of the range. Here, our intuitive idea of the middle of the range is related to the median.

The median, in this case, gives a better idea of the typical house price than would the mean. This is because a small number of exceptionally high data values can affect the mean strongly while the median is unaffected by extreme values.

Another difficulty occurs when a data set contains low values and high values but fewer values in between. Neither the mean nor the median would give a satisfactory impression of the centre in this situation. We might describe the data set as being bimodal.

When this occurs we look for an explanation in the possibility that the data has been drawn, perhaps inadvertently, from two different populations. The weights of the babies in the maternity hospital example may be of this kind, the two populations being the males and the females.

Measures of spread also need to be used carefully. The idea of range seems broadly applicable but it conveys no information about how the data is distributed.

Often associated with the mean is the measure of spread called the standard deviation. Many useful computations are done conveniently using the standard deviation, yet it can be misleading in the same way that the mean can be misleading. That is, when the data set contains exceptional values or has an asymmetrical distribution.

In other cases the interquartile range  may give a better indication of the distribution of the data about the centre. We associate measures of spread that use the quartiles with the median. For example, a box-and-whisker plot makes use of the five-number summary: minimum, first quartile, median, third quartile, maximum.  

Again, the five-number summary tends to hide the truth in the case of bimodal data. To use it effectively, we assume the existence of a single central location. Asymmetry in the distribution is reflected in unequal distances between the first quartile and the median and the median and the third quartile.

Example

Examine the following data by constructing a box-plot display.

{21, 21, 23, 24, 24, 25, 30, 36, 45, 46, 46, 47, 47}

The five-number summary consists of the values 21 (minimum), 23.5 (Q1), 30 (median), 46 (Q3), 47 (maximum)

We see that the distribution is positively skewed. More importantly, the whiskers are very short compared with the size of the box. This could indicate that the data set is bimodal and a closer inspection of it shows that this is indeed the case.

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3