Measures of Central Tendency

What does “typical” mean?

A data set may contain many observations, but it is often helpful to summarise its centre using one number. The mean, median and mode are different measures of central tendency. They do not necessarily give the same answer because each describes “typical” in a different way.

Before choosing a measure, think about the kind of data and what the result should communicate. The mean balances all the values, the median marks the halfway point, and the mode identifies what occurs most often. For grouped data, exact individual values are unavailable, so a modal class may replace a single mode and the mean must be estimated.

The mean: sharing the total equally

To calculate the mean, add all the values and divide by the number of values:

$$\text{mean}=\frac{\text{sum of the values}}{\text{number of values}}$$

The symbol $\bar{x}$ is often used for the mean. For the values $4,8,6,10,2$, the total is $30$ and there are five values, so the mean is $30\div5=6$.

This can be understood as a balancing or equal-sharing value. If the total of 30 were shared equally among five observations, each would receive 6. Consequently,

$$\text{total}=\text{mean}\times\text{number of values}.$$

This rearrangement is useful when a mean or a missing value is known. For example, if five values have a mean of 8, their total must be $5\times8=40$. If four known values total 32, the missing value is $40-32=8$.

The mean uses every observation, which is valuable when all the numerical values matter. However, an unusually high or low value can pull it towards that extreme. It is also not normally meaningful for non-numerical categories: the “mean type of transport” cannot be calculated.

The median: locating the middle

The median divides an ordered data set into two halves. The data must first be arranged in numerical order.

For an odd number $n$ of values, the median is at position

$$\frac{n+1}{2}.$$

Consider the nine values $12,19,12,15,11,15,12,13,17$. In order, they are

$$11,12,12,12,13,15,15,17,19.$$

The median is the fifth value, 13.

For an even number of values, there are two central observations. Find the values in positions $\frac n2$ and $\frac n2+1$, then calculate their . Thus the median of is

$$\frac{3+6}{2}=4.5.$$

The median depends on positions rather than the sizes of all the observations. It is therefore less affected by extreme values than the mean and is often suitable for a skewed distribution—for example, house prices containing a small number of exceptionally expensive homes.

The mode: finding what is most common

The mode is the value with the greatest frequency. In $3,7,5,7,2,7,9$, the value 7 occurs three times, so the mode is 7.

A data set can have more than one mode. For example, $2,2,4,4,7$ has modes 2 and 4. If no value occurs more frequently than the others, there is no mode. It is not correct to say that the mode is 0 unless 0 is genuinely the most frequent data value.

Unlike the mean and median, the mode can describe categorical data. If “bus” is the most common response in a survey of travel methods, bus is the mode. In a bar chart, it corresponds to the category with the greatest frequency; in a pie chart, it corresponds to the largest sector.

Central tendency in a discrete frequency table

A frequency table records repeated values compactly. Suppose the value $x$ occurs with frequency $f$. There are $\sum f$ observations altogether, while multiplying each value by its frequency and adding gives the total $\sum fx$. Therefore,

$$\bar{x}=\frac{\sum fx}{\sum f}.$$

A reliable working table has columns for $x$, $f$ and $fx$. Multiply across each row, add the $f$ column, add the $fx$ column, and divide. Do not divide by the number of rows: a row may represent many observations.

The mode is the $x$-value beside the greatest frequency. To find the median, use cumulative frequencies to locate the central observation or observations. For instance, a cumulative frequency of 12 means that the first 12 ordered observations have been accounted for. This avoids writing every repeated value separately.

The same ideas apply when information is displayed graphically. Read the frequencies carefully from a bar chart, frequency diagram or stem-and-leaf diagram, then apply the appropriate method. A stem-and-leaf diagram already displays the values in order if it has been constructed correctly.

Grouped data: what can and cannot be known

Grouped data gives intervals rather than exact observations. In the interval $20\leq f<22$, for example, every value is at least 20 but less than 22. The table does not reveal the individual values inside that interval, so some exact measures cannot be recovered.

The modal class is the complete class interval with the greatest frequency. For the foot-length data below, the greatest frequency is 12, so the modal class is $22\leq f<24$ cm.

A modal class is an interval, not its midpoint. When grouped data is represented by a histogram with unequal class widths, bar height represents frequency density rather than frequency itself, so the tallest bar alone does not necessarily identify the greatest frequency; frequencies are represented by bar areas.

The interval containing the median can be found using cumulative frequency. There are 40 adults, so halfway through the data is around the 20th observation. The cumulative frequencies are $3,9,19,31,40$, so the halfway point lies in $22\le f<$. The exact median is not known from this table because the exact foot lengths within that class have been lost.

Estimating the mean of grouped data

To estimate a grouped mean, represent every observation in a class by that class midpoint:

$$\text{midpoint}=\frac{\text{lower boundary}+\text{upper boundary}}{2}.$$

For the foot-length table, the midpoints are $17,19,21,23,25$. Multiplying each midpoint by its frequency gives

$$(17\times3)+(19\times6)+(21\times10)+(23\times12)+(25\times9)=876.$$

The estimated mean is therefore

$$\bar{x}\approx\frac{876}{40}=21.9\text{ cm}.$$

The answer is an estimate because the calculation assumes that every value in each class is equal to its midpoint. The real observations may lie elsewhere within their intervals. Using more decimal places does not make the result exact.

Choosing an appropriate measure

The most useful measure depends on the data and the purpose:

• Use the mean when numerical data are reasonably balanced and every value should affect the summary.
• Use the median when extreme values or a skewed distribution would make the mean misleading.
• Use the mode when the most common value or category is important.
• Use the modal class when grouped data only identify the most frequent interval.

When comparing two distributions, use the same appropriate measure for both and interpret it in context. For example, if one team has a higher median score, its central score is higher. A comparison of centre alone does not describe how consistent the results are; that requires an appropriate measure of spread as well.