Measures of Central Tendency and Spread

Describing a distribution with numbers

A data set may contain many observations, so it is helpful to summarise its distribution. A measure of central tendency gives a value intended to represent its centre or a typical observation. The mean, median and mode do this in different ways. The range describes spread instead: it indicates how far the data extends from its lowest to its highest value.

These summaries should always be interpreted in context. For example, a value of 12 might represent 12 goals, 12 minutes or 12 centimetres. The meaning and unit matter as much as the calculation.

The data itself may be primary data, collected first-hand for the investigation, or secondary data, previously collected by somebody else. Numerical data can be discrete, taking separate countable values, or continuous, able to take any value within an interval. These distinctions affect how data may be recorded and grouped, but the same central questions remain: what is typical, and how spread out are the observations?

The mean uses every value

To calculate the arithmetic mean, add all the values and divide by how many values there are:

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

For the data $4,6,7,8,10$, the total is $35$ and there are five values, so the mean is $35\div5=7$.

The mean need not be one of the observed values. If five families have a mean of 1.8 children, this does not suggest that any family has 1.8 children; it is a numerical summary of the whole group.

Because every observation contributes to the total, an unusually large or small value can move the mean considerably. Changing the last value in the earlier set from 10 to 30 changes the mean from 7 to 11, even though four of the five values have not changed.

Finding a mean from a frequency table

A frequency tells you how many times a value occurs. Rather than writing every observation separately, multiply each value $x$ by its frequency $f$. Add these products, then divide by the total frequency:

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

Suppose the numbers of books read are shown by values $0,1,2,3$ with frequencies $2,5,4,1$. The products $fx$ are $0,5,8,3$, so . There are people, giving a of books.

Here, $\sum f$ is the number of observations. The number of rows in the table is not the correct divisor.

Estimating the mean of grouped data

Grouped continuous data does not show every individual value. If a class is $20\leq t<30$, for instance, the observations could lie anywhere from 20 up to but not including 30. Its midpoint is

$$\frac{20+30}{2}=25.$$

To estimate the mean, use each class midpoint as a representative value, multiply it by the class frequency, and apply

$$\text{estimated mean}=\frac{\sum(\text{midpoint}\times\text{frequency})}{\sum\text{frequency}}.$$

The answer is an estimate because the actual values within each class are unknown. The class boundaries themselves should not be used in place of the midpoint.

The median depends on position

The median is the middle of the ordered data. It depends on position rather than the total of all the values, so ordering the list is essential.

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

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

For $3,5,6,9,12$, there are five values, so the median is the third value: 6.

For an even number of observations, there are two middle positions, $n/2$ and $n/2+1$. Find the mean of the values in those positions. In $2,4,7,11,13,20$, the middle values are 7 and 11, so the median is .

In a frequency table, cumulative frequencies can locate the middle observation without expanding the whole data set. For example, if the first two values have frequencies 3 and 8, positions 1–3 contain the first value and positions 4–11 contain the second. Both the 10th and 11th observations would therefore be that second value.

With grouped data, the exact individual values are unavailable. You can identify the median class containing the middle observation, but the original ungrouped median cannot usually be calculated exactly from the grouped table alone.

Mode and modal class

The mode is the value or category occurring most often. In $2,3,3,4,6$, the mode is 3. A data set can have two or more modes if several values share the greatest frequency, and it can have no mode if no value occurs more often than the others.

Unlike the mean and median, the mode can describe non-numerical categories. If red is the most commonly selected colour, then red is the mode; taking a mean of colours would make no sense.

When data has been grouped into intervals, the individual values are hidden, so an exact mode generally cannot be found. Instead, state the modal class: the class interval with the greatest frequency. The modal class is an interval, so it must be written with its complete boundaries, such as $150\leq h<160$ cm.

Range and the effect of outliers

The range measures overall spread:

$$\text{range}=\text{greatest value}-\text{least value}.$$

For $4,6,7,8,10$, the range is $10-4=6$. A larger range means the data covers a wider span, although it does not reveal how the values are distributed within that span.

An outlier is a value unusually high or low compared with the rest. In $10,11,11,12,12,13,40$, the value 40 is visibly separated from the main group. It raises the mean and greatly increases the range. The median remains 12 because it depends on the middle position, not on how far the largest value lies from the rest.

An outlier is not automatically an error. It might be a genuine but unusual result, or it might come from a recording or measurement mistake. Its cause should be considered before deciding whether to exclude it.

Choosing an informative summary

The mean uses all the numerical data, which is often useful when the distribution has no influential extreme values. The median is often more representative when an outlier or a strongly uneven distribution would distort the mean. The mode identifies the most common value or category, while the modal class identifies the most frequent interval in grouped data.

No measure tells the entire story. Two sets can have the same mean but very different ranges, so a measure of centre is often reported alongside a measure of spread. A sound statement names the measure, gives its value and unit, and explains what it represents in the situation.