Measures of Location

What are Mean, Median and Mode?

In descriptive statistics, a measure of location tells you where the "center" of a distribution lies. The three most important measures of location are the mean, the median and the mode. They are often called measures of central tendency or measures of central location. Some textbooks also use the broader term location parameters.

Measures of location summarize a whole list of data with a single value. For example, instead of listing how long every single sports student took to finish their degree, you can simply say: the average duration of study is 11.1 semesters.

Together with the measures of dispersion, the measures of location describe a distribution in statistics: the location tells you where the data lie, the dispersion tells you how spread out they are. Mean, median and mode each describe the center of a distribution in a different way. Which measure of location you should use depends on the level of measurement of the variable and on how robust the measure needs to be to outliers.

Mean (Arithmetic Mean)

The arithmetic mean is appropriate for metric variables, that is, variables for which differences between values are meaningful. If a metric scale of measurement is given, it indicates the balance point of the distribution. In everyday language, it is often called the "average".

To calculate the mean, add up all the values of a variable and then divide the sum by the number of observations.

Calculate the Mean

A group of 5 statistics students was asked how many cups of coffee they drink per week. The answers are 21, 25, 10, 8 and 11 cups. The sum is 75, divided by 5 observations this gives a mean of 15 cups.

Tip: You can easily calculate the mean, median and mode of your data online with the numiqo statistics calculator.

Geometric Mean and Quadratic Mean

When people talk about the mean or the average, they usually mean the arithmetic mean. However, there are also other types of means, for example the geometric mean and the quadratic mean, also called Root Mean Square (RMS).

• Geometric mean: For n positive numbers, the geometric mean is the nth root of their product. It is typically used for growth factors and other multiplicative processes, such as calculating an average annual growth rate.
• Root mean square (RMS): The RMS is obtained by squaring each value, calculating the mean of those squares and then taking the square root. It is mainly used in physics and engineering, for example for alternating currents.

Median

If the values of a variable are sorted by size, the value in the middle is the median. The median is therefore the "middle value" of a distribution. At least half of the observations are less than or equal to the median, and at least half are greater than or equal to it.

Since the data must be sorted to calculate the median, the variable must have an ordinal or metric scale level.

For the median to be calculated, the scale level must be ordinal or higher. Ordinal scaling means that there is a ranking order between the values of a variable. This applies, for example, to school grades (ordinal) or salary (metric). For a variable such as place of birth, however, no ranking can be created, so the median cannot be calculated.

If there is an odd number of values, the median is a value that actually occurs in the data.

If there is an even number of metric values, there are two middle values. The usual numerical median is their arithmetic mean. For ordinal categories, averaging category labels may not be meaningful, so the two middle categories should instead be reported or handled according to the chosen convention.

Mean vs. Median

Compared to the mean, the median is much more robust to outliers. A single extreme value usually has little influence on the median, but it can pull the mean strongly in its direction. For skewed data with outliers, such as income, the median is therefore often the better measure of location.

Mode (Modal Value)

The most frequently observed value in a data set is called the mode or modal value. In other words, the mode is the value that occurs most often. It can be a useful description of what is most common, although it is not necessarily near the center of the data.

The mode can be used for both metric and categorical (nominal or ordinal) variables. For continuous metric data, however, exact values may rarely repeat, so a mode often depends on grouping the values into intervals.

Calculate the Mode

Example: In a sample of 70 managers from Berlin, 20 drive a Daimler, 25 a BMW, 10 a VW and 15 an Audi. The car brand BMW is the most common. Thus the mode is "BMW".

The mode can easily be read from a frequency table: it is simply the value with the highest frequency.

Attention: There can also be more than one mode. If two or more values occur with the same greatest frequency, the distribution has several modes and is called bimodal (two modes) or multimodal (more than two modes).

Advantages and Disadvantages of the Mean, Median and Mode

In an exactly symmetric distribution, the mean and median lie at the center, provided the mean exists. In a symmetric, unimodal distribution whose peak is at that center, the mode is there as well. In real samples, however, the three measures often differ. Which measure is most useful depends on the variable's level of measurement, the shape of the distribution and the purpose of the analysis.

Mean: The mean is widely used and includes every observation. However, it is sensitive to outliers, it does not have to be a value that occurs in the data, and its interpretation requires a metric scale.

Median: The median is robust to outliers and requires only ordered data. It does not reflect the size of every value, however.

Mode: The mode is an observed value and can be calculated even for nominal data that cannot be ordered. Its disadvantages are that a data set can have no mode or several modes, and it does not use the sizes of the other values.

The following table summarizes the three measures of location:

Example: Calculate Mean, Median and Mode

With the Online Statistics Calculator lets you calculate the mean, median and mode of your data.

How it works in numiqo: This example uses scores from a statistics exam. Copy the data into the Statistics Calculator, click on Descriptive Statistics and select the variable "Score".

The result then looks like this:

Calculate the mean:

The mean is calculated by dividing the sum of all values by the number of values.

Calculate the median:

Due to the even number of values, the median is obtained by adding the two middle values. The sum is then divided by two.