 # Mean, Median and Mode

## What are mean, median and mode?

In descriptive statistics, the mean, median and mode are location parameter (measures of central tendency). Based on data collected in a sample, the location parameter provide information about where the "center" of the distribution lies.

Measures of location can be used to summarize or describe a list of data with only one parameter. An example would be that the average duration of studies of sports students at the university XY is 11.1 semesters. Together with the dispersion parameter, the location parameter therefore describe a distribution in the statistics. The most commonly used location parameter are the mean, the mode and the median. All these measures describe the centre of distribution in different ways. Which location parameter is used depends on the level of measurement of the variable and the robustness to outliers.

## Mean (arithmetic mean)

The mean value can only be calculated for metric variables, i.e. if a metric scale of measurement is given. It indicates where the center of gravity of a distribution can be found. In everyday life it is also called the "average".

##### Definition:

The arithmetic mean is the sum of all observations divided by their number n.

The mean value can be calculated by adding up all the values of a variable and then dividing the sum by the number of characteristic values. ### Calculate mean value

A group of 5 statistics students was asked how many cups of coffee they drink per week. The result is 21, 25, 10, 8 and 11 cups. The average is thus 15. Tip: You can easily calculate the mean value or the desired Location parameter for your data here on DATAtab in the statistics calculator , it's very easy.

## Geometric mean and quadratic mean

When talking about mean or average, mostly the arithmetic mean is meant, but there are also other types of mean values. Other mean values are, for example, the geometric mean and the quadratic mean also called Root Mean Square (RMS). • Geometric mean: If there are n positive numbers, the geometric mean is the nth root of the product of the n values.
• Root Square Mean: The root square mean is obtained by dividing the sum of the squares by the number of values and taking the square root.

## Median

If the measured values of a variable are ordered by size, the value in the middle is the median. The median is therefore the "middle value" of a distribution. It leads to a division of the series into two parts: one half is smaller and one is larger than the median.

Since for the calculation of the median the data are ordered, the variables must have ordinal or metric scale level.

##### Definition:

In an ordered series, the median is the value that divides the series into an equal upper and lower range. For the median to be calculated, the variable must be ordinally scaled. Ordinal scaling means that there is a ranking order between the values of a variable. This applies, for example, to school grades, height or salary. However, it is not possible to create a ranking for a variable Place of birth and therefore the median cannot be calculated here.

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

If there is an even number of characteristic carriers (persons), the two middle characteristics are added together and their sum is divided by two. ## Mean vs Median

Compared to the mean, the median is much more robust against scattering. An outlier usually has no influence on the median, but it has a more or less large influence on the mean. ## Mode (Modal Value)

The mode is the most common value. The mode is therefore the most frequent value in a distribution and corresponds to the highest value in the distribution. It is therefore the value that is "typical" for a distribution.

The mode can be used for both metric and categorical (nominal or ordinal) variables.

##### Definition:

The mode is the value of a distribution that occurs most often. ### Calculate 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". Therefore the mode can easily be read in a frequency table, it is the most frequent observed value.

Attention: There can also be several mode values. If two or more points occur with the greatest frequency, then there are several mode values. In this case one speaks then of a bimodal or multimodal distribution.

If the distribution is symmetric, the mean and median are equal, and if the distribution is symmetric and unimodal, all three measures are equal. As a rule, however, the three measures have different values. Now, of course, the question is which of the measures of central tendency to use. Unfortunately, there is no clear rule for this, only a few decision aids.

Mean: The mean value is by far the most used. The disadvantages of the mean are that it is sensitive to outliers, the value does not have to exist in the data and for the interpretation to be meaningful, the data should have metric scale level.

Median: The great advantage of the median is that it is very robust against outliers and that the data only have to be scaled ordinally.

Mode: The mode is the value that occurs most frequently, which has the advantage that the value actually occurs. Furthermore, the mode can also be calculated for data that cannot be ordered and thus have a nominal scale level. The disadvantage is that the mode does not take into account the other existing data.

## Example location parameter

With the online Statistics Calculator on DATAtab you can calculate the mean, median and mode of your data.

That's how it works with DATAtab: As an example, the score for a statistics exam can be used. To do this, copy the data into the Statistics Calculator, click on Descriptive Statistics and select the variable "Score".

Student Score
1 4
2 5
3 5
4 8
5 9
6 12
7 14
8 16
9 17
10 20

The result then looks like this:

Score
Mean 11
Median 10,5
Mode 5
##### Calculate mean:

The mean value is calculated by dividing the sum of all values by the number of values. ##### Calculate 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. ##### Calculate Modus:

To maintain the mode, the frequency of occurrence of each individual value is counted. The value that occurs most frequently is the mode. In this case, the value 5 is the only one that occurs twice, so the mode in this example is 5.

Cite DATAtab: DATAtab Team (2023). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net