Mean, median and mode - Definition and examples

This article will be about The measures of central tendency, with definition and examples of how and when to use them.

What are the mean, median and mode

The mean, the median and the mode are measures of central tendency, this measures have the objective of finding a central number in a series of numbers, and from there the rest of the numbers are distributed. This 3 measures of central tendencies use different parameters to find the central number. This measures are very useful to express a whole series of numbers in just one value, even when the mean, the median and the mode are not always given a clear perspective of the data.

Each of this measures of tendency define the number that is in the center of the rest of the numbers using different criteria and sometimes the mode gives a better result than the median or the mean, or vice versa, this is why we are going to learn how to use this 3 measures, so we can understand in which situation is most convenient to use each of this measures.

The arithmetic mean, also now just as the mean, is use to find an average value of the data series, the mean is used only when we are working with quantitative variables (variables expressed by numbers), because we can not find an average of qualitative variables (variables expressed by letters). The arithmetic mean is represented with an “x” with a line above X.

The mean is really useful when we want to compare two data series, for example, if two people wants to know how much money do they spend daily, and they take data of how much money did they spend in the last month, when we obtain the mean of each person, the result would be the that the higher mean is the person that spends the most daily.

Now, something where the mean is not that accurate is when we try to know the composition of a data set, for example, if we want to find the mean of the obtained grades of two group of students, and we have that the first group have the following grades {7,7,7,7}, while the second group got the following grades {10,10,4,4}. When we find the mean of each group, we can see that both means were 7, this average defines exactly the composition of the first group, because every student got “average” grades, but in the second group there are no “average grades”, instead, there are two excellent grades, and 2 bad grades.

The previous example happens because the arithmetic mean is very influenced by the extreme data, by extreme data we mean really high or really low data, and this is one of the main drawbacks of using the mean. The mean is calculated by adding all the elements of a series, and dividing it by the total number of elements.

The mean is calculated by dividing the sum of every data by the quantity of data.

Example of how to find the mean.

Find the following mean: A study was made to determinate the average age when the people started going to the gym, if this question was asked to 10 people ¿What is the mean of the age when this people started going to the gym?

The results were: {18, 22, 31, 20, 25, 41, 19, 44, 17, 39}

The median is found when we order a number series from highest to lowest, and we find the number that is exactly in the middle of the series. The median defines the number that is in the center of the other values, no matter how many difference there is between each of the values. For example, if we have the following numbers: {1,2,100} the median would be the number 2, even when the difference between the first number and the median is of only 1 value, and the difference between the median and the third number is of 98.

The median of a number series that has an odd number of number, is found by finding the central number when the series is ordered, but, when we have an even number of numbers, then this series will have two middle numbers, in this cases we are going to find an average between this 2 central numbers and the result will be the median. Just as the mean, the median can only be found with quantitative variables.

Example of how to find the median.

Calculate the median: A study was made about how many glasses of water an adult person drinks, this question was asked to 16 people and these were the answers.

{3, 4, 2, 7, 9, 5, 7, 3, 1, 4, 7, 4, 5, 8, 3, 6}

Knowing this, calculate the median.

First of all we have to order the numbers from highest to lowest and we find the middle number.

{1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 7, 8, 9}

When we order the numbers, we can see that there are two central numbers, so, as we said before, when there are two central numbers we have calculate the average of this numbers, so we are going to find it.

The mode is the most repeated value in a series of numbers. Unlike the mean and the median, the mode can be used in both quantitative and qualitative variables, this is because we do not use any mathematical process to find it, we only find the frequency of the elements. When we have a series of numbers where two elements are the most repeated, this would be a bimodal series, and in case there are three or more modes, this series would be a multi-modal series.

Example of how to find the mode.

Find the mode: In a work group, where there are 11 people, somebody asked them about the domestic animal they like the most, and this were the results.

{dog, cat, dog, parrot, dog, cat, bunny, cat, cat, dog, bunny}

In this case, we are talking about a bimodal series, because there are two animals that were the most repeated, these were the dog and the cat.

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