## Lesson 5: Statistics. Mean, Median and Mode (III)

January 18, 2012

### 4. Averages or central tendency measurements

When you are carrying out a statistical study your first task is organising data. Secondly we will try to interpret data and draw some conclusions. Average or central tendency measurements are different ways of determining or indicating which value from the information is the central value. We are going to study:

• Arithmetic mean
• Median

### 4.1. Arithmetic mean

Arithmetic mean, or mean for short, of a set of data is the quotient of the sum of all the data divided by the number of data. If the variable considered is named  , it is represented by .

As a consequence of its definition it can only be calculated in quantitative variables.

To calculate the mean we have to:

• Add up all the values of the variable if a value is repeated, include both or all the values in the sum.
• Divide the sum above  by the total number of data

Abbreviated calculations for finding the mean

Usually some values of the variable are repeated, so we can calculate their frequencies and make calculations easier.

Let xi, , x2 ,…xn ,  be the (different) values of the variable. Let, fi, , f2 ,…fn  be their corresponding absolute frequencies.

To calculate the mean:

• Multiply each different data by its frequency.
• Add up all of them (last column at the bottom)
• Divide the sum before by the total number of data

This is:

x = (xi,· fi, +  x·, f2 + …+ xn·fn)/N

### 4.2. Median

The median of a set of data Me, is the central value of the data, this means there are as many values less than the median as there are greater than it.

It can only be calculated in quantitative variables, it is unique and it can’t coincide with any other data.

Calculation of the median

First, we order data from smallest to biggest (in ascending order). There are two cases:

• If the total number of data is odd, the median is the value that occupies the central position.
• If the total number of data is even, the median is the arithmetic mean of the two values that occupy the central positions.

### 4.3. Mode

The mode in a set of data, Mo refers to the value of the set of data that occurs most frequently.

This measurement can be calculated with any variable, qualitative or quantitative.

It is important to note that there can be more than one mode and if no number occurs more than once in the set, then there is no mode for that set of numbers.

Calculation of the Mode

• Given a set of data, the mode is the value of the set of data that occurs most frequently. If we are given a table of frequencies, the mode is the value of the variable whose absolute frequency is the greatest. If there are two values with maximum frequency, then the distribution is bimodal, if there are several values with maximum frequency, then the distribution is multimodal.
• If we are given a bar chart, the mode is the data corresponding to the highest bar

If we are given a pie chart, the mode is the data corresponding to the circular sector of largest degree of angle.

The links below provide you some practice:

The video below gives you a quick explanation about the averages by using a song. Try to understand it! Post your translation the first and you will get an extra grade!