Central Tendency
In order to understand the differences between mean and average, we need to think beyond just their definitions. We need to understand where these terms are used and the different things that they can imply. Both mean and average are used to find the central tendency of a data set.
The central tendency of a data set is a single value that is used to represent all values in the data set. The central tendency of a data set can be calculated in many ways; mean or average is just one of them. Let us look at the definition of mean and average and learn the differences between them.
What is ‘Average’?
Average is a method of finding the central tendency of a data set. Average is calculated by adding all the data values and dividing them by the total number of values.
Example: The average of 1, 2, 3, 4, 5, 6 and 7 is \(\frac{1~+~2~+~3~+~4~+~5~+~6~+~7}{7}=4\)
The average 4 is near the central value of the set of numbers.
What is Importance of ‘Mean’?
There are different types of means: arithmetic mean, geometric mean and harmonic mean. But we generally refer to arithmetic mean when we use the term mean. The definition of arithmetic mean is the same as the definition of average. Arithmetic mean is calculated by dividing the sum of all data values by the number of values.
Example: The average of 2, 4, 6, 8, 10, 12 is \(\frac{2~+~4~+~6~+~8~+~10~+~12}{6}=7\)
The average 7 is near the central value of the set of numbers.
Difference Between Mean and Average
We use the terms ‘mean’ and ‘average’ interchangeably on many occasions. This is because average has the same definition as arithmetic mean. The most commonly used type of mean is the arithmetic mean. The term ‘average’ is not only used in math but also in colloquial language. Mean is a more precise word that is preferred in the field of statistics. Let’s explore some more differences between mean and average.
Rapid Recall
The average value, or the arithmetic mean, of a set of values always lies between the smallest and the largest values in the data set.
Solved Examples
Example 1: Find the average of the following values:
14, 18, 11, 15, 12, 18, 7, 9
Solution:
The values are 14, 18, 11, 15, 12, 18, 7, 9
Sum of all values = 14 + 18 + 11 + 15 + 12 + 18 + 7 + 9 = 104
Number of values = 8
\(Mean=\frac{Sum~of~all~values}{Number~of~values}\)
\(=\frac{104}{8}\)
= 13
So, the average of the given values is 13.
Example 2: Find the mean of 1, 4, 12, 27, 32, 44
Solution:
The values are 1, 4, 12, 27, 32, 44
Sum of all values = 1 + 4 + 12 + 27 + 32 + 44 = 120
Number of values = 6
\(Mean=\frac{Sum~of~all~values}{Number~of~values}\)
\(=\frac{120}{6}\)
= 20
So, the mean of the given values is 20.
Example 3: The number of trekking suits printed by the machine in the last 7 days is as follows: 25, 28, 27, 29, 31, 24, 25.
Find the mean number of trekking suits printed by the machine every day.
Solution:
Number of suits that were printed each day of the week are: 25, 28, 27, 29, 31, 24, 25
Total number of suits printed in a week = 25 + 28 + 27 + 29 + 31 + 24 + 25 = 189
Number of days = 7
\(Mean=\frac{Sum~of~all~values}{Number~of~values}\)
\(=\frac{189}{7}\)
= 27
Therefore, the mean value of the number of trekking suits printed in a day is 27.
Central tendency is the value that is used to represent all values of a data set.
What are the different methods of calculating central tendency?
There are three methods to calculate central tendency: mean, mode and median.
Average refers to the arithmetic mean of a data set. Mean can be either arithmetic mean, geometric mean, or harmonic mean.
To find the average of a set, divide the sum of all the values by the total number of values. The result of this operation is called the arithmetic mean or average.
The values of average and arithmetic mean are always the same. But, the value of geometric mean and harmonic mean can differ from the value of average.