Central Tendency
Central tendency is a term that is commonly used in the field of statistics. It is a statistical measure that represents a single value for the entire distribution or a dataset. Central tendency provides us an accurate explanation of the data in the distribution.
We use the concept of mean quite often to find the central tendency of data sets. There are multiple ways to find the mean of a data set. But arithmetic mean or average is the most commonly used method for finding mean. Hence, the term ‘mean’ is synonymous with arithmetic mean. We will learn the formula used to calculate the arithmetic mean of a data set.
List of Formulas
- \(\text{Mean of data set}=\frac{\text{Sum of all data}}{\text{Number of values}}\)
The mean of a data set is the sum of all values in the data set divided by the number of values in the data set. To find the mean of a data set, we first need to know all values present in the data set. We need to find the number of values and the sum of all values in the data set. Mean of the data set is the sum of all data values divided by the number of values in the data set.
Solved Examples
Example 1: Find the mean of the values: 5, 8, 12, 24, 4, 21, 51, 6, 1, 14.
Solution:
Data values: 5, 8, 12, 24, 4, 21, 51, 6, 1, 14
Number of data values = 10
Sum of data values = 5 + 8 + 12 + 24 + 4 + 21 + 51 + 6 + 1 + 14
= 146 [Add]
\(\text{Mean of data set}=\frac{\text{Sum of all data}}{\text{Number of values}}\)
= \(\frac{146}{10}\) [Substitute values]
= 14.6 [Divide]
So, the mean of the given data set is 14.6.
Example 2: Find the mean of the following set of values:
45, 43, 56, 78, 102, 213, 42, 29, 79, 11, 3, 91
Solution:
The values are 45, 43, 56, 78, 102, 213, 42, 29, 79, 11, 3 and 91.
Total number of values = 12
Sum of all values = 45 + 43 + 56 + 78 + 102 + 213 + 42 + 29 + 79 + 11 + 3 + 91
= 792 [Add]
\(\text{Mean of data set}=\frac{\text{Sum of all data}}{\text{Number of values}}\)
= \(\frac{792}{12}\) [Substitute values]
= 66 [Divide]
Therefore, the mean of the given values is 66.
Example 3: The height of 10 students in a class are as follows.
145, 152, 143, 144, 156, 138, 144, 137, 140 and 132
All measurements are in centimeters. Find the mean height of the students.
Solution:
Height of the students: 145, 152, 143, 144, 156, 138, 144, 137, 140 and 132
Number of students = 10
Mean of data set = \(\frac{\text{Sum of all data}}{\text{Number of values}}\)
= \(\frac{145~+~152~+~143~+~144~+~156~+~138~+~144~+~137~+~140~+~132}{10}\) [Substitute values]
= \(\frac{1431}{10}\) [Add]
= 143.1 cm [Divide]
So, the mean height of the 10 students is 143.1 centimeters.
Example 4: Two values, a and b, are not marked on the line plot. If the mean of the completed data set is 24 and the value of a is one more than the value of b, find the missing terms.
Solution:
As per question a = b + 1, and the mean is 25.
\(\text{Mean of data set}=\frac{\text{Sum of all data}}{\text{Number of values}}\)
24 = \(\frac{~22~+~23~+~23~+~23~+~24~+~25~+~26~+27~+~28~+~a~+~b}{11}\)
\(11~\times~24=221~+a~+~b\) [Multiply both sides by 11]
264 = 221 + a + b [Multiply]
264 – 221 = a + b [Subtract 221 from both sides]
a + b = 43
But
So, b + b + 1 = 43 [Substitute b+1 for a]
2b = 43 – 1 [Subtract 1 from both sides]
2b = 42
b = \(42~\div~2\) [Divide both sides by 2]
b = 21
So, a = b + 1
a = 21 + 1 [Substitute 21 for b]
a = 22 [Add]
Hence the missing terms are 22 and 21.
Example 5: A basketball player scores an average of 14 points in a match. If the player has scored 672 points in total, find the number of matches he played.
Solution:
Average points scored per match = 14 points
Total points scored = 672
Number of matches played = ?
We can use the mean formula to calculate the number of matches he has played. Let the number of matches be x.
\(\text{Mean of data set}=\frac{\text{Sum of all data}}{\text{Number of values}}\)
14 = \(\frac{672}{x}\) [Substitute values]
x = \(\frac{672}{14}\) [Multiply both sides by x, and divide both sides by 14]
x = 48
So, the basketball player has played in 48 matches.
Central tendency is a value that is used to represent all values of a data set. Mean is one of the most commonly used methods used to calculate the central tendency of data sets.
Mean, median and mode are the common methods used for calculating the central tendency of a data set.
Knowing the central tendency of a data set allows us to represent the whole data set with a single number. We can use central tendency as an approximate value where accuracy is not important.
The mean of a data set can be calculated in two steps. We need to find the sum of all values and then divide it by the number of values. The result is known as the mean of the data set.