The term 'average' refers to the ‘middle’ or ‘central’ point. The term refers to a number that is a typical representation of a group of numbers or data set. Averages can be calculated in different ways, here we are covering the most commonly used ones: the mean, median and mode. When the term ‘average’ is used in a mathematical sense, it usually refers to the mean, especially when no other information is given. Central tendency is a more appropriate word to refer to mean, median and mode. A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.
The mean of the ungrouped or raw data is obtained by adding all the observations and dividing the total by the number of observations. If x1, x2, x3,...xn be n observations, then their mean is denoted by \(\bar x\).
\(\bar x = \frac{x_1+x_2+x_3+...+x_n}{n} = \frac{\sum x}{n}\)
Mean = Sum of the observations/Number of observations
Example 1: The marks scored by class eight students are 3, 5, 7, 10, 4, 6, 8 and 9. Determine the mean marks.
Number of students in the class is 8.
Sum of marks scored ,\(\sum x = 3 + 5 + 7 + 10 + 4 + 6 + 8 + 9 = 52\)
∴\(\bar x = \frac{52}{8} = 6.5\)
Example 2: If the mean of 9, 14, x + 3, 12, 2x - 1 and 3 is 9. Find value of x.
Number of observation is 6
\(\sum x = 9 + 13+ x + 3 + 12 + 2x - 1 + 3\\ \sum x =39 + 3x\)
\(\frac{39 + 3x} {6} = 9\) => 3x = 54 - 39 => 3x = 15
∴ x = 5
Arithmetic Mean of Tabulated Data
If the frequency of n observations x1, x2, x3,...xn be f1, f2, f3,...fn respectively, then their \(\bar x\) is
\(\bar x = \frac{f_1x_1+f_2x_2+f_3x_3+...+f_nx_n}{f_1+f_2+f_3+...+f_n} \)
\(\bar x= \frac{\sum fx}{\sum f}\)
Example 1: Find the mean for the following distribution
| x | 5 | 6 | 7 | 8 | 9 |
| f | 4 | 8 | 14 | 11 | 3 |
Solution:
| x | f | fx |
| 5 | 4 | 20 |
| 6 | 8 | 48 |
| 7 | 14 | 98 |
| 8 | 11 | 88 |
| 9 | 3 | 27 |
| \(\sum f = 40\) | \(\sum fx = 281\) |
\(\bar x= \frac{281}{40} = 7.025\)
Mode of statistical data is the variate that occurs most frequently. Thus, the mode is the value of that variable which has a maximum frequency. For example, in the following data 2, 3, 4, 5, 4, 4, 5, 3, 7
Number 4 occurs 3 times(maximum) so 4 is the mode of this series.
It is not necessary that in data there can be only one mode. Let us see few examples:
Example 1: Find the mode of the following data: 2, 3, 8, 9, 4
As each number occurs only once and hence it has no mode.
Example 2: In the data 2, 2, 2, 3, 4, 4, 6, 6, 6, 7- 2 and 6 are both modes.
Example 3: Find the mode of the following data:
| Shirt size(in inches) | 32 | 34 | 36 | 40 |
| Number of shirts sold | 45 | 35 | 15 | 40 |
In a frequency distribution, the mode is the value of that variate that has the highest frequency. The mode of this distribution is 32" shirt.
If the given observations are arranged in an order, preferably from the smallest to the largest, the median is defined as the middle observation if the number of observations is odd. If the number of observations is even then the mean of the two middle observations is the median. Therefore there will be an equal number of observations above and below the median.
If number of observations is n then
Median = value of \(\frac{(n + 1)}{2 }\)th observation if n is odd
= mean of \(\frac{n}{2}\)th and \((\frac{n}{2} + 1)\)th observations if n is even
Example 1: Determine the median of the values: 15, 6, 7, 14, 8, 10, 12
Arrange data in ascending order: 6, 7, 8, 10, 12, 14, 15.
As n is 7 therefore median is the value of (7+1)∕2 = 4th observation. 10 is the median.
Example 2: Find the median of values: 30, 32, 36, 25, 28, 29, 31, 40
Arrange data in ascending order: 25, 28, 29, 30, 31, 32, 36, 40
As n is 8 therefore median is the mean of 4th and 5th observation. = (30 + 31) ∕ 2 = 61/2 = 30.5
What is the best measure of central tendency?
The mean is the most commonly used measure of central tendency because it uses all values in the data set to calculate the average. But in cases where your data has outliers, the median is a better option. Outliers are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. The mode is the only measure you can use for categorical data that can’t be ordered.
