The measures of central tendency depict the central point of the given data distribution. These measures indicate where the major portion of the data falls for a given distribution. In statistics, there are three basic measures of central tendency, which shows how the distribution is clustered around the middle value of the data set. The three basic measures of central tendencies are—mean, median and mode. Each of these measures shows the middle characteristics of the data sets.
The measures of central tendencies describe
- A summary of the given data distribution.
- The central value or middle value over a set of data.
- To which extent the data is scattered from the middle value.
- In which range most of the data falls for a given dataset.
Mode
This measure of central tendency provides information about which type of data occurs the most within a given data distribution. For example—below is the data distribution of marks obtained by the students of a certain standard,
| Percentage | Number of Students |
| 30 – 40% | 6 |
| 40 – 50% | 4 |
| 50 – 60% | 10 |
| 60 – 70% | 25 |
| 70 – 80% | 15 |
| 80 – 90% | 5 |
| 90 – 100% | 2 |
We observe that most students have scored between 60 – 70%. Hence, the mode class of the given data distribution is 60 – 70%.
| The mode of the given dataset is that value whose frequency is the most among the set of values. It describes the most occurring data of the data distribution. |
Learn more about the mode with examples.
Median
The median describes the middle value of the ordered data distribution. To find the median of the given data distribution, we first have to arrange it in ascending or descending order, then find the middle or central value. Let there are n numbers of data; if n is an even number, then the middle values are (n/2)th and [(n/2) + 1]th, and the median is the average of (n/2)th and (n/2 + 1)th data. If n is an odd number, the median is the [(n + 1)/2]th data.
Consider a data distribution, 1, 2, 4, 12, 3, 4, 5, 13, 7, 9, 15. Arranging the data in ascending order, 1, 2, 3, 4, 4, 5, 7, 9, 12, 13, 15. There is an odd number of data, n = 11.
Hence, median of the data distribution = (11 + 1)/ 2 th entry = 6th entry = 5.
Again, consider the data distribution with an even number of data values, 22, 24, 26, 23, 25, 26, 27, 28, 29, 23. Arranging the data in ascending order, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29. The median will be the average of (n/2)th and [(n + 1)/ 2]th value, where n = 10. Hence, the median will be the average of the 5th and 6th values,
Median = (25 + 26)/ 2 = 25.5.
Therefore, the median of the given data distribution is 25.5.
Mean
The mean is the average of all the data in the given data distribution. It is generally calculated by dividing the sum of the numbers by the total number of data in the data set.
| Mean = Sum of all the data ÷ Total number of data |
If there are n of number data x1, x2, …, xn in a data distribution, then the mean of data is given by
X̄ = (x1 + x2 + …+ xn)/ n.
There are different types of mean—arithmetic mean, geometric mean, and harmonic mean. Usually, to find the mean of ungrouped data distribution, we calculate the arithmetic mean.
Learn: what is mean in math?
For example, we find the mean of 18, 17, 15, 23, and 25.
Sum of all the numbers = 18 + 17 + 15 + 23 + 25 = 98
Total number of data = 5
Mean = Sum of all the data ÷ Total number of data = 98/5 = 19.6
Hence, the mean or average of the given dataset is 19.6.
