Mean, median and mode are statistical measures used to describe the central tendencies of a set of data. They provide insight into the typical or representative value within a dataset. Here’s an explanation of each of these measures:
Mean:
The mean, also known as the average, is calculated by adding up all the values in a dataset and then dividing by the total number of values.
The formula for the mean (average):
\begin{aligned}\text{Mean} &= \dfrac{\text{Sum of all values}} {\text{Number of values}}\\\end{aligned}
The mean represents the arithmetic center of the dataset. It is sensitive to extreme values (outliers) and can be influenced by large deviations in the data.
Median:
The median is the middle value in a dataset when all the values are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values.
The median is not affected by extreme values or outliers and is a good measure of central tendency when data is skewed.
It is calculated by finding the middle value in the dataset.
Mode:
The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode at all (if all values are unique).
The mode is useful when you want to identify the most common value or category in a dataset.
In some datasets, there may be no mode, while in others, there may be multiple modes.
Here’s an example to illustrate these concepts:
Question 1:
Consider the dataset: 5, 8, 7, 8, 12, 5, 3, 8, 6, 7. Find the mean, median and mode of this dataset.
Solution:
Mean:
Add up all the values:
\begin{aligned}\text{Sum of all values}&= 5 + 8 + 7 + 8 + 12 + 5 + 3 + 8 + 6 + 7\\ &= 69\\\end{aligned}
We have, the total number of values:
\begin{aligned}\text{Number of values} = 10\\\end{aligned}
Divide the sum by the number of values:
\begin{aligned}\text{Mean} &= \dfrac{\text{Sum of all values}} {\text{Number of values}}\\ &= \dfrac{69}{10}\\ &= 6.9\\\end{aligned}
Median:
Arrange the values in ascending order:
3, 5, 5, 6, 7, 7, 8, 8, 8, 12
Since there are 10 values (an even number), the median is the average of the two middle values, which are the 5^{th} and 6^{th} values:
\begin{aligned}\text{Median } &= \dfrac{7+7}{2}\\ &= 7\\\end{aligned}
Mode:
Here, the value 8 appears most frequently (three times) in the dataset, and so the mode is 8.
In the above example, the mean is 6.9, the median is 7, and the mode is 8. These measures provide different perspectives on the central tendency of the data, and the choice of which one to use depends on the characteristics of the dataset and the specific questions being addressed.