Introduction to Mean, Median, Mode
Mean, median and mode are some of the measures of central tendency. These are three different properties of data sets that can give us useful, easy to understand information about a data set to see the big picture and understand what the data means about the world in which we live.
Mean
“Mean” and “average” are just two different terms for the same property of a data set. It is also known as the arithmetic mean. The mean or average is beneficial to property and one of the most significant, easy and most used calculations out of all the three central tendencies. The mean is basically the summation of all the values in the set of data after it is divided by the total number of values in the set of the data.
There are three methods of taking out averages – or mean in this case – and they are: direct method, assumed mean approach and step deviation method.
The above definition is of Arithmetic Mean, one of the many types of Mean. In detail, the types of mean are explained although most of them are out of scope for elementary Statistics
Arithmetic Mean
Arithmetic Mean is the average of all the observations. Generally, if the mean is mentioned without any adjective, it is assumed to be Arithmetic Mean.
Example- We have a set of observations-x=1,3,5,7,91,3,5,7,91,3,5,7,9. The Arithmetic Mean is computed as (x/n) where n is the number of observations which is equal to 5 in this case. Thus x=25 in this case and n=5 so the mean comes out to be 5
Weighted Mean
Weighted mean is almost the same as Arithmetic Mean, the difference being that in weighted Mean, some values contribute more than the others. 2 Cases arise while calculating Weighted Mean. The weighted mean is useful in situations when one observation is more important than others.
Case 1- When the sum of weights is 1- Simply multiply each weight by its corresponding value and sum it all up.
Example- In the previous example, let us assume that w=0.2 for all the observations, then the weighted mean is- W_mean= (0.2*1)+(0.2*3)+(0.2*5)+(0.2*7)+(0.2*9)=5 which is the same as Arithmetic Mean but if we change the weights then the mean also changes.
Case 2- When the sum of weights is not equal to 1- In this case it is beneficial to make a table that shows each weight against each observation. Then calculate the product of each observation and its corresponding weight.
Harmonic Mean
Harmonic Mean is calculated by dividing the total number of observations by the reciprocal of each observation. It is quite useful in Physics and has many other applications
(example- average speed when the duration of several trips is known).
It is given by the formula- \[H.M= \frac{ n}{(1/x1)+(1/x2)+(1/x3)+.....(1/xn)}\]
Geometric Mean
The Geometric Mean indicates the central tendency using the product of the observations rather than their sum(which is used in calculating Arithmetic Mean). It is used in the field of finance and social sciences. In finance, it is used to calculate the average growth rates. The Geometric Mean is most useful when the observations are dependent on each other or they have large fluctuations. It is given by(INSERT EQUATION)
Solved Example of Mean
1. Find the mean for the following frequency table:
Solution :
Arithmetic mean = \[\Sigma \frac{fx}{N}\]
Arithmetic mean = \[\Sigma \frac{fx}{N}\] = 1165 / 40
= 29.125
Hence the required arithmetic mean for the given data is 29.125.
Median
As the name suggests, the median is nothing but the middle – or “mid” – of all the values presented in the data set. This shows what the middle of the data is. For example: in a data set of 5, 10, 15, 20, 25, 15 is the median.
There are two different methods of finding out the mean. They are the odd number of values and even numbers of values.
Solved Example of Median
1. Find the median for the following frequency table:
Solution:
Here, the total frequency, N = \[\Sigma f\] = 40
N/2 = 40 / 2 = 20
The median is (N/2)th value = 20th value.
Now, the 20th value happens in the cumulative frequency 22, whose corresponding x value is 25.
Hence, the median = 25.
Mode
is defined as the value that is found mostly in a data set. When the frequencies in the data keep repeating, the mode takes place. This is mainly used for taking out most of the averages. For example, if you want to calculate the average of how many students scored the most, you might want to use the mode.
Solved Example of Mode
1. Find the mode for the following frequency table:
By observing the given data set, the number 40 occurs more often. That is 10 times.
Hence the mode is 40.
Mean = 29.125
Mode = 25 and
Mode = 40.
FAQs on Measures of Central Tendency: Mean, Median, and Mode
1. What are the key measures of central tendency in statistics?
The key measures of central tendency are **mean**, **median**, and **mode**. These values help to describe the center or typical value of a data set, providing a single number that represents the entire set of observations.
2. How is the **mean** calculated for a given set of data?
The **mean**, also known as the average, is calculated by summing all the values in a data set and then dividing this sum by the total number of values. It is the most commonly used measure of central tendency.
3. What is the **median** and how is it determined for a data set?
The **median** is the middle value in a data set when the values are arranged in ascending or descending order. If the data set has an odd number of values, the median is the single middle value. If there's an even number of values, the median is the average of the two middle values.
4. What is the **mode** and how is it identified in a data set?
The **mode** is the value that appears most frequently in a data set. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode at all if all values appear with the same frequency.
5. Why is understanding mean, median, and mode important in data analysis?
Understanding **mean**, **median**, and **mode** is crucial because they provide different insights into the distribution and characteristics of data. They help to summarize complex data, identify typical values, and understand variability, which is essential for making informed decisions and drawing conclusions in various fields like economics, science, and everyday life.
6. How do mean, median, and mode differ in their representation of data?
Mean, median, and mode represent data differently:
- The **mean** represents the arithmetic average and is sensitive to every value, including extreme ones.
- The **median** represents the middle point, making it less affected by extreme outliers.
- The **mode** represents the most frequent value, indicating the most popular or common observation.
The best measure to use depends on the data's distribution and the goal of the analysis.
7. Can a data set have multiple modes or no mode at all?
Yes, a data set can indeed have multiple modes or no mode:
- A data set with two modes is called **bimodal**, and with more than two modes, it's **multimodal**. This occurs when two or more values share the highest frequency.
- A data set has **no mode** if every value appears only once or if all values have the same frequency.
8. How are measures of central tendency applied in real-life situations?
Measures of central tendency are widely applied in real life:
- The **mean** is used to calculate average scores in exams, average daily temperatures, or average income.
- The **median** is often used for income or property values to avoid distortion by extremely high or low figures.
- The **mode** is useful for determining the most popular product size, the most frequent blood type in a population, or common customer preferences.
9. What are the general formulas for calculating mean, median, and mode?
The general concepts for calculating these are:
- **Mean:** Sum of all observations / Number of observations.
- **Median:** For sorted data, it's the middle value (or average of two middle values).
- **Mode:** The value that occurs with the highest frequency in the data set.
Specific formulas may vary depending on whether the data is grouped or ungrouped, as per CBSE 2025-26 syllabus.
10. How do extreme values or outliers affect the mean, median, and mode differently?
Extreme values or **outliers** impact the measures of central tendency differently:
- The **mean** is highly sensitive to outliers because every value contributes to the sum, pulling the average towards the extreme value.
- The **median** is robust to outliers since it only depends on the position of values, not their magnitude.
- The **mode** is generally not affected by outliers unless the outlier itself becomes the most frequent value, which is rare.
