

Mean, Median, and Mode
Mean, Median and Mode are the three statistical measures often used to encapsulate data sets. They are mainly known by the expected name “average”. There are many “averages” in statistics, but these three are the most common.
The “mean” is mainly the “average” where all the numbers are added up and then divided by the number of digits. The "median" is the "middle" value in the number series, and the "mode” is the value that occurs on numerous occasions. Out of these three measures, the mode formula in statistics opens different ranges and aspects for the student's
Formulas of Mean and Median
The first theory to understand from the mean, median, and mode is mean. As described above, the mean is the ratio of summation of all the values to the number of items. There are two types of Mean - simple arithmetic mean, and weighted arithmetic mean. If you want to know what is the formula of mean, go through the section given below:
\[ Mean= \frac{\sum X}{ N}\]
Here, \[ \sum X\]= Sum of individual values and =Total number of Items
The simple arithmetic mean examine values in data as equal and permits equal importance to each value. On the other hand, in weighted arithmetic mean, weights or importance is assigned to the values.
To find the Median, the numbers should be arranged in numerical order from smallest to highest. Taking the number of observations as (n), and “th” signifies the (n)th number -
Median Formula = \[\frac{(n+1)}{2} th\]
Students who are looking for mean, median, mode formula pdf can wish to visit the official website of Vedantu.
Mode Formula in Statistics for Discrete Series:
Mode comes third in the concept of mean, median, and mode. As discussed earlier, it merely refers to the value that occurs most frequently in a number series. In discrete series, the values of items with their equivalent frequencies are found. Essentially, the value of the items with the highest frequency will be the mode for the distribution. Mode formulae in statistics help the students to solve different complicated statistical problems in their textbooks. Students can follow this method of mode in discrete series from an example shown below -
Example 1 - Find Mode from the following data
Ans: Arranging data in ascending order
The term that is occurring the maximum number of times is considered as Mode (Z). Here we get 16 four times, 12 and 20 two times each, and other terms once only. Thus Z=16
Students can find more examples of mode calculation in discrete series in Vedantu’s online classes.
Illustrating Measures of Central Tendency
Mean, median, and mode are together known as measures of central tendency. It signifies a synopsis of data that represents the center point or quintessential value of a dataset. These measures specify where most values in a distribution fall and are also referred to as the central location of a distribution. Best measures of central tendency can be chosen depending on the data you have. A “measure of central tendency” is either a location framework or a statistic used to evaluate a location parameter.
Several Measures of Central Tendency
Arithmetic mean
Median
Mode
Geometric mean
Harmonic mean
Generalized mean
Weighted mean
Truncated mean
Interquartile mean
Midrange
Students can solve their statistical problems with measures of central tendency formulas that are given below -
Arithmetic Mean \[ (\overline{X} = \frac{\sum X}{N})\]
(Where \[\sum X\] is the summation of all the observations provided in a dataset and “n” is the number of observations)
Median formula= \[ \frac{{(n+1)}}{2}\]
Mode Calculation for Continuous Series
While demonstrating the mode formula for continuous series, it is only one step forward of the method for discrete series. We get the value of Mode by interposing as in the case with a median. Mode formula in statistics is easy to understand and pretty simple to calculate Formula provided below has been used to calculate Mode (Z):
Mode (Z) = \[L + \frac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}} * i\]
Where L= lower limit of modal interval
\[f_{1}\]= frequency correlating to modal interval
\[f_{0}\]= frequency heading modal interval
\[f_{2}\]= frequency succeeding modal interval
\[i\]= length of modal interval
Several Points While Calculating Mode -
Class intervals must be absolute
Length of the classes should be equal
Series should be arranged in ascending order
If the series is increasing, then convert it into continuous series
If the first class is considered as a modal class, then f0 will be zero
Similarly, if the last class is considered as a modal class, then f2 is zero
Students can have access to online resources where they can get innumerable solutions considering the formula of mode in continuous series. To know more about this topic, download Vedantu’s app now!
The mean, median, and mode are the three classifications of central tendency.
Mean
The arithmetic average of the given data set which is found by the division of the data set by the total number of observations in the data set is referred to as the mean of the data set.
\[ \overline{x} = \frac{\sum x}{N}\]
where the summation is represented by \[ \sum\]
observations are represented by x
number of observations is represented by N
If the data is presented in a tabular form, in this case, the formula used to compute the mean is
Mean=\[ \frac{\sum f x}{\sum f} \]
Where \[ \sum f = N\]
Median
The middle number in the data set is referred to as the median when the numbers are listed in either ascending or descending order.
In the case when the total number of observations (n) is an odd number, then the formula of the median is given by-
\[Median = \left ( \frac{n+1}{2} \right )^{th}observation\]
In the case when the total number of observations (n) is an even number, then the formula for the median is given by-
\[ Median = \frac{(\frac{n}{2})^{th}observation + (\frac{n}{2} + 1)^{th} observation}{2} \]
When the data is continuous and is presented in form of the frequency distribution, then the median given by-
\[Median=1 + \left [ \frac{(\frac{n}{2}) - c}{f} \right ] \times h\]
where the lesser limit which belongs to the median class= l
cumulative frequency value before the median class= c
frequency that is possessed by the median class= f
size of the class= h
Mode
The value in a data set that has occurred the most number times is referred to as the mode and the difference between the highest and lowest values of the data set is referred to as the range.
To calculate mode in a data set, the following steps are followed-
Determination of the modal class that possesses the maximum frequency.
Calculation of the mode by the formula- \[Mode= 1+ \left [ \frac{f_{m}-f_{1}}{2fm-f_{1}-f_{2}} \right ] \times h\]
where frequency possessed by modal class= \[f_{m}\]
frequency possessed by the class that is before the modal class= \[f_{1}\]
frequency possessed by the class that is after the modal class= \[f_{2}\]
Lesser limit of modal class= l
Width of the class= h
FAQs on How to Calculate Mean, Median, and Mode
1. What is the mean, and how is it calculated for any given set of numbers?
The mean, often referred to as the average, is a measure of central tendency. To calculate it, you add up all the numbers in a dataset and then divide the sum by the total count of numbers.
- Formula: Sum of all observations / Total number of observations.
- Example: For the numbers 2, 7, 9, the mean is (2 + 7 + 9) / 3 = 18 / 3 = 6.
2. How is the median determined for a dataset, considering both an even or odd number of values?
The median is the middle value of a dataset when all the numbers are arranged in either ascending or descending order.
- If the total number of values (N) is odd, the median is the single middle value at the position ((N+1)/2)th.
- If the total number of values (N) is even, the median is the average of the two middle values, found at the (N/2)th and ((N/2)+1)th positions.
3. What is the mode in statistics, and is it possible for a dataset to have multiple modes or no mode at all?
The mode is the value that appears most frequently in a dataset. It indicates the most common observation.
- A dataset can have one mode (unimodal), multiple modes (bimodal if two, multimodal if more than two values share the highest frequency), or no mode at all if every value appears only once.
4. What is the empirical relationship that connects the mean, median, and mode?
For distributions that are moderately asymmetrical or skewed, there's an empirical relationship often used to approximate one measure if the other two are known:
- Formula: Mode = 3 Median - 2 Mean
This formula is particularly useful in statistics for understanding the central tendency of datasets that are not perfectly symmetrical.
5. Why is understanding the mean, median, and mode important for analyzing data?
Understanding the mean, median, and mode is crucial because each provides a unique insight into the central tendency of a dataset. They help describe where the 'center' of the data lies but in different ways.
- The mean is good for evenly distributed data.
- The median is robust against extreme values.
- The mode identifies the most common observation.
Together, they offer a comprehensive picture of the data's distribution, essential for making informed conclusions in various fields.
6. How do extreme values or outliers affect the mean, median, and mode differently?
Extreme values, known as outliers, have varying impacts on these measures:
- The mean is highly sensitive to outliers because every value contributes to its calculation, pulling the average towards the extreme numbers.
- The median is much less affected by outliers as it only considers the position of the values, not their magnitude.
- The mode is generally unaffected by outliers unless an outlier itself happens to be the most frequent value.
7. How do you choose the most appropriate measure of central tendency (mean, median, or mode) for a specific dataset?
Choosing the best measure depends on the **nature and distribution** of your data:
- Use the **mean** for symmetrical distributions without significant outliers, especially for numerical data where arithmetic operations are meaningful.
- Choose the **median** for skewed distributions or datasets containing outliers, as it provides a more representative center by minimizing the influence of extreme values.
- The **mode** is ideal for categorical (nominal) data or when you need to identify the most popular or frequent item/category in any type of data.
8. What is the 'range' of a dataset in basic statistics?
The range is a simple measure of data spread or variability. It is calculated by finding the difference between the highest (maximum) value and the lowest (minimum) value within a dataset.
- Formula: Range = Maximum Value - Minimum Value.
9. Can you illustrate the calculation of mean, median, and mode with a single example?
Certainly! Let's calculate the mean, median, and mode for the dataset: **13, 16, 12, 14, 19, 12, 14, 13, 14**.
1. Mean:
Sum = 13 + 16 + 12 + 14 + 19 + 12 + 14 + 13 + 14 = 127
Count = 9
Mean = 127 / 9 = 14.11 (approximately)
2. Median:
First, arrange the data in ascending order: 12, 12, 13, 13, 14, 14, 14, 16, 19
Since there are 9 values (an odd number), the median is the ((9+1)/2) = 5th value.
Median = 14
3. Mode:
Count the frequency of each number:
- 12 appears 2 times
- 13 appears 2 times
- 14 appears 3 times
- 16 appears 1 time
- 19 appears 1 time
The number that appears most frequently is 14.
Mode = 14

















