Mean Median Mode Formula

Mean Median Mode Formula

When analyzing data, understanding the measures of central tendency—mean, median, and mode is crucial. These statistical measures provide insights into the distribution and central value of a dataset. Mean is useful for datasets with values that are evenly distributed and without extreme outliers. Median is preferred for skewed datasets or those with outliers, as it better represents the central value without being affected by extreme values.
Mode is ideal for categorical data or to identify the most common value in a dataset. Learn more about the mean, median, and mode formulas in this article by Extramarks.

What is the Mean Median Mode Formula?

In statistics, the three measures of central tendency are the mean, median, and mode. While describing a set of data,  students identify the core position of any data set. This is known as the central tendency measure.

The measurements of central tendency mean median, and mode are used to investigate the various properties of a particular collection of data. A measure of central tendency defines a set of data by identifying the data set’s central location as a single number.  Students may think of it as data clustering around a middle value. The three most prevalent measures of central tendency in statistics are mean, median, and mode.

Mean: The mean, often called the average, is the sum of all values in a dataset divided by the number of values.

Median: The median is the middle value of a dataset when it is ordered from least to greatest.

Mode: The mode is the value that appears most frequently in a dataset.

Mean Formula

The sum of all observations divided by the number of observations is the arithmetic mean of a set of data. For example, a cricketer’s five ODI scores are as follows: 12, 34, 45, 50, and 24.  Students use the mean formula to compute the arithmetic mean of data to estimate their average score in a match:

Mean = Sum of all observations divided by the number of observations.

Mean = (12 + 34 + 45 + 50 + 24)/5

Mean = 165/5 = 33

x represents the mean (pronounced as x bar).

Mean formula for grouped data

\[ \text{Mean} (\mu) = \frac{\sum_{i=1}^{k} f_i \cdot x_i}{\sum_{i=1}^{k} f_i} \]

Where:
\( f_i \) is the frequency of the \(i\)-th class or interval.
\( x_i \) is the midpoint of the \(i\)-th class or interval.
\( k \) is the number of classes or intervals.

Median Formula

The median of the data is the value of the middlemost observation obtained after organising the data in either ascending or descending order.

To get the median,  students must arrange the data in either ascending or descending order. Get the total number of observations in the data after organising it.

• The median is (n+1)/2 if the number is odd.
• The median formula for even numbers is as follows: ((n/2)th term + ((n/2) + 1)th term)/2 is the median.

Median for Grouped Data

For grouped data, the median is found using the following formula:

\[ \text{Median} = L + \left(\frac{\frac{n}{2} – CF}{f}\right) \cdot h \]

Where:
\( L \) = Lower boundary of the median class
\( n \) = Total number of observations
\( CF \) = Cumulative frequency of the class preceding the median class
\( f \) = Frequency of the median class
\( h \) = Class width

Mode Formula

A Mode is a value or number that appears the most frequently in a data collection. The mode for ungrouped data is simply the value that appears most frequently in the dataset. There is no complex formula for this; it is determined by counting the frequency of each value.

Mode for Grouped Data

For grouped data, the mode is found using the following formula:

\[ \text{Mode} = L + \left( \frac{f_m – f_1}{2f_m – f_1 – f_2} \right) \cdot h \]

Where:
\( L \) = Lower boundary of the modal class
\( f_m \) = Frequency of the modal class (class with the highest frequency)
\( f_1 \) = Frequency of the class preceding the modal class
\( f_2 \) = Frequency of the class succeeding the modal class
\( h \) = Class width

Examples of Mean Median Mode Formula

Example 1: Find mean of the dataset representing the ages of participants in a workshop: \( \{22, 25, 30, 28, 24, 26, 27\} \).

Solution:

List the Values:
\[ 22, 25, 30, 28, 24, 26, 27 \]

Calculate the Sum of the Values:
\[ 22 + 25 + 30 + 28 + 24 + 26 + 27 = 182 \]

Count the Number of Values:
\[ n = 7 \]

Calculate the Mean:
\[ \text{Mean} (\mu) = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{182}{7} = 26 \]

So, the mean age of the participants is 26.

Example 2: Find median of the dataset representing the number of books read by a group of students in a month: \( \{5, 3, 8, 7, 2, 4, 6\} \).

Solution:

Order the Data:
\[ 2, 3, 4, 5, 6, 7, 8 \]

Identify the Number of Values:
\[ n = 7 \] (Since there are 7 values, \( n \) is odd)

Calculate the Median:
For an odd number of values, the median is the middle value.

\[ \text{Median} = x_{\left(\frac{n+1}{2}\right)} = x_{\left(\frac{7+1}{2}\right)} = x_4 \]

So, the median is the 4th value in the ordered dataset:

\[ \text{Median} = 5 \]

Thus, the median number of books read is 5.

Example 3: Find the mode of the dataset representing the shoe sizes of a group of people:

Solution:

List the Values and Their Frequencies:

Identify the Mode:

• • The mode is the value that appears most frequently.
  • In this dataset, shoe size 7 appears 4 times, which is more frequent than any other value.

So, the mode of the shoe sizes is

$7$

7.