Mean, Median, Mode - Definition, Differences | How to Find?

Mean, Median, Mode

 Mean, median, and mode are the fundamental measures of central tendency in statistics. It helps to summarise data into a single representative value. While they each serve a similar purpose, they do so in different ways and can provide unique insights into the data's distribution. 

 The mean is calculated by adding all values in the data set and dividing by the number of values, while the median represents the middle value when the data is arranged in order. The mode is the most frequently occurring value in the data set. Understanding the nature of each concept helps students with data distribution. 

What is Mean Median and Mode in Statistics?

In statistics, the three most common measures of central tendency are mean, median, and mode.  A measure of central tendency is a statistical value that describes the centre value of a dataset. They provide a single representative number that summarizes a large set of data. It makes it easier to understand the overall pattern and trend of data. 

• Mean: The mean, or average, in statistics is calculated by adding all the values in a data set and dividing by the total number of values. It represents the central value or typical value of the data.
• Median: The median is the middle value in an ordered set of data. It divides the data into two halves. Median helps in understanding the data's central tendency when the data is skewed or has outliers.
• Mode: The mode is the value that appears most frequently in a data set. It shows the most common or popular value and can be used with both numerical and categorical data.

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Mean

The mean, often simply called the average, is the most widely used measure of central tendency. It’s calculated by summing up all the values in a dataset and then dividing that sum by the total number of values.

 Mean = Sum of Data Points/Total Number of Data Points.

 It is also denoted as xˉ (pronounced as x bar)

 xˉ=n∑x​

Where:

• xˉ (pronounced "x-bar") is the symbol for the sample mean.
• ∑ (sigma) is the symbol for summation, meaning "the sum of.
• x represents each value in the dataset.
• n is the total number of values in the dataset.

Example 

 Let's find the mean of the following set of numbers: {10,15,20,25,30}

• Step 1: Add all the numbers together.10
  10+15+20+25+30=100
• Step 2: Count the total number of values in the dataset.11
  There are 5 numbers in the set.
• Step 3: Divide the sum by the count.12
  5100​=20

Therefore, the mean of the dataset {10,15,20,25,30} is 20.

Read More: Construction in Maths

Mean of Ungrouped Data

The mean of ungrouped data is the simplest form of the average. It's calculated for a set of raw data that has not been organized into frequency tables or class intervals. It gives you a single value that represents the center of the dataset.

The formula for calculating the mean is straightforward:

 xˉ=n∑x​

 Where:

• xˉ (pronounced "x-bar") represents the mean of the sample data.
• ∑x (read as "sigma x") means the sum of all the individual data values.
• n is the total number of values in the dataset.

In simple terms, you just add up all the numbers and divide by how many numbers there are.

 Example

Let's find the mean number of points a basketball player scored in their last 5 games. The scores are: 15, 20, 10, 25, 30.

• Step 1: Find the sum of all values (∑x).

Add all the points scored together:

15+20+10+25+30=100

• Step 2: Find the total number of values (n).

Count how many games there were. There are 5 scores, so n=5.

• Step 3: Apply the formula.

Divide the sum of the scores by the number of games.

xˉ=5100​=20

The mean number of points scored is 20.

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Mean of Grouped Data

The mean of grouped data is an estimate of the average for a dataset that has been organized into a frequency distribution with class intervals. Since the individual data points are unknown, we use the midpoint of each class to represent the values within that interval.

Formula for Mean of Grouped Data

The most common method for calculating the mean of grouped data is the direct method, which uses the following formula:

 xˉ=∑f∑f⋅xm​​

 Where:

• xˉ = Mean of the grouped data.
• ∑f⋅xm​ = The sum of the products of each class frequency (f) and its midpoint (xm​).
• ∑f = The sum of all frequencies, which is the total number of data points.

 Example: 

Let's find the estimated mean score for a class of students based on the following grouped data:

 Step 1: Find the Midpoint (xm​) for each class.

The midpoint is the average of the lower and upper limits of the class interval.

• For 10-20: (10+20)÷2=15
• For 20-30: (20+30)÷2=25
• For 30-40: (30+40)÷2=35
• For 40-50: (40+50)÷2=45

Step 2: Calculate the product of frequency and midpoint (f⋅xm​) for each class.

• 3×15=45
• 5×25=125
• 8×35=280
• 4×45=180

Step 3: Find the sum of all frequencies (∑f) and the sum of the products (∑f⋅xm​).

• ∑f=3+5+8+4=20
• ∑f⋅xm​=45+125+280+180=630

Step 4: Apply the formula.

xˉ=∑f∑f⋅xm​​=20630​=31.5

The estimated mean score for the class is 31.5.

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Median

The median is the middle value in a dataset when the numbers are arranged in ascending or descending order. It is not influenced by outliers (extremely high or low values) as the mean is. The method for calculating the median depends on whether the dataset has an odd or even number of values.

The first step for any median calculation is to sort the data from least to greatest.

1. For an odd number of data points (n):

• The median is the single middle value. 
• The formula to find its position is:
  Median=(2n+1​)th value

 2. For an even number of data points (n):

• There isn't one middle value, so the median is the average of the two middle values. 
• The formula to find their positions is:
  Median=2(2n​)th value+(2n​+1)th value​

Example 1: Odd Number of Values

Let's find the median of the following dataset: {13,22,5,41,30}

1. Arrange the data in ascending order:
   {5,13,22,30,41}
2. Count the number of values (n):
   There are 5 values, so n=5.
3. Find the middle value:
   Using the formula for an odd number of values: (25+1​)th value=3rd value.
   The 3rd value in the sorted list is 22.
   The median is 22.

Example 2: Even Number of Values

Let's find the median of the following dataset: {7,10,2,15,8,11}

1. Arrange the data in ascending order:
   {2,7,8,10,11,15}
2. Count the number of values (n):
   There are 6 values, so n=6.
3. Find the average of the two middle values:
   Using the formula for an even number of values: The middle values are the 26th and (26​+1)th values, which are the 3rd and 4th values.
   The 3rd value is 8, and the 4th value is 10.
   Now, calculate their average: 28+10​=218​=9.
   The median is 9.

Read More: Median - Formula, Meaning, Example

Mode

The mode is the value that appears most frequently in a dataset. It is the only measure of central tendency that can be used with categorical data. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if no value repeats.

Formula and Calculation

There is no mathematical formula for calculating the mode of ungrouped data because it's determined by observation. You simply count the occurrences of each value.

• Step 1: List all the values in the dataset.
• Step 2: Count the frequency (number of times) each value appears.
• Step 3: The value(s) with the highest frequency are the mode.

Example 1:

Let's find the mode of the following dataset, representing the number of goals scored by a soccer team in their last 10 matches:

{2,0,1,3,2,4,2,1,3,2}

1. Count the frequency of each value:

• 0 appears 1 time
• 1 appears 2 times
• 2 appears 4 times
• 3 appears 2 times
• 4 appears 1 time

2. Identify the highest frequency:
The highest frequency is 4, which corresponds to the value 2.

The mode of this dataset is 2.

Examples 2: 

• Bimodal Dataset: {5,8,8,9,11,11,15}
  Both 8 and 11 appear twice, which is the highest frequency. This dataset is bimodal, with modes of 8 and 11.

• No Mode: {1,2,3,4,5,6}
  Each value appears only once. Since no value is repeated, this dataset has no mode.

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Relation Between Mean Median and Mode

The relation between mean, median, and mode in statistics is given by the empirical formula:

Mode = 3×Median−2×MeanMode = 3×Median−2×Mean

 This formula is particularly useful for moderately skewed distributions and helps estimate one measure if the other two are known. It indicates that the difference between the mean and mode is approximately three times the difference between the mean and median:

 Mean−Mode=3(Mean−Median)Mean−Mode=3(Mean−Median)

In symmetrical distributions, mean = median = mode, while in positively skewed distributions, 

 mean > median > mode

 In negatively skewed distributions:

 mean < median < mode

 The three measures of central values, i.e., mean, median, and mode, are closely connected by the following relations (called an empirical relationship).

 Mean - Mode = 3 (Mean - Median) (or) 2Mean + Mode = 3Median

Examples:

1. If the mean of a data set is 12 and the median is 10, the mode can be found using the formula:

Mode=3×10−2×12=30−24=6Mode=3×10−2×12=30−24=6

     2. For a positively skewed distribution where mean = 30 and mode = 20, the median lies between:

20<Median<3020<Median<30

    3. In a symmetrical distribution where mean = 25, the median and mode are also:
Median=25

            Mode=25

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Difference Between Mean and Average

This can be a point of confusion for many. The mean is a specific, statistical term for the arithmetic average. In most everyday conversations, when people say "average," they are referring to the mean. However, in a more formal context, the term "average" can also encompass other measures of central tendency, such as the median or mode. 

Here is the difference between Mean and Average:

Difference Between Mean and Median

The core difference between the mean and median lies in their sensitivity to outliers. The following table highlights the key difference between Mean and Median. 

Mean Medium Mode Formula for Grouped & Ungrouped Data 

In statistics, the mean, median, and mode are measures of central tendency calculated differently for grouped and ungrouped data. Ungrouped data consists of individual raw values, while grouped data is organized into class intervals or groups with frequencies. 

The formulae for finding mean, median, and mode differ based on this classification to effectively summarize the data distribution.

Mean

Case 1: Ungrouped Data

 The formula for the mean of ungrouped data is:

xˉ=n∑x​

where ∑x is the sum of all values, and n is the number of values.

 Case 2: Grouped Data

 The formula for the mean of grouped data is:

xˉ=∑f∑f⋅xm​​

where f is the frequency of each class and xm​ is the midpoint of each class.

Also Read: Successor and Predecessor

Median 

Case 1: Ungrouped Data

The median is found by arranging the data in order and locating the middle value.

• If the number of values (n) is odd, the median is the value at the (2n+1​)th position.
• If the number of values (n) is even, the median is the average of the values at the 2n​th and (2n​+1)th positions.

 Case 2: Grouped Data

 Median=L+(f2n​−cf​)×h

 Where:

• L = lower limit of the median class.
• n = total frequency.
• Cf = cumulative frequency of the class before the median class.
• f = frequency of the median class.
• h = class size.

Mode 

Case 1: Ungrouped Data

There is no formula. Value that appears most frequently in the dataset is the mode.

 Case 2: Grouped Data

 Mode=L+(2fm​−f1​−f2​fm​−f1​​)×h

Where:

• L = lower limit of the modal class.
• Fm​ = frequency of the modal class.
• F1​ = frequency of the class preceding the modal class.
• F2​ = frequency of the class succeeding the modal class.
• h = class size.

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