Measures of Central Tendency

Measures of central tendency are statistical tools that are used to identify the center or typical value of a given data set. The three main measures of central tendency are: mean, median and mode.

In simple terms, mean is the average value of a dataset. It can be found by adding the values in a dataset and dividing the resultant by the number of values.
Mean has different types like:

• Arithmetic mean
   
• Geometric Mean
   
• Harmonic Mean
   

Median is the middle value of an ordered data. It divides the data into two equal halves. The types of median are given below:

• Simple Median
  
   
• Median for Grouped Data
  
   
• Weighted Median
  
   
• Interquartile Median
   

Mode is the number that appears most frequently in a dataset. In other words, the observation of the data that occurs the maximum number of times in a dataset. The types of mode are:

• Unimodal
   
• Bimodal
   
• Multimodal
   
• No Mode (Amodal)
   
• Mode in Grouped Data
   

The mean, median, and mode are closely connected and follow a pattern called empirical relationship:
2Mean + Mode = 3Median.

Say, for example, if we are asked to calculate the mean, median and mode of a grouped data which is continuous, we can calculate the mean and median using the formula given to them respectively. Then using the above empirical relationship, we can find the mode.

Let us use an example:

The median and mode of a given data set are 56 and 54 respectively. Calculate the value of the mean from the data set given.

Answer:  Using the empirical relation formula:

2Mean + Mode = 3Median

We are given the values of mode and median

Median = 56

Mode = 54

Substitute the values in the formula:

2Mean + 54 = 3(56)

2Mean + 54 = 168

2Mean = 168 – 54

2Mean = 114

Mean = 114/2 = 57.
 

While using measures of central tendency, the use depends on the given type of distribution. Some of the types of distribution are given below:

• Normal Distribution
• Skewed Distribution
  
   

Let us see how each of the following types of distribution affects the measures of central tendency.

Measures of Central Tendency for Normal Distribution:

To understand how measures of central tendency affect the normal distribution, let us take an example:
Given below is a normal distribution table:

For the above normal distribution table, we can see the histogram given below:

The above histogram represents a symmetric distribution of the data. In the histogram given above, we can see that the center of the distribution graph has all the measures of central tendency.

Therefore, it can be said that the mean and median are equal in a perfectly symmetric distribution. As the above distribution is unimodal (has one mode) the mode is also same as the mean and median. If the symmetric has more than one mode (bimodal) then the mode would be different from the mean and median.

Measures of Central Tendency for Skewed Distribution:

Skewed distribution has two types:

 

• Left Skewed Distribution 
• Right Skewed Distribution
  
   

Let us see how measures of central tendency change with each type of skewed distribution:

Measures of Central Tendency for Left Skewed Distribution:

For this type of distribution, let us consider the following distribution table:
 

In the histogram given above, it is clear that the data set indicates a right skewed distribution or positive skewed distribution. In the measure of central tendency, after calculation, we find mean = 10, median = 9 and mode = 9. From this we can conclude that in a right skewed distribution, the mode is lesser than the mean. And the median lies in between the values of mode and mean.

Measures of Central Tendency for Left Skewed Distribution:

For this type of distribution, let us consider the following distribution table:

For the above distribution, the histogram is given below:

From the above histogram, we can see that the histogram is a negative or left skewed distribution. In the measure of central tendency, after calculation, we find mean = 15.75, median = 18 and mode = 21. Hence, we can conclude that the distribution is left skewed. Here the mode is greater than the median, and the median is greater than the mean.
 

The measures of central tendency have numerous applications across various fields. Let us explore how the measures of central tendency is used in different areas:

Education:

We use measures of central tendency in education; where mean is used to calculate the student's average scores to determine the overall class performance, median helps in identifying the middle performing students, mode determines the most frequently occurring grade in a class.

Business and Finance:

We use measures of central tendency in business and finance; where mean is used to calculate the average salary of employees in a company, median gives a good representation of the typical salary, mode helps in determining the most common salary range.

Healthcare:

We use the measures of central tendency in healthcare; where mean is used to calculate the blood pressure, cholesterol levels for a population, median is useful when data is skewed due to extreme cases, mode helps us in identifying the most common health conditions among patients.