Central Tendency

What is Central Tendency?

To understand what is central tendency, let us consider a set of data representing the scores of the top five students in the class (45, 40, 35, 47, 32). The central tendency of this score will be the central value of the range of scores. Central tendency is the central value of a set of grouped or ungrouped data. Central tendency does not depict any individual value of the data set. However, it gives the overall summary of the set of data. Central tendency can be measured by calculating the mean or median or mode of the particular data set. This value depicts the nature and characteristics of an entire set of data. 


Measures of Central Tendency:

The central tendency of any data set can be measured by anyone of the following ways.

  • Mean

  • Median

  • Mode

In general, the mean is the average of the data in the dataset. Median is the midpoint of the set of grouped or ungrouped data. Mode is the value that is repeated a maximum number of times in the set of data. 


Central Tendency in Statistics:

According to experts, a good measure of central tendency in statistics should have the following features.

  • Measures of Central tendency definition should be easy and rigid. The calculations should be simpler. 

  • The calculation should give one particular value. There should be no ambiguity as to the value of the central tendency obtained. 

  • The measures of central tendency should represent the entire set of data. To be more specific, the value should lie between the upper limit and the lower limit of the set of data. 

  • The measured value of central tendency should be flexible enough to fit into the further statistical calculations. 

  • The measures of central tendency should not alter with the alteration is the extreme values of the data set. 


Measures of Central Tendency Definition:

There are three measures of central tendency namely mean, median, and mode.


Mean:

Mean is the measure of the average of a set of values which can be calculated by dividing the sum of all the observations by the number of observations. There are four different ways to measure the mean of a data set. They are arithmetic mean, geometric mean, harmonic mean, and weighted arithmetic mean. Usually, arithmetic mean is calculated because it is easy to calculate. 


\[Mean = \frac{{x1 + x2 + x3 + ............xn}}{n}\]


Median:

Median is the middle value of the given data set. Usually, the data is arranged in ascending order before determining the median. If there are an odd number of observations, the median is the value represented by the middle of the data set. However, when there are an even number of observations, the median will be calculated as the mean of the two middle values. 


\[Median = \left( {\frac{n}{2}} \right)th{\text{ observation for odd number of observations}}\]


Mode:

Mode is the observation in the data set which occurs most frequently. In other words, the value that has repeated more times in a data set is called its mode. Mode can sometimes have multiple values. In certain cases where all the values are unique, the mode does not exist at all. 


Relationship Between Mean,Median, and Mode of a Data Set:

The relation between the values of mean, median, and mode of a given data set is established by empirical formula. Empirical formula states that the mode of a distribution is equal to the subtraction of 2 times its mean from 3 times its median.


\[Mode = 3 \times Median - 2 \times Mean\]


Central Tendency Measures Example:

1. Consider the scores of 10 students in class X in a mathematics examination as 45, 30, 75, 90. 64, 72, 85, 62, 45, 97. Calculate the mean and median of the data set and also find the mode using an empirical formula. 

Solution:

The given data set should be arranged in ascending order.

30, 45, 46, 62, 64, 72, 75, 85, 90, 97


The mean of the given data is calculated by adding all the numbers and dividing it by 10.


\[Mean = \frac{{30 + 45 + 46 + 62 + 64 + 72 + 75 + 85 + 90 + 97}}{{10}} = \frac{{666}}{{10}} = 66.6\]

Median is calculated as the mean of 2 middle values because the number of values is 10 which is an even number. 

\[Median = \frac{{64 + 72}}{2} = \frac{{136}}{{10}} = 68\]

Empirical formula is given as:

\[Mode = 3 \times Median - 2 \times Mean\]

Mode = 3 (68) - 2 (66.6)

Mode = 204 - 133

Mode = 70.8


Therefore, the mean, median and mode of the given data set are 66.6, 68 and 70.8


Fun Facts about Central Tendency Definition:

  • Mean is usually used to measure the central tendency of a symmetric data set. It is not used to measure the central value of skewed continuous data set because, in this type of data, the mean is pulled away from the center due to extreme values. 

  • The choice of central tendency which is to be measured for a given set of data is based on the type of data.

FAQ (Frequently Asked Questions)

1. Why are Measures of Central Tendency Important?

A. 

  • Central tendency helps to compress a data set to a single value which represents all the values in the data. 

  • It, therefore, helps to work with a larger set of grouped or ungrouped data.

  • Central tendency also can be used in comparing one set of data with the other. It also helps in easier and effective pictorial representation of the given data set. 

  • Central tendency in statistics can be used to compare a single value of data in a set of data with the other values in the data set. 

  • Had it been we did not have the measures of central tendency, it would have been very difficult to perform statistical computations accurately.

2. What is the Central Tendency and the Empirical Formula Used for?

A. 

  • Central tendency is used in statistical computations to simplify a large set of data. This is because the value of central tendency is a representative of all the values in the data set. 

  • Central tendency can also be suitably used to analyze the type of data and compare one set of data to another data set. It can also be used to compare the values within a particular data set. 

  • Empirical relationships can be used in cases where the entire values in a dataset are not available. However, when any two of the central tendency values are known, the value of the third one can be determined. 

  • Empirical formula can also be used to measure the skewness of any given data set because skewness is the measure of the difference between the mean and mode of the data set.