What is Measurement of Central Tendency?

Measurement of Central Tendency is used to describe a larger set of data by identifying a central value in that dataset and is sometimes also called summary statistics. There are 3 common measures of central tendency - Mean, Median and Mode which are considered to be accurate but in certain situations, some of them become more suitable to use than others.

1.1 Arithmetic Mean 

The arithmetic mean is the average value of an entire data set and is among the most common measures of central tendency. Arithmetic mean can further be divided into the following - 

1.2 Simple arithmetic mean 

The arithmetic mean is derived after adding the values in a data set together and dividing it by the total number of observations. In case of raw data, mean is calculated using below formula - 

1.3. Weighted arithmetic mean 

When some components in an equation contribute more weight to the overall equation, it is known as weighted arithmetic mean. To find the weighted mean in a given problem, we'll first multiply the numbers within the data set with weights and then add the results. 

2. Median
Median is defined as a central value that divides a series of data into two halves, even as the data has been arranged in descending or ascending order of succession.

3. Mode 

Mode is the measure that calculates the number of times a single value occurs in a distribution.

Let us explore the three measures of central tendency, namely mean, median and mode. 

1. Mean

It represents the average value of a dataset that can be calculated as the sum of all values in dataset divided by number of values. It is considered as arithemetic mean in general. Other measures of mean for finding central tendency include geometric mean, harmonic mean and weighted mean. IISER exam often ask questions based on this:

In case the data has variability, then mean value will differ. The following formula is used for calculating the mean value:

$$\mathrm{Mean}=\frac{x_1+x_2+\cdots +x_n}{n}$$

In a symmetric distribution of data, mean value is located accurately at centre. However, in the case of skewed continuous data distribution, extreme values in extended tail pull away the mean value from the centre. Mean should be mainly used for symmetric distributions.

2. Median

It is the middle value of dataset where dataset is arranged in either ascending or descending order. In the case where dataset contains even number of values, median value of dataset will be determined by taking the mean of middle two values.

First, let us arrange them in ascending order, i.e. 5, 11, 15, 16, 18, 21, 24, 25, 34. 

The number of observations is 9, i.e, an. odd number of observations
The middle value in this case will be position (9+1)/2 = 5
The 5th value in the sorted order is 18

3. Mode

This represents the frequently occurring value in dataset. In some cases, the dataset may contain multiple modes and in some cases, no mode might be there. Let us consider a dataset 6, 7, 8, 6, 9, 5

As we know, the mode represents the most common value of a dataset. In this case, the most frequently repeated value is 6. 

Exams like JEE Main and IIT JAM ask questions that require using different measures of central tendency. Let us look at some examples as mentioned below.

1.  A class 12th B, has 25 students who took an English test. Among them, 10 students had an average score of 80, while the other students had an average score of 60. What will be the average score of the whole class?

Solution. In the first step, we will multiply average score with number of students who had that score and then add them. 

Thus:

80x10+ 60x15 = 800+900 = 1700 
Number of observations = 25

1700/25 = 68

Ans: The average score of the class was 68. 

2: Find the median of the following series- 

5, 7, 6, 1, 8, 10, 12, 4, and 3.

Solution. 
To find the median, we need to follow the steps mentioned below: 

Step 1: Arranging the numbers in ascending order 

=  1, 3, 4, 5, 6, 7, 8, 10, 12.                     
Step 2: Find the central value; hence, it becomes the median

Since 6 is the central value in the series, the median = 6. 

3. Calculate the value of the modal worker family’s monthly income from the following data: 

Solution.
Firstly, considering that the table is a cumulative frequency distribution, we will have to convert it into an exclusive series & an ordinary frequency table - 

Hence, we derive the following from the above formula - 

L = 25, D1 = (30 – 18) = 12, D2 = (30 – 20) = 10, h = 5 

= 25+12/12+10x5 

= Rs. 27.273 

The following points highlight the advantages of different measures of central tendency. NEET entrance exam and CUET exam often ask questions that leverage the use case specific to one of the measures of central tendency.:

1. Mean: 

• It takes into account all values in the dataset, which makes it a comprehensive measure of central tendency.
• Mean is used in many statistical tests and methods such as standard deviation, variance calculation and hypothesis testing.
• Since the mean considers all data points, it is sensitive to changes in data that are useful to detect shifts in the dataset. 

2. Median

• Median is less affected by outliers and skewed data as compared to the mean.
• It makes this method one of the better measures of central tendency for datasets that have extreme values.
• A median divides a dataset in two equal halves which indicates the central point of data distribution.

3. Mode

• It is a straightforward measure of central tendency.
• Mode is especially suitable for small datasets since it requires only identifying the most frequently occurring value.
• This can be used with nominal data where values represent distinct categories without numerical order.

Just like a coin has two sides, each measure of central tendency has its own set of advantages and disadvantages. Knowing these is important from the GATE exam point of view.  Let us take a look at each one of them.:

1. Mean

• It is highly sensitive to both extreme values and outliers. A very high or low single value may skew the mean.
• The mean may not accurately reflect central tendency in datasets that have a skewed distribution. This happens because the mean can be pulled in the direction of skew.
• It is not possible to calculate the mean for categorical or non-numeric data. This limits the use of mean only to quantitative datasets.

2. Median

• Calculation of the median is time-consuming and intensive for large datasets since it requires sorting the data.
• The median is less sensitive to changes in data, which means that the median may not notice even significant changes.
• It is not possible to use the median with categorical (nominal) data since it needs an ordered dataset.

3. Mode

• A dataset may have more than one node or even no mode when all values are unique. This makes the mode ambiguous as a measure of central tendency.
• Mode may be less informative for continuous data since the frequency of each value is often either unique or low.
• It only considers the most frequent values while it ignores the remaining dataset, which is misleading in some contexts.

Let us discuss some of the important central tendency measures that are important for students of CBSE board: