## What is Measures of Central Tendency?

Central tendency refers to the value derived out of the random variables from the set of data that reflects the center of the distribution of the data and which generally can be described using different measures like mean, median and the mode.

It is a single value that attempts to describe a set of data by identifying the middle of the central position within the given dataset. Sometimes these measures of the central tendency are called the measures of middle or the central location. The mean (otherwise known as the average) is the most commonly used measure for central tendency, but there are other methodologies such as the median and the mode.

### Measures of Central Tendency Formula

The formula for Central Tendency is as per below

For Mean x,

Where,

- ∑x is the sum of all the observations in a given dataset
- n is the number of observations

The median will be the center score for a given dataset which when arranged in order of the magnitude.

The mode will be the most frequent score in the given set of data. A histogram chart can be used to identify the same.

### Explanation

The mean or the average is the sum of all the observations in the given set of data and that is then divided by the number of observations in the given set of data. So, if there are n observations in a given set of data and they have observations such as x1, x2, …, Xn, the taking some of those is total and dividing the same by observations is mean which tries to bring central point. Median is nothing but the middle value of the observations and is mostly reliable when the data has outliers while the mode is used when the number of observations is recurring frequently and hence will be preferred over mean only when there are such samples where values repeat them the most.

### Examples

#### Example #1

**Consider following sample : 33, 55, 66, 56, 77, 63, 87, 45, 33, 82, 67, 56, 77, 62, 56. You are required to come up with a central tendency.**

Solution:

Below is given data for calculation of the measures of central tendency

Using the above information, the calculation of mean will be as follows,

- Mean = 915/15

**Mean will be –**

**Mean = 61**

Calculation of Median will be as follows-

**Median =62**

Since the number of observations is odd, the middle value which is the 8^{th} position will be the median which is 62.

Calculation of Mode will be as follows-

**Mode = 56**

For more, we can note from the above table that a number of observations that are recurring most times is 56. (3 times in the dataset)

#### Example #2

**Ryan international school is considering selecting the best players to represent them in the ****inter-school Olympics competition to be organized soon. However, they have observed that their players are spread across the sections and standards. Hence before putting a name in any of the contests, they would like to study the central tendency of their students in terms of height and then weight.**

Height qualification is at-least 160cm and weight should not be more than 70 kgs. You are required to calculate what is the central tendency for their students in terms of height and weight.

Solution

Below is given data for the calculation of measures of central tendency.

Using the above information, the calculation of mean of height will be as follows,

= 2367/15

**Mean will be – **

**Mean = 157.80**

A number of observations are 15, hence the mean of height would be 2367/15 = 157.80 respectively.

Therefore, the median of height can be calculated as,

**Median = 155**

The median would be the 8^{th} observation as the number of observations is odd, which is 155 for weight.

Therefore, the mode of height can be calculated as,

**Mode = 171**

Calculation of mean of weight will be as follows,

= 1047.07/15

**Mean of weight will be –**

**Mean = 69.80**

Therefore, the median of weight can be calculated as,

**Median =69.80**

The median would be the 8^{th} observation as the number of observations is odd, which is 69.80 for weight.

Therefore, the mode of weight can be calculated as,

**Mode = 77.00**

Now mode will be the one which occurs more than one time. As it can be observed from the above table, it would be 171 and 77 for height and weight respectively.

Analysis: It can be observed that average height is less than 160 cm, however, weight is less than 70 kgs which could mean Ryan’s school students might not qualify for the race.

The mode is not showing the proper central tendency and is biased upwards, the median is still showing good support.

#### Example #3

**The universal library has got the following count of the most to read books from different clients, and they are interested to know the central tendency of books read in their library. Now you need to do the calculation of central tendency and use mode to decide the no 1 reader.**

Solution:

Below is given data for calculation of measures of central tendency

Using the above information, the calculation of mean will be as follows,

Mean =7326/10

**Mean will be – **

**Mean = 732.60**

Therefore, the median can be calculated as follows,

Since the number of observations is even, there would be 2 middle values which is the 5^{th} and 6^{th} position will be the median which is (800 + 890)/2 = 845.

**Median = 845.00**

Therefore, the model can be calculated as follows,

**Mode = 1101.00**

We can use below the histogram, to find out mode which is 1100 and readers are Sam and Matthew.

### Relevance and Uses

Below is the high-level summary to know which measure of central tendency should be used in a given sample.

All the measures of central tendency are widely used and are very useful to extract the meaning of the data which gets organized or if someone is presenting that data in front of a large audience and wishes to summarize the data. Fields like in statistics, finance, science, education, etc. everywhere these measures are used. But commonly you would be hearing more of the use of mean or average on a daily basis.

### Recommended Articles

This has been a guide to what is Central Tendency and its definition. Here we discuss the top 3 measures of central tendency – mean, mode and median and its formula along with excel examples & templates. You can learn more from the following articles –

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