Central Tendency

What are 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 are called the standards 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

For Mean x,

mean formula
Central-Tendency-Formula

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For eg:
Source: Central Tendency (wallstreetmojo.com)

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 chartHistogram ChartThe histogram excel chart is a built-in data analysis chart that displays data in histograms. The data compared in a histogram chart is classified into ranges.read more 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, then 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 frequently recurring and hence will be preferred over mean only when there are such samples where values repeat them the most.

Examples

You can download this Central Tendency Excel Template here – Central Tendency Excel Template

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.

central tendency formula example 1

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

central tendency formula example 1.1jpg
  • Mean = 915/15

Mean will be –

central tendency formula example 1.2jpg

Mean = 61

The calculation of the Median will be as follows-

central tendency formula example 1.3jpg

Median =62

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

Calculation of Mode will be as follows-

central tendency formula example 1.4jpg

Mode = 56

For more, we can note from the above table that the 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.

central tendency formula example 3jpg

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

central tendency formula example 2.1jpg

= 2367/15

Mean will be – 

central tendency formula example 2.2
  • Mean = 157.80

The number of observations is 15. Hence the mean height would be 2367/15 = 157.80, respectively.

Therefore, the median of height can be calculated as,

central tendency formula example 2.3
  • Median = 155

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

Therefore, the mode of height can be calculated as,

central tendency formula example 24jpg
  • Mode = 171

Calculation of the mean of weight will be as follows,

central tendency formula example 3.1jpg

= 1047.07/15

The mean of weight will be –

example 3.2jpg
  • Mean = 69.80

Therefore, the median of weight can be calculated as,

example 3.3jpg
  • Median =69.80

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

Therefore, the mode of weight can be calculated as,

central tendency formula example 3.4jpg
  • Mode = 77.00

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

Analysis: It can be observed that the 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 does now show 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 one reader.

Solution:

Below is given data for calculation.

example 4jpg

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

example 4.1jpg

Mean =7326/10

Mean will be –  

central tendency formula example 4.2jpg
  • Mean = 732.60

Therefore, the median can be calculated as follows,

central tendency formula example 4.3jpg

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

  • Median = 845.00

Therefore, the model can be calculated as follows,

example 4.4jpg
  • Mode = 1101.00

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

central tendency formula graph

Relevance and Uses

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 statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more, finance, science, education, etc. everywhere these measures are used. But commonly, you would hear 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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