What are Measures of Central Tendency?

It is a single value that describes a data set 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,

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Where,

  • ∑x is the sum of all the observations in a given datasetn is the number of observations

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

The mode will be the most frequent score in the data set. One can use 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 to identify the same.

Explanation

The mean or the average is the sum of all the observations in the given data set divided by the number of observations in the given data set. 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 a central point. The median is the middle value of the observations and is mostly reliable when the data has outliers. At the same time, the mode is used when the number of observations is frequently recurring and hence will be preferred over the mean only when there are such samples where values repeat them the most.

Examples

Example #1

Consider the 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 the given data for calculation.

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

  • Mean = 915/15

Mean will be –

Mean = 61

The calculation of the median will be as follows:

Median =62

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

The calculation of mode will be as follows:

Mode = 56

From the above table, we can note that the number of recurring observations most often 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 like to study the central tendency of their students in height and weight.

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

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

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

= 2367/15

Mean will be – 

  • Mean = 157.80

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

Therefore, one can calculate the median of height as:

  • Median = 155

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

Therefore, one can calculate the mode of height as,

  • Mode = 171

Therefore, one can calculate the mode of height as:

= 1047.07/15

The mean of weight will be –

  • Mean = 69.80

Therefore, one can calculate the median of weight as:

  • Median =69.80

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

Therefore, one can calculate the mode of weight as:

  • Mode = 77.00

Now, the mode will be the one that 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. However, the median is still showing good support.

Example #3

The universal library has the following count of the most to read books from different clients. They are interested in the central tendency of books read in their library. Now, you need to calculate central tendency and use mode to decide the no one reader.

Mean =7326/10

Mean will be –  

  • Mean = 732.60

Therefore, the median one can calculate as follows,

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

  • Median = 845.00

Therefore, one can calculate the model as follows:

  • Mode = 1101.00

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

Relevance and Uses

All the measures of central tendency are used widely. They 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 it. These measures are used everywhere in fields like 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. But commonly, you would hear more of the use of mean or average daily.

This article has been a guide to 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 and templates. You can learn more from the following articles: –

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