Mean Formula: Understanding the Basics of Central Tendency

In this article, learn how to calculate the mean using the mean formula. The mean is a measure of central tendency that is widely used in statistics to describe the average value of a set of numbers. Discover the steps involved in computing the mean, as well as its strengths and limitations as a statistical measure.

Let us consider you are a teacher, and you have to find the average height or average score of all the students in a class, or you are a sales manager, and you have to find the average sales by all the employees for the last year. Then how will you calculate it?

Don’t worry; in statistics, there is a Mean statistics formula that will help you to get the answer to your above queries. It can also help you to make decisions or predictions.

This article will explore the concept of mean statistics, formulas to find the mean, its different types, calculation methods, some real-life applications, and its limitations.

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So, without further delay, let’s get started.

Table of Content

• What is a Mean?
• Types of Mean in Statistics
  • Arithmetic Mean
  • Geometric Mean
  • Harmonic Mean
• How to calculate the Mean?
  • Ungrouped Data
  • Grouped Data
• Real-Life Application of Mean
• Limitations of Mean

Let’s start with the formal definition of Mean.

What is a Mean?

The mean in statistics refers to the measure of central tendency representing the average value of a numerical data set. It is a useful tool to summarize (or describe) a dataset and to detect patterns( or trends) in the data. 

Apart from summarizing and detecting patterns, it is used in finance, economics, and healthcare for decision-making.

It is defined as the ratio of the sum of all observations to the total number of observations.

Must Read: What is Maths for Data Science? A Beginner Guide to learn Maths for Data Science in 2023

Mean Formula

Mean = Sum of all Observations / Total Number of Data Points (Observations)

Let’s take a simple example to calculate the mean of the data:

Example-1: Let the score of five students in a class is 10, 15, 20, 25, and 30. Find the mean score of the class.

Solution: 

Sum of all Observations = 10 + 15 + 20 + 25 + 30 = 100

Total Number of Observations = 5

=> Mean = Sum of all Observations / Number of Observations = 100 / 5 = 20

Hence, the Mean Score = 20

Whenever the mean is referred to in the context of statistics, it always means Arithmetic Mean. But two more types of mean exist, i.e., Geometric and Harmonic Mean.

The next section will explore different types of mean in statistics.

Types of Mean

There are 3 different types of Mean in Statistics: Arithmetic Mean, Geometric Mean, and Harmonic Mean.

We have already discussed arithmetic mean (the above formula and definition of mean), so now, we will explain only geometric and Harmonic Mean.

What is Geometric Mean?

A statistical measure which is used to find the central tendency of a set of values. Mainly it is useful for calculating the average rate of change. It can also provide a more accurate representation of central tendency when the data is skewed or has outliers.

Dissimilar to the arithmetic mean, it is calculated by multiplying all the data points (n-value) and then taking the n-th root of it.

Geometric Mean Formula

If x₁, x₂ , x₃ , …, x_n are n-data points, then

G.M = ( x₁ * x₂ * x₃ * …. * x_n )^1/n

What is Harmonic Mean?

Similar to the arithmetic mean, it is a type of average that is defined as the reciprocal of the average of the reciprocal of the data values.

Confused!!

Let’s get the mathematical formula that will help you out.

Harmonic Mean Formula

If x₁, x₂ , x₃ , …, x_n are n-data points, then

HM = n / ( (1/x₁) + (1/ x₂ ) + (1/ x₃ ) + …. + (1/ x_n ))

Now, let’s take an example to find the harmonic mean value.

Example: Find the Harmonic Mean for the data: 2, 4, and 8.

Solution: First, take the reciprocal of all three data points and add them, i.e.

1/2 + 1/4 + 1/8 =  (4 + 2 + 1)/8 = ⅞

=> HM = 3 / (⅞) = 3 * (8/7) = 24/7 = 3.42

Hence, the Harmonic Mean for the data: 2, 4, and 8 is 3.42.

Relation Between Arithmetic Mean, Geometric Mean, and Harmonic Mean

Harmonic Mean <= Geometric Mean <= Arithmetic Mean

How to Calculate the Mean

To make it simple and understandable, we divide the data values (or dataset) into Ungrouped and Grouped data.

Mean for Ungrouped Data

Ungrouped data is raw data that is not organized or sorted into categories or intervals.

Example:

• Height of student in a class: 150cm, 175cm, 130cm, 145cm, 160cm.
• Marks of students: 10, 14, 15, 13, 12, 8, 7, 10.

To calculate the mean of ungrouped data, you have to follow three simple steps:

• Add all data values
• Count the number of data values
• Divide the sum of data values by the number of data values.

It is as simple as you have done in the very first example.

Isn’t it??

Now, it’s time to take an example to understand better how to calculate the mean of ungrouped data.

Example: Calculate the mean score of 10 students: 

If the marks of 10 students in a math exam are: 85, 78, 92, 89, 93, 87, 81, 90, 95, 88.

Then, calculate the mean score.

Solution: 

Sum of Data Values = 85 + 78 + 92 + 89 + 93 + 87 + 81 + 90 + 95 + 88 = 878

Number of Data Values = 10

Mean = Sum of Data Values / Number of Data Values = 878/10 = 87.8

Hence, the mean score of the 10 students in the mathematics exam is 87.8.

Mean for Grouped Data

Grouped Data is a set of data values that have been grouped into categories or intervals.

Example: 

• Sales in Each Quarter
• The number of hours workers work per week can be grouped into intervals of 0-8 hours, 8-12 hours, and 12-16 hours.

To calculate the mean of the grouped data, you have to follow these simple steps:

• If the data values (xi) are given in intervals, find the midpoint ((upper-class limit + lower-class limit)/2).
• Multiply data values (or mid-points) by their corresponding frequency (fi), i.e., xi*fi
• Add up all products from step-2, i.e., x1*f1 + x2*f2 + … + xn*fn.
• The mean is calculated by dividing the above result by the number of observations (f1 + f2 + f3 + … + fn). 

Now, let’s take an example to understand better how to calculate the mean of grouped data.

Example: Consider the below data on the number of hours that students study per week.

Find the mean.

Solution:

=> ∑xi* fi =  x1*f1 + x2*f2 + x3*f3+ x4*f4 = 20 + 120 + 150 + 70 = 360 

and, ∑fi =  f1 + f2 + f3 + f4 =  10 + 20 + 15 + 5 = 50

=>  Mean = ∑xi* fi / ∑fi = 360 / 50 = 7.2

Hence, the mean number of hours students’ study per week is 7.2.

Real-life Application of Mean

Mean is used in different domains such as Finance, Medicine, Quality Control, and Sports to calculate average return on investment, to describe the average value of a group of measurements, to determine the average performance of a manufacturing process, and to calculate the average performance of the team or an individual player respectively.

Limitations of Mean

• Influenced by Sample Size: Mean may not be a good estimate of the population mean if the sample size is small.
• Sensitive to Outliers: Mean is extremely sensitive to outliers. Even a single outlier can significantly affect the value of the mean.
• Inappropriate by Ordinal Data: Mean is inappropriate for the data where the values have a natural order but not numerical values. 
• Limited Usefulness for Skewed Data: The mean is not a robust measure of central tendency for skewed distributions. If the data is highly skewed, the mean may not accurately reflect the typical value of the data.

Conclusion

In conclusion, the mean (arithmetic mean) formula is one of the fundamental tools in statistics, which is used to calculate the average value of the data points. While the mean is a useful measure of central tendency, it has limitations too. As it is highly vulnerable to outliers and skewed distributions. Therefore, it is essential to understand the importance of the mean formula.
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