Mean: Formula, Types & Application

The mean is a fundamental measure of central tendency that represents the average of a set of numerical values. It is calculated by summing all values and dividing by the total count. The mean provides a representative value that reflects the distribution's centre. 

• It is widely used in statistics and various mathematical applications. 
• The mean is a valuable tool for summarising data and gaining insights into the typical or central value within a dataset. 
• It is essential for understanding the characteristics of numerical information.

Key Terms: Mean, Central Tendency, Arithmetic Mean, Geometric Mean, Harmonic Mean.

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Mean

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In mathematics, the mean is often referred to as the arithmetic mean or average. It is a measure of central tendency calculated by summing up a set of numerical values (observations) and then dividing the sum by the total count of values.

• Mean = (Sum of all the observations/Total number of observations)
• The symbol of mean is ‘x̄’.

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Formula of Mean

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The formula for calculating a mean is as follows.

• X̄ = (x₁ + x₂ + x₃ +….+x_n)/n OR x̄=∑ x/n.

Consider the following example to understand the mean (average).

Example

Observations: 2, 4, 6, 8, 10

Mean (x̄) = 2 + 4 + 6+ 8 + 10/5

X̄ = 6

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How to Find Mean?

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To determine the mean of a given dataset, it's crucial to distinguish between grouped and ungrouped data. The formulas for calculating the mean differ based on the data's organisation. For ungrouped data, the mean is found by summing the values and dividing by the total count. On the other hand, with grouped data, the mean involves multiplying each midpoint by its corresponding frequency, summing these products, and dividing by the total frequency.

Mean for Ungrouped Data

To find the mean for ungrouped data, follow these steps:

1. Add up the values:

• Sum all the individual values in the dataset.

2. Count the number of values:

• Determine the total count (n) of values in the dataset.

3. Calculate the mean:

• Use the formula for the mean i.e. x̄=∑ x/n.
• Where, x̄ is mean
• ∑ x is sum of values
• n is the total number of values.

Mean of Grouped Data

To find the mean for grouped data, follow these steps:

1. Identify Midpoints:

• If the data is presented in intervals (classes), identify the midpoint of each interval. The midpoint is the average of the upper and lower class limits.

2. Calculate the Product of Midpoints and Frequencies:

• Multiply each midpoint by its corresponding frequency.

3. Sum the Products:

• Add up all the products obtained in the previous step.

4. Sum the Frequencies:

• Add up all the frequencies in the dataset.

5. Calculate the Mean:

Mean = Total Frequency/Sum of Products.

Mean of Grouped Data

\(\bar{X} = \frac{\sum fx}{n}\)

Where, \(\bar{X}\) = mean

f = frequency of each class

x = mid-interval value of class

n = total frequency

∑ fx = sum of the product of mid – interval values and their corresponding frequencies

There are three methods to calculate the mean of grouped data.

• Direct Method
• Assumed Mean Method
• Step Deviation Method

Direct Method

\(\bar{X} = \frac{\sum fx}{\sum f}\)

Assumed Mean Method​

\(\bar{X} = A + \frac{\sum fx}{\sum f}\)

d = x – A

Step Deviation Method​

\(\bar{X} = A + [\frac{\sum fx}{\sum f} \times c]\), where d = \(\frac{x- A}{c}\)

Example

consider the following grouped data:

First, Identify Midpoints:

• Midpoint of 10-20 is (10+20)/2=15
• Midpoint of 20-30 is (20+30)/2=25
• Midpoint of 30-40 is (30+40)/2=35
• Midpoint of 40-50 is (40+50)/2=45

Calculate the Product of Midpoints and Frequencies:

1. 15×5=75
2. 25×8=200
3. 35×12=420
4. 45×7=315

Sum the Products:

• 75+200+420+315 = 1010

Sum the Frequencies:

• 5+8+12+7 = 32

Calculate the Mean:

Mean = 1010/32 = 31.56

Mean of Negative Numbers

When calculating the mean of negative numbers, the process is the same as finding the mean of any set of numbers. You add up all the values and then divide by the total count of values.

For example, let's find the mean of the set of negative numbers:

−3,−7,−10,−2.

(−3)+(−7)+(−10)+(−2)=−22

There are four values in the set.

Therefore, Mean = -22/4 = -5.5

However, the mean is not affected by the sign of the numbers; it is simply the sum divided by the count.

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Types of Mean

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There are mainly three different types of mean values - 

Arithmetic Mean

The arithmetic mean is determined by adding all provided values and subsequently dividing the sum by the total count of given numbers. 

• To compute it, sum all the provided numerical values and then divide this total by the quantity of numbers given in the dataset. 
• This process yields the arithmetic mean, a central measure of tendency in a set of numerical data.

Geometric Mean

The geometric mean for a pair of numbers, x and y, is obtained by multiplying them together (xy). Extending this concept to three numbers, x, y, and z, their geometric mean is calculated as the product of the three numbers (3xyz).

• For example, Find the geometric mean of 4 and 3.
• Geometric Mean = √4 * 3 = 2 √3 = 3.46

Harmonic Mean

The harmonic mean serves as a means to average ratios. In the case of two numbers, x and y, the harmonic mean is expressed as 2xy/(x + y). For three numbers, namely x, y, and z, the harmonic mean is calculated as 3xyz/(xy + xz + yz).

Quadratic Mean

The root mean square or quadratic mean is used in many engineering and statistical applications, especially when there are data points that can be negative.

Contraharmonic mean

The contraharmonic mean is also known as the reciprocal quadratic mean or the second power mean. It is a statistical measure used to characterise the relationship between pairs of numbers. For two numbers, x and y, the contraharmonic mean is defined as (x² + y²) / (x + y).

• It essentially involves summing the squares of the numbers and dividing by their sum. 
• This mean is sensitive to extreme values and is not as commonly used as the arithmetic or geometric means in statistical analysis.

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Applications of Mean

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In financial analysis, the mean is employed to calculate the average annual returns of an investment portfolio. By summing the annual returns over several years and dividing by the number of years, investors gain insights into the portfolio's typical performance. 

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Things to Remember

• Mean is a central tendency measure.
• There are 3 main types of mean in maths.
• Formula of mean is x̄=∑ x/n.
• The type of data set needs to be recognised before computing its mean.

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