Mean

Introduction

Mean is one of the important and most commonly used measures of central tendency. There are several types of means in mathematics. In statistics, the mean for a given set of observations is equal to the sum of all the values of a collection of data divided by the total number of values in the data. In other words, we can simply add all the values in a data set and divide it by the total number of values to calculate mean. However, the general method and formulas vary depending upon the type of data given, grouped data, or ungrouped data.

Grouped data is the data set formed by aggregating individual observations of a variable into different groups, while ungrouped data is a random set of observations. Let us understand the different mean formulas and methods to find the mean of the given set of observations using examples.

What is Mean in Statistics?

“Mean” is commonly known as “average”. For example, if we say, the average height of the class of students of grade 9 is 150 cm, then it means that the mean of their heights is 150 cm. Mean is a statistical concept that carries a major significance in finance and is used in various financial fields and business valuation. Mean, median, and mode are the three statistical measures of the central tendency of data.
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Mean Formula

The mean formula in statistics for a set is defined as the sum of the observations divided by the total number of observations. But the formula is different if the data is grouped (i.e., if the data is seggrated as categories). We are going to study two types of mean formulas in this page:

• Mean formula of grouped data
• Mean formula of ungrouped data

But the common formula of mean (of ungrouped data) is:

Mean = (Sum of all data points) ÷ (Number of data points)

Mean of Ungrouped Data

Ungrouped data is the raw data gathered from an experiment or study. In other words, an ungrouped set of data is basically a list of numbers. To find the mean of ungrouped data, we simply calculate the sum of all collected observations and divide by the total number of the observations. Follow the below-given steps to find the mean of a given set of data,

• Note down the given set of data whose mean is to be calculated.
• Apply any of the following formulas based on the type of information available.

x̄ = (x₁+ x₂+ x₃+ … + x_n )/n where x₁, x₂, . . ., x_n are n observations.

Mean of Grouped Data

Grouped data is a set of given data that has been bundled together in categories. For a mean of grouped data, a frequency distribution table is created, which shows the frequencies of the given data set. We can calculate the mean of the given data using the following methods:

• Direct Method
• Assumed Mean Method
• Step Deviation Method

Calculating Mean Using Direct Method

The direct method is the simplest and the most popular method to find the mean of the grouped data. The steps that can be followed to find the mean for grouped data using the direct method are given below,

Create a table containing four columns as given below,

• Column 1- Class interval.
• Column 2- Class marks (corresponding), denoted by xi. The class mark is the middle value of the interval. i.e., xi = (upper limit) + (lower limit) / 2.
• Column 3- Frequencies (fi) (corresponding)
• Column 4- xifi (corresponding product of column 2 and column 3)

Calculate mean by the formula ∑xifi/∑fi

Calculating Mean Using Assumed Mean Method

We apply the assumed mean method to find the mean of a set of grouped data when the direct method becomes tedious. We can follow the below-given steps to find mean using the assumed mean method,

Create a table containing five columns as stated below,

• Column 1- Class interval.
• Column 2- Classmarks (corresponding), denoted by xi. Take the central value from the class marks as the assumed mean and denote it as A.
• Column 3- Calculate the corresponding deviations using, i.e. di = xi – A
• Column 4- Frequencies (fi) (corresponding)

Finally, calculate the mean by the formula, A + ∑fidi/∑fi

Calculating Mean Using Step Deviation Method

Step deviation is also called the shift of origin and scale method. We apply the step deviation method to reduce the tedious calculations while calculating the mean for grouped data. Steps to be followed while applying the step deviation method are given below,

Create a table containing five columns as given below,

• Column 1- Class interval.
• Column 2- Classmarks (corresponding), denoted by xi. Take the central value from the class marks as the assumed mean (A).
• Column 3- Calculate the corresponding deviations, i.e. di = xi – A
• Column 4- Calculate the values of ui using the formula, ui = di/h, where h is the class width.
• Column 5- Frequencies (fi) (corresponding)

Finally, calculate the mean by using the formula A + h (∑fiui / ∑fi).

How to Find Mean?

Mean is the most common central tendency we know about and use. It is also commonly used as average. We can calculate the mean for a given set of data using different methods based on the type of given data. Let us see how to find mean for a few different cases.

Case 1: Let there be “n” number of items in a list. {x1, x2, x3, … , xn }

Mean can be calculated using the formula given below,

x̄ = (x₁+ x₂+ x₃+ … + x_n )/n

or

x̄ = Σxi/n

Case 2: Let there be n number of items in a list, given as, {x1, x2, x3, … , xn } and the frequency of each item be {f1, f2, f3, … , fn } respectively.

Mean can be calculated using the formula given below,

x̄ = (f₁x₁ + f₂x₂ + f3x3 + . . . + f_nx_n)/(f₁ + f₂ + f₃ + . . . + f_n)

or

x̄ = Σfixi/Σfi

Case 3: When the items in a list are written in the form of intervals, for example, 10 – 20, we need to first calculate the class mark using xi = (lower limit + upper limit) / 2.

Then, the mean can be calculated using the formula given below,

x̄ = Σfixi/Σfi

Types of Mean in Math

There are different types of means in mathematics, which are arithmetic mean, weighted mean, geometric mean (GM), and harmonic mean (HM). If mentioned without an adjective (as mean), mean generally refers to the arithmetic mean in statistics. Some of the types of the mean are explained in brief as given below,

• Arithmetic Mean
• Weighted Mean
• Geometric Mean
• Harmonic Mean
• Arithmetic Mean

Arithmetic mean is often referred to as the mean or arithmetic average, which is calculated by adding all the numbers in a given data set and then dividing it by the total number of items within that set. The general formula to find the arithmetic mean is given as,

x̄ = Σxi/n (or) x̄ = Σfixi / Σfi.

where,

x̄ = the mean value of the set of given data.

x_i = data value

f_i = corresponding frequency

n = total number of data values

Weighted Mean

The weighted mean is calculated when certain values that are given in a data set are more important than the others. A weight wi is attached to each of the values xi. The general formula to find the weighted mean is given as,

Weighted mean = Σwixi/Σwi

where,

xi = data value

wi = corresponding weight

Geometric Mean

The geometric mean is defined as the nth root of the product of n numbers in the given data set. The formula to find the geometric mean for a given set of data, x1, x2, x3, … , xn ,

G.M. = n√(x₁ · x₂ · x₃ · … · x_n)

Harmonic Mean

For a given set of observations, harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations, given using the formula,

Harmonic mean = 1/[Σ(1/xi)]/N = N/Σ(1/xi)

Solved Examples on Mean

Example 1: The following lists the ages of a group of 10 people. A = {45, 39, 53, 45, 43, 48, 50, 40, 40, 45}. Calculate the mean age of the group.

Solution: The total number of people in the group, n = 10

x1 = 45, x₂ = 39, x₃ = 53, … , x_n = 45

We will use the formula given below.

x̄ = (x₁+ x₂+ x₃+ … + x_n )/n

The average age of the group = (45 + 39 + 53 + 45 + 43 + 48 + 50 + 40 + 40 + 45)/10 = 448/10 = 44.8

Example 2: There are 30 students in Grade 8. The marks obtained by the students in mathematics are tabulated below. Calculate the mean marks.

Solution: The total number of students in Grade 8 = 2 + 7 + 10 + 6 + 5 = 30

x₁ = 100, x₂ = 95, x₃ = 88, x₄ = 76, x₅ = 69, f₁ = 2, f₂ = 7, f₃ = 10, f₄ = 6, f5 = 5

x₁f₁ = 100 × 2 = 200

x_2f2 = 95 × 7 = 665

x₃f₃ = 88 × 10 = 880

x¹f₁ = 76 × 6 = 456

x1f1 = 69 × 5 = 345

Σfixi = f1x1 + f2x2 + f3x3 + f4x4 + f5x5

= 200 + 665 + 880 + 456 + 345

= 2,546

Σfi = f1 + f2 + f3 + f4 + f5

= 2 + 7 + 10 + 6 + 5

= 30

We will use the formula given below.

x̄ = Σfixi/Σfi

Mean marks = 2546/30 = 84.87