Mean

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.

"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.

Mean Meaning

The mean is the average or a calculated central value of a set of numbers and is used to measure the central tendency of the data. Central tendency is the statistical measure that recognizes the entire set of data or distribution through a single value. It provides an exact description of the whole data. In statistics, the mean can also be defined as the ratio of sum of all observations to the total number of observations.

⇒ Given a data set, X = {x₁,x₂, . . . ,x_n}, the mean (or arithmetic mean, or average), denoted x̄, is the mean of the n values x₁,x₂, . . . ,x_n.

Mean Symbol: The mean is represented as x-bar, x̄.

Examples:

Examples of mean in real life are:

• Mean of the runs scored by a cricketer in test matches.
• Mean price of houses in a particular area calculated by real estate agents.

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)

Example: Find the mean of the first five natural odd numbers, using the mean formula.

Solution:

The first five natural odd numbers = 1, 3, 5, 7, and 9

Using mean formula,

Mean = (1 + 3 + 5 + 7 + 9) ÷ 5 = 25/5 = 5

Answer: The mean of the first five natural odd numbers {1, 3, 5, 7, 9} is 5.

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.

Example: The heights of five students are 161 in, 130 in, 145 in, 156 in, and,162 in respectively. Find the mean height of the students.

Solution: To find: the mean height of the students.

The heights of five students = 161 in, 130 in, 145 in, 156 in, and,162 in (given)

Sum of the heights of five students = (161 + 130 + 145 + 156 + 162) = 754

Using mean formula,

Mean = 754/5 = 150.8

Answer: The mean height of the students is 150.8 inches.

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 x_i. The class mark is the middle value of the interval. i.e., x_i = (upper limit) + (lower limit) / 2.
  Column 3- Frequencies (f_i) (corresponding)
  Column 4- x_if_i (corresponding product of column 2 and column 3)
• Calculate mean by the formula ∑x_if_i/∑f_i

Example: There are 100 members in a basketball club. The different age groups of the members and the number of members in each age group are tabulated below. Calculate the mean age of the club members.

Solution:

In this case, we first need to calculate the Class Mark for each age group.

We will use the formula given below and calculate the Class Mark for each age group.

Class mark = (Upper Limit + Lower Limit)/2. For example, for the first interval 10-20, class mark = (10 + 20) / 2 = 30/2 = 15.

Now,

Σf_ix_i = f₁x₁ + f₂x₂ + f₃x₃ + f₄x₄ + f₅x₅
= 255 + 550 + 700 + 945 + 1100
= 3550

Σf_i = f₁ + f₂ + f₃ + f₄ + f₅
= 17 + 22 + 20 + 21 + 20
= 100

We will use the formula given below.

x̄ = Σf_ix_i/Σf_i

The mean age = 3550/100

= 35.5

The mean age of the members = 35.5

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 x_i. 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. d_i = x_i - A
  Column 4- Frequencies (f_i) (corresponding)
• Finally, calculate the mean by the formula, A + ∑f_id_i/∑f_i

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 x_i. Take the central value from the class marks as the assumed mean (A).
  Column 3- Calculate the corresponding deviations, i.e. d_i = x_i - A
  Column 4- Calculate the values of u_i using the formula, u_i = d_i/h, where h is the class width.
  Column 5- Frequencies (f_i) (corresponding)
• Finally, calculate the mean by using the formula A + h (∑f_iu_i / ∑f_i).

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, … , x_n }

Mean can be calculated using the formula given below,

x̄ = (x₁+ x₂+ x₃+ … + x_n )/n

or

x̄ = Σx_i/n

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

Mean can be calculated using the formula given below,

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

or

x̄ = Σf_ix_i/Σf_i

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 x_i = (lower limit + upper limit) / 2.

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

x̄ = Σf_ix_i/Σf_i

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 = Σx_i/n (or) x̄ = Σf_ix_i / Σf_i.

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 w_i is attached to each of the values x_i. The general formula to find the weighted mean is given as,

Weighted mean = Σw_ix_i/Σw_i

where,

• x_i = data value
• w_i = corresponding weight

Geometric Mean

The geometric mean is defined as the n^th 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, … , x_n ,

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/x_i)]/N = N/Σ(1/x_i)

Related Topics on Mean:

• Categorical Data
• Range in Statistics
• Average

What is the Definition of Mean?

Mean, one of the important and most commonly used measures of central tendency is the average or a calculated central value of a set of numbers. The process of calculating the mean is different based on the type of data (grouped or ungrouped data).

What is Mean Formula?

There are different formulas to find the mean of a given set of data, as given below,

• For ungrouped data, mean, x̄ = (sum of observations) ÷ (number of observations)
• For grouped data, mean, x̄ = Σfx/Σf
  where,
  f = frequency of each class
  x = mid-interval value of each class

What is Mean Formula for Grouped Data?

The mean formula to find the mean of a grouped set of data can be given as, x̄ = Σf_ix/_iΣf_i, where, x̄ is the mean, f_i is frequency of a class and x_i is mid-interval value of corresponding class.

What is Population Mean Symbol?

The population mean refers to the mean of all the data values that are there in the population. It is represented by the symbol μ.

What is the Mean Formula for Ungrouped Data?

The mean formula to find the mean for an ungrouped set of data can be given as, mean = (sum of data values) / (number of data values).

What is the Difference Between Mean and Median?

The mean formula is given as the average of all the observations. It is expressed as mean = (sum of observations) ÷ (total number of observations). Whereas, the median formula is totally dependent on the number of observations (n). If the number of observations is even then the median formula is [median = ((n/2)^th term + ((n/2) + 1)^th term)/2] and if n = odd then the median formula is [median = {(n + 1)/2} ^th term].

How To Find the Mean Using Mean Formula?

If the data is not grouped, then just divide the sum of observations by total number to find the mean. If the data is grouped, then to find the mean, first multiply the mid point of each class by the corresponding frequency first, then divide the sum of all products obtained by the sum of all frequencies.

What are Different Types of Mean?

The different types of means in mathematics are,

• Arithmetic Mean
• Geometric Mean
• Harmonic Mean
• Weighed Mean

What is the Difference Between Arithmetic Mean and Weighted Mean?

Arithmetic mean is calculated when all data values have equal importance, while weighted mean is calculated when certain values that are given in a data set are more important than the others.

What are the Applications of Mean in our Daily Lives?

Mean is used to understand the whole data by a single number. For example, to analyze how high the students of a class are, it is enough to just find the mean instead of looking into each student's height.

How To Use the Mean Formula?

The usage of mean formula is shown using an example here.
Example: Find the mean of (1, 2, 3, 4, 5, 6, 7).
Solution: Total number of observations = 7
Mean = (sum of observations) ÷ (total numbers of observations)
Mean = (1 + 2 + 3 + 4 + 5 + 6 + 7) ÷ 7 = 28/7 = 4
Mean of (1, 2, 3, 4, 5, 6, 7) is 4.

What is Sample Mean Symbol?

The sample mean refers to the mean of only selected data values from the population. It is represented by the symbol x̄.