How to find the mean in math

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 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 Definition

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 sum of all observations to the total number of observations.

⇒ Given a data set, \( X = x_{1},x_{2}, . . . ,x_{n}\), the mean (or arithmetic mean, or average), denoted x̄, is the mean of the n values \(x_{1},x_{2}, . . . ,x_{n}\).

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

Mean Example

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.

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. The formula to calculate the mean will be helpful in solving a majority of the topics related to the mean.

Mean Formula for Ungrouped Data:

The mean formula for a set of given observations can be expressed as,

Mean = (Sum of Observations) ÷ (Total Numbers of Observations)

Mean Formula for Grouped Data:

Similarly, we have a mean formula for grouped data, which is expressed as
x̄ = Σfx/Σf
where,

• x̄ = the mean value of the set of given data.
• f = frequency of each class
• x = mid-interval value of each class

Hence, the average of all the data points is termed as mean.

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 = {Sum of Observation} ÷ {Total numbers of Observations}
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.

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. {\({x_1, x_2, x_3, … , x_n }\)}

Mean can be calculated using the formula given below,

x̄ = \((x_1, x_2, x_3, … , x_n )/n\)

or

x̄ = Σx\(_i\)/n

Case 2: Let there be n number of items in a list, given as, {\({x_1, x_2, x_3, … , x_n }\)} and the frequency of each item be {\({f_1, f_2, f_3, … , f_n }\)} respectively.

Mean can be calculated using the formula given below,

x̄ = (f\(_1\)x\(_1\) + f\(_2\)x\(_2\) + f\(_3\)x\(_3\) + . . . + f\(_n\)x\(_n\))/(f\(_1\) + f\(_2\) + f\(_3\) + . . . + f\(_n\))

or

x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)

Case 3: When the items in a list are written in the form of a range, for example, 10 - 20, we need to first calculate the class mark.

Then, the mean can be calculated using the formula given below, where x\(_i\) will be the classmark for each item.

x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)

Let us see these methods for different cases in detail in the following sections.

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 = \((x_1, x_2, x_3, … , x_n )/n\) or x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\), where x\(_1\), x\(_2\), . . ., x\(_n\) are n observations and if given f\(_1\), f\(_2\), . . . f\(_n\) are respective frequencies.

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 = {Sum of Observation} ÷ {Total numbers of Observations} = 754/5 = 150.8

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

Mean of Grouped Data

Grouped data is a set of given data that has been bundled together in categories. It is a set of data formed by aggregating individual observations of a variable into groups. 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 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\(_i\).
  Column 3- Frequencies (fi) (corresponding)
  Column 4- x\(_i\)f\(_i\) (corresponding product of column 2 and column 3)
• Calculate Mean by the Formula Mean = ∑x\(_i\)f\(_i\)/∑f\(_i\)

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 given as, i.e. di = x\(_i\) - A
  Column 4- Frequencies (fi) (corresponding)
  Column 5- Mean of d\(_i\), using formula, Mean of d\(_i\) = ∑x\(_i\)d\(_i\)/∑d\(_i\)
• Finally, calculate the Mean by adding the assumed mean to the mean of the d\(_i\)

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 and denote it as A.
  Column 3- Calculate the corresponding deviations given as, i.e. di = 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 (fi) (corresponding)
• Find the Mean of u\(_i\) = ∑x\(_i\)u\(_i\) / ∑u\(_i\)
• Finally, calculate the Mean by adding the assumed mean A to the product of class width(h) with mean of u\(_i\)

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

Now,

\(x_1\) = 15, \(x_2\) = 25, \(x_3\) = 35, \(x_4\) = 45, \(x_5\) = 55
\(f_1\) = 17, \(f_2\) = 22, \(f_3\) = 20, \(f_4\) = 21, \(f_5\) = 20

\(x_1f_1\) =15 × 17 = 255
\(x_2f_2\) = 25 × 22 = 550
\(x_3f_3\) = 35 × 20 = 700
\(x_1f_1\) = 45 × 21 = 945
\( x_1f_1\) = 55 × 20 = 1100

f\(_1\)x\(_1\) + f\(_2\)x\(_2\) + f\(_3\)x\(_3\) + f\(_4\)x\(_4\) + f\(_5\)x\(_5\) = 255 + 550 + 700 + 945 + 1100 = 3550
f\(_1\) + f\(_2\) + f\(_3\) + f\(_4\) + f\(_5\) = 17 + 22 + 20 + 21 + 20 = 100

We will use the formula given below.

x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)

The mean age = 3550/100

= 35.5

The mean age of the members = 35.5

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 = Σf\(_i\)/N

where,

• x̄ = the mean value of the set of given data.
• f = frequency of the individual data
• N = sum of frequencies

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\(_i\)x̄/Σw\(_i\)

where,

• x̄ = the mean value of the set of given data.
• w = corresponding weight for each observation

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, \(x_1, x_2, x_3, … , x_n \),

G.M. = n√(x\(_1\) · x\(_2\) · x\(_3\) · … · 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
• Geometric Mean

FAQs on Mean

What is Meant By Mean in Statistics?

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.

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 = (Sum of Observations) ÷ (Total Numbers of Observations)

For grouped data, x̄ = Σfx/Σf
where,

• x̄ = the mean value of the set of given data.
• 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 = Σfx/Σf, where, x̄ is the mean value of the set of given data, f is frequency of each class and x is mid-interval value of each class

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 Observations) ÷ (Total Numbers of Observations)

What is the Difference Between Mean Formula and Median Formula?

The mean formula is given as the average of all the observations. It is expressed as Mean = {Sum of Observation} ÷ {Total numbers of Observations}. Whereas, the median formula is totally dependent on the number of observations (n). If the number of observation 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 Calculate the Mean Using Mean Formula?

If the set of 'n' number of observations is given then the mean can be easily calculated by using a general mean formula that is, Mean = {Sum of Observation} ÷ {Total numbers of Observations}.

What are Different Types of Mean?

The different types of means in mathematics are,

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

What is the Difference Between Arithmetic Mean and Weighted 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, 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 a statistical concept that carries a major significance in finance and is used in various financial fields like in business valuation.

How To Use the Mean Formula?

The general mean formula is mathematically expressed as Mean = {Sum of Observation} ÷ {Total numbers of Observations}. Let us consider an example to understand its use.
Example: Find the mean of (1, 2, 3, 4, 5, 6, 7)
Solution: Total number of observation = 7
Mean = {Sum of Observation} ÷ {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 the Mean Formula for n Observations?

Mean formula for 'n' observations is expressed as
Mean of n observations = {Sum of 'n' Observation} ÷ {Total numbers of 'n' Observations}

How do you solve mean median and mode?

How do u find the median?