In statistics, the mean is one of the measures of central tendency, apart from the mode and median. Mean is nothing but the average of the given set of values. It denotes the equal distribution of values for a given data set. The mean, median and mode are the three commonly used measures of central tendency. To calculate the mean, we need to add the total values given in a datasheet and divide the sum by the total number of values.The Median is the middle value of a given data when all the values are arranged in ascending order. Whereas mode is the number in the list, which is repeated a maximum number of times.

Learn: Central tendency

In this article, you will learn the definition of mean, the formula for finding the mean for ungrouped and grouped data, along with the applications and solved examples.

**Table of contents:**

- Definition
- Mean Symbol
- Mean Formula
- How to Find Mean
- For Ungrouped data
- For Grouped data

- Types
- Arithmetic Mean
- Geometric Mean
- Harmonic Mean
- Root Mean Square
- Contraharmonic Mean

- Applications
- Practice problems
- FAQs

## Definition of Mean in Statistics

**Mean**is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.

Mean = (Sum of all the observations/Total number of observations)

**Example:**

What is the mean of 2, 4, 6, 8 and 10?

**Solution:**

First, add all the numbers.

2 + 4 + 6 + 8 + 10 = 30

Now divide by 5 (total number of observations).

Mean = 30/5 = 6

In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x) and then adding all these products together.

### Mean Symbol (X Bar)

The symbol of mean is usually given by the symbol ‘x̄’. The bar above the letter x, represents the mean of x number of values.

X̄ = (Sum of values ÷ Number of values)

X̄ = (x_{1} + x_{2} + x_{3} +….+x_{n})/n

**Read more:**

- Mean Median Mode
- Mean definition
- Mean formula
- Median

## Mean Formula

The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of the sum of all the terms and the total number of terms. Hence, we can say;

**Mean = Sum of the Given Data/Total number of Data**

To calculate the arithmetic mean of a set of data we must first add up (sum) all of the data values (x) and then divide the result by the number of values (n). Since∑ is the symbol used to indicate that values are to be summed (see Sigma Notation) we obtain the following formula for the mean (x̄):

## How to Find Mean?

As we know, data can be grouped data or ungrouped data so to find the mean of given data we need to check whether the given data is ungrouped. The formulas to find the mean for ungrouped data and grouped data are different. In this section, you will learn the method of finding the mean for both of these instances.

### Mean for Ungrouped Data

The example given below will help you in understanding **how to find the mean** of ungrouped data.

**Example:**

In a class there are 20 students and they have secured a percentage of 88, 82, 88, 85, 84, 80, 81, 82, 83, 85, 84, 74, 75, 76, 89, 90, 89, 80, 82, and 83.

Find the mean percentage obtained by the class.

**Solution:**

Mean = Total of percentage obtained by 20 students in class/Total number of students

= [88 + 82 + 88 + 85 + 84 + 80 + 81 + 82 + 83 + 85 + 84 + 74 + 75 + 76 + 89 + 90 + 89 + 80 + 82 + 83]/20

= 1660/20

= 83

Hence, the mean percentage of each student in the class is 83%.

### Mean for Grouped Data

For grouped data, we can find the mean using either of the following formulas.

Direct method:

\(\begin{array}{l}Mean, \overline{x}=\frac{\sum_{i=1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\end{array} \)

Assumed mean method:

\(\begin{array}{l}Mean, (\overline{x})=a+\frac{\sum f_id_i}{\sum f_i}\end{array} \)

Step-deviation method:

\(\begin{array}{l}Mean, (\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\end{array} \)

Go through the example given below to understand how to calculate the mean for grouped data.

**Example:**

Find the mean for the following distribution.

x_{i} | 11 | 14 | 17 | 20 |

f_{i} | 3 | 6 | 8 | 7 |

**Solution:**

For the given data, we can find the mean using the direct method.

x_{i} | f_{i} | f_{i}x_{i} |

11 | 3 | 33 |

14 | 6 | 84 |

17 | 8 | 136 |

20 | 7 | 140 |

∑f_{i} = 24 | ∑f_{i} x_{i}= 393 |

Mean = ∑f_{i}x_{i}/∑f_{i} = 393/24 = 16.4

## Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers till now. But what if the numbers in the observation list include negative numbers. Let us understand with an instance,

**Example:**

Find the mean of 9, 6, -3, 2, -7, 1.

**Solution:**

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

## Types of Mean

There are majorly three different types of mean value that you will be studying in statistics.

- Arithmetic Mean
- Geometric Mean
- Harmonic Mean

### Arithmetic Mean

When you add up all the values and divide by the number of values it is calledArithmetic Mean.To calculate, just add up all the given numbers then divide by how many numbers are given.

**Example: What is the mean of 3, 5, 9, 5, 7, 2?**

Now add up all the given numbers:

3 + 5 + 9 + 5 + 7 + 2 = 31

Now divide by how many numbers are provided in the sequence:

316= 5.16

### Geometric Mean

The geometric mean of two numbers x and y is xy. If you have three numbers x, y, and z, their geometric mean is 3xyz.

\(\begin{array}{l} Geometric\;Mean=\sqrt[n]{x_{1}x_{2}x_{3}…..x_{n}}\end{array} \)

Example: Find the geometric mean of 4 and 3 ?

\(\begin{array}{l}Geometric Mean = \sqrt{4 \times 3} = 2 \sqrt{3} = 3.46\end{array} \)

### Harmonic Mean

The harmonic mean is used to average ratios. For two numbers x and y, the harmonic mean is 2xy(x+y). For, three numbers x, y, and z, the harmonic mean is 3xyz(xy+xz+yz)

\(\begin{array}{l} Harmonic\;Mean (H) = \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+……\frac{1}{x_{n}}}\end{array} \)

### Root Mean Square (Quadratic)

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

\(\begin{array}{l} X_{rms}=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}….x_{n}^{2}}{n}}\end{array} \)

### Contraharmonic Mean

The contraharmonic mean of x and y is (x2 + y2)/(x + y). For n values,

\(\begin{array}{l} \frac{(x_{1}^{2}+x_{2}^{2}+….+x_{n}^{2})}{(x_{1}+x_{2}+…..x_{n})}\end{array} \)

### Real-life Applications of Mean

In the real world, when there is huge data available, we use statistics to deal with it.Suppose, in a data table, the price values of 10 clothing materials are mentioned. If we have to find the mean of the prices, then add the prices of each clothing material and divide the total sum by 10. It will result in an average value. Another example is that if we have to find the average age of students of a class, we have to add the age of individual students present in the class and then divide the sum by the total number of students present in the class.

### Practice Problems

Q.1: Find the mean of 5,10,15,20,25.

Q.2:Find the mean of the given data set: 10,20,30,40,50,60,70,80,90.

Q.3: Find the mean of the first 10 even numbers.

Q.4: Find the mean of the first 10 odd numbers.

## Frequently Asked Questions – FAQs

### What is mean in statistics?

In statistics, Mean is the ratio of sum of all the observations and total number of observations in a data set. For example, mean of 2, 6, 4, 5, 8 is:

Mean = (2 + 6 + 4 + 5 + 8) / 5 = 25/5 = 5

### How is mean represented?

Mean is usually represented by x-bar or x̄.

X̄ = (Sum of values ÷ Number of values in data set)

### What is median in Maths?

Median is the central value of the data set when they are arranged in an order.

For example, the median of 3, 7, 1, 4, 8, 10, 2.

Arrange the data set in ascending order: 1,2,3,4,7,8,10

Median = middle value = 4

### What are the types of Mean?

In statistics we learn basically, three types of mean, they are:

Arithmetic Mean, Geometric Mean and Harmonic Mean

### What is the mean of the first 10 natural numbers?

The first 10 natural numbers are: 1,2,3,4,5,6,7,8,9,10

Sum of first 10 natural numbers = 1+2+3+4+5+6+7+8+9+10 = 55

Mean = 55/10 = 5.5

### What is the relationship between mean, median and mode?

The relationship between mean, median and mode is given by:

3 Median = Mode + 2 Mean.

### What is the mean of the first 5 even natural numbers?

As we know, the first 5 even natural numbers are 2, 4, 6, 8, and 10.

Hence, Mean = (2 + 4 + 6 + 8 + 10)/5

Mean = 6

Thus, the mean of the first 5 even natural numbers is 6.

### Find the mean of the first 5 composite numbers?

The first 5 composite numbers are 4, 6, 8, 9 and 10.

Thus, Mean = (4 + 6 + 8 + 9 + 10)/5

Mean = 37/5 = 7.4

Hence, the mean of the first 5 composite numbers is 7.4.

## FAQs

### How do you calculate mean with examples? ›

Mean: The "average" number; found by **adding all data points and dividing by the number of data points**. Example: The mean of 4, 1, and 7 is ( 4 + 1 + 7 ) / 3 = 12 / 3 = 4 (4+1+7)/3 = 12/3 = 4 (4+1+7)/3=12/3=4left parenthesis, 4, plus, 1, plus, 7, right parenthesis, slash, 3, equals, 12, slash, 3, equals, 4.

**How do you find the mean formula? ›**

How do I find the mean? You can find the mean, or average, of a data set in two simple steps: **Find the sum of the values by adding them all up.** **Divide the sum by the number of values in the data set**.

**How do you explain the answer of mean? ›**

A mean in maths is **the average of a data set, found by adding all numbers together and then dividing the sum of the numbers by the number of numbers**. For example, with the data set: 8, 9, 5, 6, 7, the mean is 7, as 8 + 9 + 5 + 6 + 7 = 35, 35/5 = 7. What is the definition of mean?

**What is mean give an example? ›**

Mean is the simple mathematical average of a set of two or more numbers. The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series, and the geometric mean method, which is the average of a set of products.

**What are the three formulas of mean? ›**

**They are:**

- Direct Method.
- Assumed Mean Method.
- Step-deviation Method.

**Why do you calculate the mean? ›**

The mean is used to summarize a data set. It is a measure of the center of a data set.

**What is the formula to calculate mean quizlet? ›**

How do you find the mean? **Add all numbers in a set of data and then divide the sum by the number of items**. Example: 4+2+3+3= 12. 12 divided by 4= 3, so 3 is the mean.

**What is mean in math? ›**

A mean in math is **the average of a data set, found by adding all numbers together and then dividing the sum of the numbers by the number of numbers**. For example, with the data set: 8, 9, 5, 6, 7, the mean is 7, as 8 + 9 + 5 + 6 + 7 = 35, 35/5 = 7.

**What is the solution to the equation mean? ›**

An equation is a mathematical statement formed by placing an equal sign between two numerical or variable expressions, as in 3x+5=11 . A solution to an equation is **a number that can be plugged in for the variable to make a true number statement**.

**What is the definition of mean *? ›**

adjective, mean·er, mean·est. **offensive, selfish, or unaccommodating; nasty; malicious**: a mean remark;He gets mean when he doesn't get his way. small-minded or ignoble: mean motives. penurious, stingy, or miserly: a person who is mean about money.

### What is the formula of mean in short method? ›

To calculate mean deviation about mean by shortcut method, *First take an appropriate 'Assumed Mean' A. *calculate the sum of the frequencies, ∑i=1nfi . *calculate ∑i=1nfidi where, di=xi−A . *calculate mean as **xˉ=A+∑fi∑fidi×h** where h = the class width =upperclasslimit−lowerclasslimit of the frequency distribution.

**What is the definition of mean in statistics? ›**

The mean (average) of a data set is found by **adding all numbers in the data set and then dividing by the number of values in the set**. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set. Created by Sal Khan.

**How do you write a mean in statistics? ›**

To calculate mean, **add together all of the numbers in a set and then divide the sum by the total count of numbers**.

**Why is the mean the most important? ›**

The mean represents the average value in a dataset. The mean is important because **it gives us an idea of where the center value is located in a dataset**. The mean is also important because it carries a piece of information from every observation in a dataset.

**How do you solve the mean assessment? ›**

To calculate the mean, **add up all of the data points and divide that result by the total number of data points**.

**What does of mean in math word problems? ›**

Answer: In Algebra, 'of' means **to multiply**.

Let's see some examples. Explanation: In math, 'of' is also considered as one of the arithmetic operations which means multiplication within the brackets. For example, we need to find one-third of 30. Mathematically, we re-write it as 1/3 of 30 = 1/3 × 30 = 10.

**What is the mean in math calculator? ›**

The mean x̄ of a data set is **the sum of all the data divided by the count n**. mean=¯x=∑ni=1xin.

**What is the formula to calculate the mean by the assumed mean method? ›**

**a = assumed mean**. **f _{i} = frequency of ith class**.

**d**.

_{i}= x_{i}– a = deviation of ith class**Σf**.

_{i}= n = Total number of observations**x**.

_{i}= class mark = (upper class limit + lower class limit)/2**What are the two meanings of mean? ›**

The adjective mean can describe someone who's stingy or ungenerous, but it also means "**unkind or unfair**," which is what a little kid intends to convey when she yells at her mother, "You're mean!" In the sentence, "She lived in a mean little house," mean has yet another meaning, this time being "shabby or poor."

**What is an mean in algebra? ›**

The mean is often called **the average**. To find the mean you take a set of data and calculate the sum of the data, after that you divide the sum by the number of pieces in the set.

### What does mean mean in math 6th grade? ›

Mean. The mean is **the typical average**. To find the mean, add up all the numbers you have, and divide by how many numbers there are in total.

**What are the 4 steps to solving an equation? ›**

We have 4 ways of solving one-step equations: **Adding, Substracting, multiplication and division**. If we add the same number to both sides of an equation, both sides will remain equal.

**What is the mean of simple equation? ›**

What is Simple Equation? **A mathematical equation which represents the relationship of two expressions on either side of the sign**. It mostly has one variable and equal to symbol. Example: 2x – 4 = 2.

**What is mean and median with example? ›**

The mean (informally, the “average“) is found by adding all of the numbers together and dividing by the number of items in the set: 10 + 10 + 20 + 40 + 70 / 5 = 30. The median is found by ordering the set from lowest to highest and finding the exact middle. The median is just the middle number: 20.

**What are the 3 ways to calculate average? ›**

There are three main types of average: **mean, median and mode**. Each of these techniques works slightly differently and often results in slightly different typical values. The mean is the most commonly used average. To get the mean value, you add up all the values and divide this total by the number of values.

**What is the mean and mode in math example? ›**

**The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set**. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

**Is mean the same as average? ›**

**The average is the sum of all values divided by the number of values.** **It is also sometimes referred to as mean**. In statistics, the mean is the average of the given sample or data set.

**How do I calculate mean and median? ›**

To find the mean, **add up the values in the data set and then divide by the number of values that you added**. To find the median, list the values of the data set in numerical order and identify which value appears in the middle of the list.

**What does median mean in math example? ›**

The median is the number in the middle {2, 3, 11, 13, 26, 34, 47}, which in this instance is 13 since there are three numbers on either side. To find the median value in a list with an even amount of numbers, one must determine the middle pair, add them, and divide by two.

**Which is the simplest way to find the average? ›**

How to Calculate Average? We can easily calculate the average for a given set of values. We just have to **add all the values and divide the outcome by the number of given values**.

### How do you solve average problems quickly? ›

**Formulas and Quick Tricks for Average Problems**

- Average = Sum of quantities/ Number of quantities.
- Sum of quantities = Average * Number of quantities.
- The average of first n natural numbers is (n +1) / 2.
- The average of the squares of first n natural numbers is (n +1)(2n+1 ) / 6.

**What are the different types of mean? ›**

There are different types of mean, viz. **arithmetic mean, weighted mean, geometric mean (GM) and harmonic mean (HM)**.