📚 Mean in Statistics | Class 11 Mathematics Notes
Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS
Are you preparing for the CBSE Board Exams or JEE Mains & Advanced? Understanding the concept of Mean in Statistics is essential for scoring well in your Class 11 Mathematics exams. This guide covers everything you need to know about the mean, including its definition, types, formulas, examples, and key concepts.
Table of Contents
🔍 What is Mean in Statistics?
In statistics, the mean is a fundamental measure of central tendency. It represents the average value of a dataset and helps summarize large amounts of data with a single number. Essentially, the mean gives you an idea of the central position of data points in a distribution.
Imagine you have marks from five subjects: 80,75,90,85,70. Instead of listing each score, you can calculate the mean to quickly understand the student’s average performance.
The mean can be categorized into two main types:
- Arithmetic Mean
- Weighted Mean
Both these means help describe the data but are used in different contexts depending on the importance (weight) of each value.
🔢 Arithmetic Mean For Ungrouped Data
Mean for ungrouped data, also known as the Arithmetic Mean (often just called the “mean”) is the most common type of average. It is obtained by adding all the observations and dividing the total by the number of observations.
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.1 : 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
✅ Example.2 : Find the mean of the numbers 10, 15, 20, 25, 30
Solution:
Mean=(10+15+20+25+30)/5=100/5=20
The mean of the data set is 20.
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 = (x1 + x2 + x3 +….+xn)/n
x̄=∑ x/n
Mean Percentage
Example.3 : 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 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
Solved Examples of Mean in Statistics
Example.1 : What is the mean of 3, 5, 9, 5, 7, 2?
Solution : 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
Example.2 : Calculate the arithmetic mean of the first 7 natural numbers.
Solution:
We know that the first 7 natural numbers are 1, 2, 3, 4, 5, 6, 7.
We know that,
Arithmetic Mean = Sum of all values / Total number of values.
Hence, the arithmetic mean of first 7 natural numbers = Sum of first 7 natural numbers/Total number of natural numbers.
Arithmetic Mean = (1+2+3+4+5+6+7)/7
AM = 28/7
AM = 4
Therefore, the arithmetic mean of the first 7 natural numbers is 4.
Example 3: Determine the mean of the first 5 prime numbers.
Solution:
The first 5 prime numbers are 2, 3, 5, 7 and 11.
Hence, the mean of the first 5 prime numbers is calculated as follows:
Mean = Sum of first 5 prime numbers/Total number of prime numbers
Mean = (2+3+5+7+11)/5
Mean = 28/5 = 5.6.
Therefore, the mean of the first 5 prime numbers is 5.6.
Frequently Asked Questions on Arithmetic Mean in Statistics
Q1
What is meant by Arithmetic Mean?
In Mathematics and Statistics, the Arithmetic Mean (AM) or Mean or Average is defined as the sum of all observations in the given data set divided by the total number of observations in the dataset.
Q2
What is the formula to calculate arithmetic mean?
The formula to calculate the arithmetic mean is:
Arithmetic Mean, AM = Sum of all Observations/Total Number of Observations.
Q3
How to calculate the arithmetic mean between two numbers?
The steps to calculate the arithmetic mean between 2 numbers are:
Step 1: Add the given two numbers.
Step 2: Divide the sum by 2.
Q4
What is the arithmetic mean between 2 and 6?
The arithmetic mean between 2 and 6 is 4.
(i.e) AM = (2+6)/2 = 8/2 = 4.
Q5
What is the arithmetic mean between 10 and 24?
The arithmetic mean between 10 and 24 is 17.
(i.e) AM = (10+24)/2 = 34/2 = 17.
🔑 Importance of Mean in Statistics
- Simple to Calculate: The mean is easy to compute and interpret.
- Basis for Advanced Calculations: Concepts like variance, standard deviation, and hypothesis testing rely on the mean.
- Representative Value: It gives a general idea of where the majority of data points lie.
💡 Applications of Mean in Real Life
- Academic Performance: Calculating the average marks of students.
- Business: Determining average sales over a period of time.
- Sports: Analyzing player performance over a season.
📥 Download PDF Notes
Students preparing for board exams and competitive exams like JEE Mains and JEE Advanced can download the complete PDF notes from ANAND CLASSES for in-depth explanations and additional solved examples.
🔖 Key Takeaways:
- Use the Arithmetic Mean when all data points carry equal importance.
- Use the Weighted Mean when different data points have varying importance.
- The mean gives an overall sense of the data distribution.