TABLE OF CONTENT
Content
page no
Objective
Historical background
Introduction
Result of finding
Real life application
Conclusion
Bibliography
6
OBJECTIVES
understand the importance of statistics in daily life.
variables.
sion equations are formed.
-life problems.
.
7
HISTORICAL
BACKGROUND
The study of how two or more variables are related has been an important part of
statistics for more than a century. The ideas of correlation and regression started
taking shape in the 19th century, when scientists wanted better methods to analyze
data and understand patterns in nature and society.
The first steps were taken by Auguste Bravais, a French mathematician, in 1846,
who explored ways to measure how two variables might be connected. His work
laid the groundwork for future studies on correlation.
Later, in the 1880s, Sir Francis Galton examined traits like human height in
families. He noticed an interesting trend: extremely tall or short parents often had
children whose heights were closer to the average. He called this trend “regression
toward the mean,” giving rise to the term regression. Galton’s observations also
helped explain how variables can relate to one another, forming the foundation for
correlation analysis.
Following Galton, Karl Pearson, a British mathematician, developed a precise
mathematical formula in the late 1890s to measure the strength and direction of the
relationship between variables. This formula, now known as the Pearson
correlation coefficient, and his work on regression equations, made these statistical
tools practical for research in many fields.
Thanks to the pioneering work of Bravais, Galton, and Pearson, correlation and
regression are now essential tools in statistics. They are widely used in areas such
as science, business, healthcare, education, and pharmacy to study data, understand
trends, and make predictions that support better decision-making.
8
INTRODUCTION
Statistics is the study of collecting, organizing, and analyzing data to
understand patterns and trends.



It helps us make sense of large amounts of information.
It is used in daily life to solve problems and make decisions.
Often, we want to know how two things are related, for example,
study hours and marks, or price and demand.
Correlation is an important tool in statistics.



It shows how strongly two variables are connected.
The relationship can be positive, negative, or sometimes not
present at all.
For example, if study hours increase, marks may also increase,
showing a positive correlation.
Regression is another method closely related to correlation.



It helps predict the value of one variable based on another variable.
Regression is widely used to make predictions and plan ahead.
For example, a teacher can estimate a student’s marks based on the
number of hours studied.
9
Types of correlation
1. Positive or negative correlation
If both variables are moving in same direction the
correlation is positive else if both variables are
changing in opposite direction then they are
negative.
2. Linear and non-linear correlation
For ratio amount of change in one variable to
another being constant it is linear and non-linear
when amount of change has no constant relation.
3. Simple partial or multiple correlation
It is simple if only 2 variable, multiple for variable
more than 2.
10
Types of regression
 Simple Regression –
This type has one independent variable and one
dependent variable. It shows a straight-line
relationship between the two. Example: Predicting
marks based on hours of study.
 Multiple Regression –
This type has two or more independent variables
and one dependent variable. It predicts the
dependent variable using several factors at the same
time. Example: Predicting marks based on hours of
study and number of practice tests.
11
RESULTS OR FINDING
Scatter Diagram Method
The scatter diagram method is used to study the relationship between two
variables. It helps to see visually whether the variables have a positive, negative, or
no correlation.
Sample Data: Study Hours vs Marks
Student Study Hours (X) Marks (Y)
1
2
45
2
3
55
3
4
65
4
5
72
5
6
80
6
7
88
Scatter Diagram (Text Version)
Marks (Y)
90 |
*
85 |
80 |
*
75 |
70 |
*
65 |
*
60 |
55 |
*
50 |
45 | *
40 |
+-----------------------------2
3
4
5
6
7
Study Hours (X)
Observation:



The points show a rising trend, which indicates a positive correlation.
As study hours increase, marks also increase.
This diagram provides a visual understanding before performing correlation
or regression calculations.
12
Regression Method
Regression is used to study the relationship between two variables and to predict
the value of one variable based on another. The regression line (or line of best
fit) represents the general trend of the data points.
Sample Data: Study Hours vs Marks
Student Study Hours (X) Marks (Y)
1
2
45
2
3
55
3
4
65
4
5
72
5
6
80
6
7
88
Regression Diagram (Text Version)
Marks (Y)
90 |
*
85 |
80 |
*
75 |
70 |
*
65 |
*
60 |
55 |
*
50 |
45 | *
40 |
+-----------------------------2
3
4
5
6
7
Study Hours (X)
Regression line: roughly follows the trend from bottom-left to topright
Observation:



The points lie close to a straight line, showing a positive relationship.
The regression line can be used to predict marks for any given number of
study hours.
For example, if a student studies 6.5 hours, we can estimate their marks
using the regression line.
13
Regression Equation (Simple Example):
Using the data, the regression line can be expressed as:
Y = 5 + 12X


Here, Y = predicted marks, X = study hours
For X = 6.5 hours: Y = 5 + 12(6.5) = 83 marks (approx.)
Example: Finding Correlation Coefficient
Problem:
The table below shows the number of hours students spend practicing a sport (X) and their
scores in a fitness test (Y). Find the correlation coefficient to see the relationship between
practice hours and fitness scores.
Student Practice Hours (X) Fitness Score (Y)
1
1
30
2
2
35
3
3
45
4
4
50
5
5
55
Step 1: Formula for Correlation Coefficient
The Pearson correlation coefficient is:
r = [ n * Σ(XY) - (ΣX)(ΣY) ] / √{ [ n * Σ(X²) - (ΣX)² ] * [ n * Σ(Y²) - (ΣY)² ] }
Where:


n = number of observations
X, Y = values of variables
Step 2: Prepare a Table for Calculation
X Y X × Y X² Y²
1 30 30
1 900
2 35 70
4 1225
14
X Y X × Y X² Y²
3 45 135 9 2025
4 50 200 16 2500
5 55 275 25 3025
Sum of Columns:






ΣX = 15
ΣY = 215
ΣXY = 710
ΣX² = 55
ΣY² = 9675
n=5
Step 3: Apply Formula
Numerator = n * Σ(XY) - (ΣX)(ΣY)
= 5 * 710 - 15 * 215
= 3550 - 3225
= 325
Denominator = √{ [ n * Σ(X²) - (ΣX)² ] * [ n * Σ(Y²) - (ΣY)² ] }
= √{ [5 * 55 - 15²] * [5 * 9675 - 215²] }
= √{ (275 - 225) * (48375 - 46225) }
= √{ 50 * 2150 }
= √107500 ≈ 328
Step 4: Calculate r
r = Numerator / Denominator
r = 325 / 328 ≈ 0.99
15
REAL LIFE
APPLICATOIN
 Daily Life and Research


Studying how social media effects students’ marks.
Seeing how population growth are linked to literacy.
 Business


Seeing how price affects the demand for a product.
Estimating sales based on money spend on advertising.
 Education


Predicting marks based on hours of study.
Finding the link between attendance and exam result.
 Agriculture


Studying how soil effects plant growth.
Estimating crop yield from rainfall or fertilizer used.
 Healthcare


Studying how medicine effects recovery.
Predicting blood pressure based on age or weight.
16
Conclusion
In this project, we learned about correlation and regression.
Correlation shows how two things are connected, while
regression helps us predict one thing from another. These
methods make it easier to understand data and see patterns.
Correlation and regression are used in many areas like school,
business, health, farming, and research. They help people solve
problems and make better decisions using real information.
Overall, these tools are very useful in real life. They help us
understand relationships between things and make predictions
that can guide our decisions.
17
Bibliography
1. Gupta, S.P., Statistical Methods, Sultan Chand & Sons,
New Delhi, 2019.
2. Sharma, J.K., Business Statistics, Pearson Education, 2020.
3. Kenney, J.F. & Keeping, E.S., Mathematics of Statistics,
Princeton Hall, 2018.
4. Triola, M.F., Elementary Statistics, Pearson Education,
2018.
5. Wikipedia contributors, “Correlation and Regression,”
Wikipedia,
https://en.wikipedia.org/wiki/Correlation_and_regression
(Accessed on 24 January 2026).
6. Online Tutorials: https://www.statisticslectures.com
(Accessed on 24 January 2026).
7. Online Tutorials:
https://www.investopedia.com/terms/r/regression.asp
(Accessed on 24 January 2026).
18