In the world of data and analytics, understanding how two variables move together is fundamental.
For example —

• Do higher temperatures increase ice cream sales?
• Does more advertising lead to higher revenue?
• Do fertilizer inputs improve crop yield?

These relationships are captured by a powerful statistical concept called Correlation.

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🔍 What is Correlation?

Correlation measures the strength and direction of a linear relationship between two variables.

In simple terms:

Correlation tells us how changes in one variable are associated with changes in another.

For instance:

• As temperature rises, ice cream sales also rise → positive correlation.
• As fuel price increases, car usage decreases → negative correlation.
• The number of pens owned and height of a person → no correlation.

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🧮 The Correlation Coefficient (r)

The degree of correlation is measured using the Pearson’s correlation coefficient, denoted by r.

Where:

• X and Y = variables
• n = number of observations

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📊 Interpretation of r

Value of r	Relationship	Strength
+1	Perfect positive	Very strong
0.7 to 0.9	Strong positive	Strong
0.3 to 0.7	Moderate positive	Moderate
0	No correlation	None
-0.3 to -0.7	Moderate negative	Moderate
-0.7 to -0.9	Strong negative	Strong
-1	Perfect negative	Very strong

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🌡️ Example 1: Positive Correlation

Let’s look at a simple dataset:

Hours Studied (X)	Marks Scored (Y)
2	40
4	50
6	60
8	70
10	80

We can calculate the Pearson’s r using the formula.

Step 1: Compute intermediate values

Step 2: Apply the formula

✅ Result: r = +1, indicating a perfect positive correlation.
As study hours increase, marks also increase in a perfectly linear way.

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🧊 Example 2: Negative Correlation

Temperature (°C)	Hot Chocolate Sales
10	90
15	80
20	60
25	40
30	30

If you compute rrr, you’ll find r ≈ -0.96
→ A strong negative correlation — as temperature rises, sales fall.

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🪞 Example 3: No Correlation

Shoe Size	Intelligence Score
5	110
6	120
7	115
8	118
9	116

If we calculate r, it will be close to 0, implying no relationship.
The size of shoes does not determine intelligence!

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📈 Scatter Diagram (Graphical Representation)

A scatter plot is the easiest way to visualize correlation.

• Positive correlation: Points rise from bottom left to top right.
• Negative correlation: Points fall from top left to bottom right.
• No correlation: Points are scattered randomly.

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🔸 Types of Correlation

Type	Description	Example
Positive Correlation	Both variables move in the same direction	Height & Weight
Negative Correlation	One variable increases, the other decreases	Price & Demand
Zero Correlation	No relationship	Shoe size & IQ
Linear Correlation	Data forms a straight-line relationship	Study time & Marks
Non-linear Correlation	Relationship curves (not straight)	Stress & Productivity

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📊 Other Correlation Measures

Measure	When Used	Notes
Pearson’s r	Both variables are continuous & normally distributed	Most common
Spearman’s rank (ρ)	Data is ordinal or not normally distributed	Based on ranks
Kendall’s tau (τ)	Small samples or tied ranks	Non-parametric

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📘 Example 4: Spearman’s Rank Correlation (ρ)

Student	Math Rank	Science Rank
A	1	2
B	2	1
C	3	3
D	4	4
E	5	5

Step 1: Compute difference in ranks (d)

Student	Math Rank	Science Rank	d	d²
A	1	2	-1	1
B	2	1	1	1
C	3	3	0	0
D	4	4	0	0
E	5	5	0	0

Step 2: Apply formula:

✅ Result: Strong positive correlation (ρ = 0.9)

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📏 Key Points to Remember

• Correlation does not imply causation.
  (E.g., ice cream sales and drowning incidents are correlated due to hot weather — not cause-effect.)
• Correlation measures association, not influence.
• Outliers can significantly distort the correlation coefficient.
• Always visualize with a scatter plot before interpreting results.

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💡 Real-World Applications

• Business: Sales vs. marketing spend
• Agriculture: Rainfall vs. crop yield
• Economics: GDP vs. employment rate
• Health: Exercise vs. body mass index (BMI)
• Education: Study time vs. exam performance

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📚 Further Reading

• Field, A. (2022). Discovering Statistics Using SPSS. Sage Publications.
• Gujarati, D. N. (2020). Basic Econometrics. McGraw Hill.
• Jim Frost, Statistics by Jim – Correlation Explained Simply
• NIST e-Handbook: https://www.itl.nist.gov/div898/handbook/eda/section3/eda35.htm