Chapter 6: ML Linear Graphs
First: What is a “Linear Graph” in Machine Learning?
In ML, a linear graph almost always means:
- A straight line drawn on a plot (scatter plot usually)
- That line tries to show the relationship between two things (variables)
- One thing (x-axis) helps predict the other thing (y-axis)
The most famous use → Linear Regression (a supervised ML algorithm)
Linear Regression = We assume the relationship between input & output is straight-line like (linear). We find the best straight line that fits our data points as closely as possible.
Why “linear graph”? Because the model’s prediction is always a straight line (or a flat plane/hyperplane in higher dimensions).
Equation of that magic line (simple version): y = mx + c (or in ML language: y = w * x + b)
- y = predicted value (what we want to guess)
- x = input/feature (what we know)
- m or w = slope (how steep the line is — how much y changes when x changes by 1)
- c or b = intercept (where the line crosses y-axis when x=0)
Real-Life Story Example Everyone Gets — Hyderabad Flat Prices
Imagine you’re looking to buy a 2BHK flat in Hyderabad (Gachibowli area).
You collect data from 99acres/Magicbricks:
- Size of flat (sq ft) → x-axis (independent variable)
- Price in lakhs → y-axis (dependent variable / what we want to predict)
You plot dots:
- 800 sq ft → ₹45 lakh
- 1200 sq ft → ₹68 lakh
- 1500 sq ft → ₹85 lakh
- 1800 sq ft → ₹102 lakh
- 2200 sq ft → ₹125 lakh
When you plot these as dots on graph paper:
- They roughly form an upward sloping pattern
- Not perfect (some flats cheaper due to age/location), but mostly straight trend
Linear Regression job: Draw the best straight line through these dots.
That line might look like: Price (₹ lakh) = 0.055 × Size (sq ft) + 5
- Slope (0.055) → every extra 100 sq ft adds ≈ ₹5.5 lakh
- Intercept (5) → a imaginary “0 sq ft” flat costs ₹5 lakh (maybe land value or base)
Now, for a new flat never seen: 1600 sq ft Model predicts: 0.055 × 1600 + 5 ≈ ₹93 lakh You can check if the seller’s price is fair!
This graph (scatter dots + red straight line) = Linear Graph in ML
Why Do We Draw This Graph? (Super Important Purposes)
- Visualize the relationship
- Is it really linear? (straight-ish)
- Positive slope? (as x increases, y increases)
- Negative? (like car age vs price — older = cheaper)
- See how good the fit is
- If dots are very close to the line → great model (high R², like 0.9)
- Dots scattered far → poor linear fit (maybe need non-linear model)
- Make predictions
- Just pick any x, go up to the line, read y
- Understand errors
- Vertical distance from each dot to line = residual/error
- Goal of linear regression: Minimize sum of squared errors (least squares method)
Classic Simple Example – Study Hours vs Exam Score
Data (10 students):
- Hours studied (x): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- Exam score % (y): 45, 50, 58, 62, 68, 75, 80, 85, 89, 95
Plot → dots go up-right, almost straight.
After linear regression: Score = 5.5 × Hours + 40 (approx)
Graph shows:
- At 0 hours → ~40% (maybe from luck/general knowledge)
- Each extra hour → +5.5% score
- For new student who studied 7.5 hours → predict ≈ 81%
This is the linear graph teachers show in every ML intro class!
What the Graph Looks Like (Picture This)
- X-axis horizontal: Independent variable (hours, size, temperature…)
- Y-axis vertical: Dependent variable (score, price, sales…)
- Blue/black dots: Actual data points
- Red/orange straight line: The fitted regression line (best fit line)
- Sometimes green dashed lines: Showing residuals (errors)
If the line goes up-right → positive correlation Down-right → negative Flat → no relationship (model useless)
When Linear Graphs Fail (Important Warnings)
Not everything is linear!
- House price vs size: mostly linear up to certain point, then flattens (very big houses don’t keep doubling price)
- Salary vs experience: increases fast at first, then slows
- Temperature vs ice cream sales: linear in summer, but zero below 0°C
In these cases → we see curve in scatter plot → don’t force straight line → use polynomial regression, decision trees, etc.
But linear is first try because:
- Simple to understand
- Fast to train
- Easy to interpret (slope tells impact)
Quick Summary Table (Keep in Your Notes)
| Term in ML Linear Graph | What it Means | Real Example (Flat Price) |
|---|---|---|
| Scatter Plot | Dots of actual data | Each dot = one flat (size vs price) |
| Regression Line / Best Fit Line | The straight line we find | Price = slope × size + intercept |
| Slope (m or w) | How much y changes per 1 unit x | +₹5500 per extra sq ft |
| Intercept (c or b) | y value when x=0 | Base price ≈ ₹5 lakh |
| Residuals | Vertical distance dot to line | Prediction error for each flat |
| R² (goodness of fit) | 0–1 score (1 = perfect line fit) | 0.92 → very good linear relationship |
Final Teacher Words (2026)
ML Linear Graphs = the visual heart of Linear Regression — the simplest, most taught supervised ML algorithm.
It teaches us:
- Assume straight-line relationship
- Find best line by minimizing errors
- Use that line to predict new things
- Always plot first — eye test tells if linear makes sense!
Understood the concept? 🌟
Want next step?
- How exactly computer finds the “best” line (gradient descent story)?
- Python code to make this graph yourself (using sklearn)?
- Non-linear examples comparison?
Just tell me — class is still going! 🚀
