Why Linear Regression Exists

1 · Observe

You are asked to price a flat nobody has sold yet

A buyer asks what a 75 sq m flat should cost. You have eight recent sales. The simplest answer ignores size: quote the average price. That is a fair opening move, and a weak one, because a 40 sq m flat and a 110 sq m flat do not sell for the same money.

Linear regression is the next simplest idea. Draw one straight line through the sales and read the price off it. Everything else in this module is about drawing that line well and knowing when to distrust it.

2 · Express

A prediction is a rule from inputs to a number

The input (feature) is area in sq m. The output (target) is price in ₹ lakh. A model is a rule that turns the first into an estimate of the second. The baseline rule says 'always the average'. The line rule says 'price depends on area'.

3 · Visualize

Eight flats, two rules

Areas 40 to 110 sq m with prices ₹30 to ₹84 lakh.

Using area cuts the typical miss from about ₹15.6 lakh to ₹4.0 lakh, a 75 percent reduction. For a 75 sq m flat, the line says 0.806 × 75 − 6.82 = ₹53.6 lakh, which happens to equal the average, because 75 is the average area and a fitted line passes through the point of averages.

4 · Discover

What 'fitted' means

Among all straight lines, we choose the one with the smallest average squared miss. The next lessons show how to write that line down, measure its misses, find it by repeated small steps, and extend it to many features.

5 · Apply

Two worked examples

Example 1, a small flat. 45 sq m: 0.806 × 45 − 6.82 = 29.5 lakh. The average rule would say 53.6, over 24 lakh too high.

Example 2, a big flat. 110 sq m: 0.806 × 110 − 6.82 = 81.8 lakh, against a recorded sale of 84. The line misses by 2.2 lakh, the average rule by 30.4 lakh.

6 · Where it goes wrong

Three ways it goes wrong

7 · In Python

The line against the average

Here is how to implement it, step by step. The code below does exactly these steps, and every line is editable.

1. Put the eight sales into two lists: area and price.
2. Make the baseline: guess the average price for every flat, and measure its average miss (15.62 lakh).
3. Fit a straight line with np.polyfit(area, price, 1). It returns the slope and the starting height.
4. Measure the line's average miss (3.98 lakh) and compare it with the baseline.
5. Use the line on a flat nobody has sold: price = slope × area + height.

import numpy as np

area = np.array([40, 50, 60, 70, 80, 90, 100, 110], float)
price = np.array([30, 34, 36, 52, 55, 58, 80, 84], float)           # lakh rupees

guess_avg = np.full_like(price, price.mean())
w, b = np.polyfit(area, price, 1)
guess_line = w * area + b
mae = lambda p: np.abs(price - p).mean()
print("average price:", round(price.mean(), 1), "lakh | MAE of always guessing it:", round(mae(guess_avg), 2))
print("fitted line: price = %.3f x area + %.2f" % (w, b), "| MAE of the line:", round(mae(guess_line), 2))
print("a 75 sq m flat -> line says", round(w * 75 + b, 1), "lakh")

average price: 53.6 lakh | MAE of always guessing it: 15.62
fitted line: price = 0.806 x area + -6.82 | MAE of the line: 3.98
a 75 sq m flat -> line says 53.6 lakh