Imagine throwing a handful of rice onto a table. The grains scatter, but you can still lay a ruler through them in a way that best represents the overall direction. Linear regression is that ruler — it finds the line that minimises the total distance from every point to the line.

The core idea:

Data points rarely sit perfectly on a line. Linear regression finds the line that is the least wrong across all points.

import matplotlib.pyplot as plt
from scipy import stats
x = [5, 7, 8, 7, 2, 17, 2, 9, 4, 11, 12, 9, 6]
y = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
slope, intercept, r, p, std_err = stats.linregress(x, y)
print(f"Slope: {round(slope, 2)}")
# Output: Slope: -1.75
print(f"Intercept: {round(intercept, 2)}")
# Output: Intercept: 103.1

What slope and intercept mean:

> 💡 Key Insight: The equation of the line is y = slope × x + intercept. This one formula is your prediction machine — plug in any x value and get a predicted y. That is the entire point of regression: turning a pattern in past data into a formula for the future.

Think of linregress() as a Swiss Army knife for linear relationships. You hand it two arrays, and it hands back five numbers that tell the complete story of the relationship.

The five return values:

from scipy import stats
x = [5, 7, 8, 7, 2, 17, 2, 9, 4, 11, 12, 9, 6]
y = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
slope, intercept, r, p, std_err = stats.linregress(x, y)
print(f"Slope:     {round(slope, 4)}")
# Output: Slope:     -1.7512
print(f"Intercept: {round(intercept, 4)}")
# Output: Intercept: 103.1045
print(f"R-value:   {round(r, 4)}")
# Output: R-value:   -0.7586
print(f"P-value:   {round(p, 4)}")
# Output: P-value:   0.0027
print(f"Std Error: {round(std_err, 4)}")
# Output: Std Error: 0.4553

Building the prediction function:

def predict(x_value):
    return slope * x_value + intercept
print(f"Predicted speed for 10-year car: {round(predict(10), 2)}")
# Output: Predicted speed for 10-year car: 85.59

> 💡 Key Insight: The p-value tells you whether the relationship is real or just a coincidence. A p-value below 0.05 means there is less than a 5% chance this relationship happened by random luck. In our car example, p=0.0027 — very strong evidence that age really does affect speed.

A scatter plot shows what happened. Adding a regression line shows the trend — and where you expect future data points to land.

Drawing the line through the dots:

import matplotlib.pyplot as plt
from scipy import stats
x = [5, 7, 8, 7, 2, 17, 2, 9, 4, 11, 12, 9, 6]
y = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
slope, intercept, r, p, std_err = stats.linregress(x, y)
def myfunc(x):
    return slope * x + intercept
mymodel = list(map(myfunc, x))
plt.scatter(x, y)
plt.plot(x, mymodel, color="red")
plt.xlabel("Car Age (years)")
plt.ylabel("Speed (km/h)")
plt.title(f"Linear Regression (r={round(r, 2)})")
plt.show()
# Output: Scatter plot with a red downward-sloping line

Reading the result:

• The dots = actual data (what really happened)
• The line = the model''s best guess (the trend)
• Dots close to the line = good fit
• Dots far from the line = poor fit or outliers

# The distance from each dot to the line = the "error"
errors = [y[i] - myfunc(x[i]) for i in range(len(x))]
print([round(e, 1) for e in errors])
# Output: [4.6, -4.8, -2.1, -2.8, 11.4, -12.3, 3.4, -0.4, -2.1, -5.8, -5.1, -2.4, 2.4]

> 💡 Key Insight: The regression line minimises the sum of squared errors — that is why it is called "least squares regression." It does not try to pass through any single point; it finds the line that is collectively closest to ALL points. Some dots will be above, some below — balance is the goal.

Linear regression finds the straight line that best fits your data. Use SciPy's linregress() to compute slope and intercept.