How Good is Your Line? Let’s Talk RSS, MSE & RMSE in Linear Regression | by Alakara | Aug, 2025

So let’s say you’re attempting to foretell how lengthy it takes to get your pizza delivered based mostly on how far-off you’re from the restaurant. You accumulate this small, humble dataset:

You, as an clever being, resolve to construct a linear regression mannequin to foretell supply time based mostly on distance. NICE! However now comes the large query:
Statistically talking, how good are your predictions?

That’s the place RSS, MSE and RMSE are available. However earlier than we perceive these three, we have to know what residuals are.

Step 1: Perceive Residuals
A residual is simply:
residual = actual_value — predicted_value
It tells you the way off every prediction was.

For instance:
In case your mannequin says supply will take 18 minutes nevertheless it truly took 20, the residual is -2 (mannequin underestimated by 2 minutes).

We’ll construct on this concept to judge how unhealthy (or good) our mannequin is utilizing three widespread metrics:

The Metrics

1️⃣ RSS: Residual Sum of Squares
That is…properly, the sum of squared residuals clearly, i.e. how a lot whole “oops” your mannequin made.
Formulation:
RSS = Σ(precise − predicted)²

If it isn’t apparent at this level, we take the residuals, sq. them, then take the sum. It’s within the title ❤️

2️⃣ MSE: Imply Squared Error
The typical squared error. With RSS, the bigger the dataset, the larger the worth. To stage the taking part in discipline, simply divide RSS by the variety of information factors. EQUALITY✨ RIGHT?
MSE = RSS / n = (1/n) * Σ(precise − predicted)²

3️⃣ RMSE: Root Imply Squared Error
That is the sq. root of MSE. It places the error again within the unique unit (minutes, on this case). A lot simpler to interpret.

NOTE: Since we squared all of the residuals in RSS and MSE, the items additionally get squared. Because of this I mentioned, RMSE places the error again within the unique unit.

RSS & MSE:

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Okay, sufficient comedy, let’s do that in code:

import numpy as np
import matplotlib.pyplot as plt

# Distance in km
X = np.array([1, 2, 3, 4, 5])
# Precise supply instances in minutes
y_actual = np.array([15, 17, 20, 22, 26])

# Let’s say your mannequin predicted:
y_predicted = np.array([14, 16, 19, 23, 25]) # a bit off, however not horrible

# Calculate residuals
residuals = y_actual — y_predicted

# RSS
RSS = np.sum(residuals ** 2)

# MSE
MSE = RSS / len(y_actual)

# RMSE
RMSE = np.sqrt(MSE)

print(f”Residuals: {residuals}”)
print(f”RSS: {RSS}”)
print(f”MSE: {MSE}”)
print(f”RMSE: {RMSE:.2f}”)

Output:

Residuals: [ 1 1 1 -1 1]
RSS: 5
MSE: 1.0
RMSE: 1.00

Interpretation:
RSS = 5 → Whole squared error throughout all predictions.
MSE = 1.0 → On common, the mannequin is off by 1 squared minute.
RMSE = 1.00 → On common, predictions are off by about 1 minute. Clear and straightforward to interpret.

So, When Do You Use What?

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TL;DR

1. Residuals are how off every prediction is (actual-prediction)
2. RSS is the full squared error (sum of squares of residuals)
3. MSE is the common squared error. (RSS/no. of rows)
4. RMSE is your go-to, sane metric for “how far off are we, on common?” (sq. root of the MSE)

Closing Ideas

Metrics like RMSE assist us quantify the standard of our fashions. Evaluating your fashions will prevent time, cash and dignity 🙂 So earlier than you begin flexing your mannequin, why don’t you strive testing it first?

Subsequent up, we are able to dive into:

• What occurs in case your errors aren’t usually distributed?
• MAE vs RMSE
• Cross-validation
• Regularized regression

Keep tuned for good vibes, good classes and ‘GOOD’ MEMES.