R squared and adjusted R squared

R squared (R²) is a performance metric for regression that measures how much of the variation in the output a model explains, and adjusted R squared is a version that penalises extra features which do not help.

Last updated: 05 Oct, 2026 · scikit-learn 1.9.1

The cost function tells training which line is best. R² tells you afterwards how good that best line is, on a scale where 1 means perfect.

Comparing the best fit line with the mean line

• SS_res, the sum of residuals, adds up the squared distances from each point to its predicted point on the best fit line.
• SS_total adds up the squared distances from each point to the mean line, a horizontal line at ȳ.
• The mean line is a worse fit, so SS_total is the higher number. Low divided by high is a small number, and 1 minus a small number is big: an R² of 90%, say, means the line fits well.
• R² can be negative, but only when the line is worse than the mean line, such as a line drawn above every point. A fitted line is at least as good as the mean, so this is rare.

Adding features that do not matter

Predict a house price. With bedrooms alone, say R² is 85%. Add location, which is correlated with price: R² rises to 90%. Now add gender, male or female, of the people living there. Gender has nothing to do with price, yet R² still creeps up, to 91%. Picking by R² alone would choose the 91% model, which carries a useless feature. Adjusted R² prevents that.

Penalising extra predictors with adjusted R²

As p grows, N − p − 1 gets smaller, so the fraction gets bigger, and 1 minus it gets smaller. Unless the new feature raises R² enough to pay for itself, adjusted R² falls. The board's example: with p = 2, R² is 90% and adjusted R² 86%; with p = 3, R² rises to 91% while adjusted R² drops to 82%. Adjusted R² is always at or below R², which answers the video's interview question at the end of the clip: which of the two is bigger?

The board's 86% and 82% are illustrative values. They do not come from one N: with N = 8 houses the formula gives 86% for p = 2 and 84% for p = 3. The direction is the point: R² up, adjusted R² down.

Computing R² and adjusted R²

R² by hand and with r2_score

On the age and weight line from Simple linear regression, the formula and scikit-learn's r2_score should agree. A flat line at 80, above every point, shows a negative R².

import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score

age = np.array([[24], [25], [21], [27]])
weight = np.array([62, 63, 72, 62])
pred = LinearRegression().fit(age, weight).predict(age)

ss_res = np.sum((weight - pred) ** 2)
ss_total = np.sum((weight - weight.mean()) ** 2)
print("1 - SS_res / SS_total:  ", round(1 - ss_res / ss_total, 4))
print("r2_score(y_true, y_pred):", round(r2_score(weight, pred), 4))
print("a line above every point:", round(r2_score(weight, np.full(4, 80)), 2))

1 - SS_res / SS_total:   0.7599
r2_score(y_true, y_pred): 0.7599
a line above every point: -13.15

Adjusted R² as a function

def adjusted_r2(r2, n, p):
    # n = number of data points, p = number of predictors
    return 1 - (1 - r2) * (n - 1) / (n - p - 1)

Checking the board's numbers

print("p = 2, R² = 0.90, N = 8:", round(adjusted_r2(0.90, n=8, p=2), 4))
print("p = 3, R² = 0.91, N = 8:", round(adjusted_r2(0.91, n=8, p=3), 4))

p = 2, R² = 0.90, N = 8: 0.86
p = 3, R² = 0.91, N = 8: 0.8425

A house table with a useless column

The video gives no house table, so the rows here are generated: price depends on bedrooms and location plus noise, and gender is random.

import numpy as np
import pandas as pd

rng = np.random.default_rng(6)
n = 20
houses = pd.DataFrame({
    "bedrooms": rng.integers(1, 6, n),
    "location": rng.integers(1, 11, n),   # a 1-10 location score
    "gender": rng.integers(0, 2, n),      # who lives there: no link to the price
})
houses["price"] = 30 * houses.bedrooms + 5 * houses.location + rng.normal(0, 12, n)

Adding bedrooms, location and gender

from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score

for cols in (["bedrooms"], ["bedrooms", "location"], ["bedrooms", "location", "gender"]):
    model = LinearRegression().fit(houses[cols], houses.price)
    r2 = r2_score(houses.price, model.predict(houses[cols]))
    print(f"{' + '.join(cols):28} R² = {r2:.4f}   adjusted R² = {adjusted_r2(r2, n, len(cols)):.4f}")

bedrooms                     R² = 0.8112   adjusted R² = 0.8008
bedrooms + location          R² = 0.9551   adjusted R² = 0.9499
bedrooms + location + gender R² = 0.9561   adjusted R² = 0.9478

What R² and adjusted R² printed

• The formula and r2_score agree on the age table: 0.7599. The line explains about three quarters of the variation in those four weights.
• The flat line at 80 scores −13.15: it is far worse than the mean line, so SS_res is 14 times SS_total.
• The board's numbers with N = 8 give 0.86 for p = 2, as the board says, and 0.8425 for p = 3, not 0.82.
• Location lifts both scores, R² from 0.8112 to 0.9551. Gender lifts R² by a hair (0.9551 to 0.9561) but lowers adjusted R² (0.9499 to 0.9478): the same pattern as the board's 90% to 91% and 86% to 82%.

R² vs adjusted R²

Where you use R² and adjusted R²

• Reporting a regression model: R² on the test set says what share of the variation it explains.
• Feature selection: keep a new column only if adjusted R² goes up.
• Interviews: why R² keeps rising with useless features, and why adjusted R² is never larger.

Related

• Previous: Ordinary least squares
• Next: MSE, MAE and RMSE
• Reference: r2_score