Accuracy and the Confusion Matrix

Definition

A confusion matrix is a table that compares a classifier's predictions with the true labels. For two classes it has four counts: true positives, false positives, true negatives and false negatives.

In simple words

Think of a fire alarm. It rings during a fire: true positive. It rings when someone burns toast: false positive. It stays silent on a normal day: true negative. It stays silent during a fire: false negative, the dangerous one. A single accuracy number can't show you this, and this table can.

Why it matters

Different errors have different costs. Missing a cancer is far worse than a false alarm, and blocking a real customer's payment has a different cost from missing a fraud. All the common metrics are calculated from these four numbers.

How it works

	Predicted: Yes	Predicted: No
Actually: Yes	True Positive (TP)	False Negative (FN): missed it
Actually: No	False Positive (FP): false alarm	True Negative (TN)

The maths

Accuracy, from the four counts

Accuracy = \frac{TP + TN}{TP + TN + FP + FN}

100 patients: 10 have the disease. The model catches 8, and flags 5 healthy people wrongly.

TP = 8    FN = 2
FP = 5    TN = 85
Accuracy = (8 + 85) / 100 = 93%

93% sounds fine, but the model still missed 2 of 10 patients. A model that says "healthy" to everyone would score 90% and catch nobody. That's why accuracy alone can't be trusted on imbalanced data.

In code

confusion.py

from sklearn.metrics import accuracy_score, confusion_matrix

actual =    [1] * 10 + [0] * 90
predicted = [1] * 8 + [0] * 2 + [1] * 5 + [0] * 85

tn, fp, fn, tp = confusion_matrix(actual, predicted).ravel()
print("TP:", tp, " FN:", fn, " FP:", fp, " TN:", tn)
print("Accuracy:", accuracy_score(actual, predicted))

everyone_healthy = [0] * 100
print("Always says healthy:", accuracy_score(actual, everyone_healthy))

Output

TP: 8  FN: 2  FP: 5  TN: 85
Accuracy: 0.93
Always says healthy: 0.9

Common mistakes

Judging a model on accuracy alone, particularly on imbalanced classes.
Mixing up false positives and false negatives. Think about what each costs in your case.

Check yourself

1. A spam filter sends a real email to spam. This is a:

2. TP = 40, TN = 50, FP = 5, FN = 5. What's the accuracy?

3. Which error is usually worst in disease screening?

Summary

Four counts: TP, FP, TN and FN.
All common classification metrics come from them.
Accuracy can mislead when classes are imbalanced.