The Confusion Matrix Explained
Learners will build a confusion matrix from predictions and ground truth, then derive true positives, false negatives, and related counts.
The Confusion Matrix Explained is a free Machine Learning Academy lesson on CoddyKit — lesson 3 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the Machine Learning Academy learning path, one of 4 lessons in the course, and your progress syncs across the web and the CoddyKit app.
Beyond Accuracy: Understanding Errors
Accuracy tells you the percentage of correct predictions, but it treats all errors as equal. In reality, different types of errors have very different consequences. A spam filter that sends a legitimate email to the spam folder is annoying; a medical diagnosis model that misses a cancer case is life-threatening.
The confusion matrix is the tool that breaks down classifier performance into a 2×2 table (for binary classification) showing every possible combination of actual and predicted labels. It reveals not just how much the model is wrong but how it is wrong.
The Four Cells of the Confusion Matrix
For binary classification with classes Positive (1) and Negative (0), the confusion matrix has four cells:
- True Positive (TP) — actual positive, predicted positive. Correct detection.
- True Negative (TN) — actual negative, predicted negative. Correct rejection.
- False Positive (FP) — actual negative, predicted positive. Type I error; false alarm.
- False Negative (FN) — actual positive, predicted negative. Type II error; missed detection.
All four counts are needed to fully characterise a classifier's behaviour.
import numpy as np
# Medical test example:
# Positive = disease present, Negative = disease absent
y_true = np.array([1, 0, 1, 1, 0, 0, 1, 0, 1, 0])
y_pred = np.array([1, 0, 1, 0, 1, 0, 1, 0, 0, 0])
TP = ((y_pred == 1) & (y_true == 1)).sum()
TN = ((y_pred == 0) & (y_true == 0)).sum()
FP = ((y_pred == 1) & (y_true == 0)).sum()
FN = ((y_pred == 0) & (y_true == 1)).sum()
print(f'TP={TP} FP={FP}')
print(f'FN={FN} TN={TN}')
print(f'Total: {TP+TN+FP+FN}')Building the Confusion Matrix with scikit-learn
Scikit-learn's confusion_matrix() function takes the actual labels and predicted labels and returns the matrix as a NumPy array. The convention in scikit-learn is rows = actual class, columns = predicted class.
For binary classification with labels [0, 1], the matrix layout is:
- Row 0, Col 0 = TN (actual 0, predicted 0)
- Row 0, Col 1 = FP (actual 0, predicted 1)
- Row 1, Col 0 = FN (actual 1, predicted 0)
- Row 1, Col 1 = TP (actual 1, predicted 1)
from sklearn.metrics import confusion_matrix
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import load_breast_cancer
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
X, y = load_breast_cancer(return_X_y=True)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
scaler = StandardScaler()
X_train_s = scaler.fit_transform(X_train)
X_test_s = scaler.transform(X_test)
model = LogisticRegression(max_iter=1000)
model.fit(X_train_s, y_train)
y_pred = model.predict(X_test_s)
cm = confusion_matrix(y_test, y_pred)
print('Confusion matrix:')
print(cm)
print('\nRows=Actual, Cols=Predicted')Visualising the Confusion Matrix as a Heatmap
A confusion matrix visualised as a heatmap makes the counts immediately readable. Large numbers on the diagonal (top-left to bottom-right) indicate correct predictions. Large off-diagonal numbers indicate errors that need investigation.
Use seaborn's heatmap() with annot=True and fmt='d' to display integer counts inside each cell. Adding target class names makes the plot self-explanatory without needing to consult a key.
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.metrics import confusion_matrix
cm = confusion_matrix(y_test, y_pred)
class_names = ['Malignant', 'Benign'] # breast cancer dataset
plt.figure(figsize=(7, 6))
sns.heatmap(
cm,
annot=True,
fmt='d',
cmap='Blues',
xticklabels=class_names,
yticklabels=class_names
)
plt.ylabel('Actual Label')
plt.xlabel('Predicted Label')
plt.title('Confusion Matrix — Breast Cancer Classifier')
plt.show()Deriving Accuracy from the Confusion Matrix
Every classification metric can be derived from the four confusion matrix cells. Accuracy is the simplest — it is the fraction of predictions that fall on the diagonal (TP + TN) out of all predictions:
Accuracy = (TP + TN) / (TP + TN + FP + FN)
All predictions are correct. All predictions = TP + TN + FP + FN. If your test set has 100 examples and you have TP=40, TN=50, FP=5, FN=5, then accuracy = 90/100 = 0.90. But the 5 FN errors (missed positives) may be far more consequential than the 5 FP errors.
from sklearn.metrics import confusion_matrix, accuracy_score
import numpy as np
y_true = np.array([1, 0, 1, 1, 0, 0, 1, 0, 1, 0])
y_pred = np.array([1, 0, 1, 0, 1, 0, 1, 0, 0, 0])
cm = confusion_matrix(y_true, y_pred)
TN, FP, FN, TP = cm.ravel() # for 2x2 matrix
# Manual accuracy
manual_acc = (TP + TN) / (TP + TN + FP + FN)
sklearn_acc = accuracy_score(y_true, y_pred)
print(f'TP={TP}, TN={TN}, FP={FP}, FN={FN}')
print(f'Manual accuracy: {manual_acc:.2f}')
print(f'sklearn accuracy: {sklearn_acc:.2f}')Type I and Type II Errors
In statistics and ML, errors have formal names that reflect their severity in different contexts:
- Type I error = False Positive. You raise the alarm when there is nothing to worry about. Example: flagging a legitimate transaction as fraud, sending someone to hospital unnecessarily.
- Type II error = False Negative. You miss a real event. Example: approving a fraudulent transaction, sending a cancer patient home as healthy.
In most real-world classification problems, Type I and Type II errors have very different costs, which is why a single accuracy number is insufficient for decision-making.
The Confusion Matrix for Multi-Class Problems
For a k-class problem, the confusion matrix is k×k. Each row represents an actual class and each column represents a predicted class. Diagonal elements are correct predictions; off-diagonal elements are misclassifications.
Multi-class confusion matrices reveal which classes the model confuses most. For example, a digit recogniser might frequently confuse 4s and 9s because they look similar. This insight guides targeted improvements: collect more training examples of confused classes, add features that distinguish them, or post-process predictions for those specific pairs.
from sklearn.metrics import confusion_matrix
from sklearn.tree import DecisionTreeClassifier
from sklearn.datasets import load_digits
from sklearn.model_selection import train_test_split
import seaborn as sns
import matplotlib.pyplot as plt
X, y = load_digits(return_X_y=True)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = DecisionTreeClassifier(max_depth=10)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
cm = confusion_matrix(y_test, y_pred)
plt.figure(figsize=(9, 7))
sns.heatmap(cm, annot=True, fmt='d', cmap='Blues')
plt.title('Digit Confusion Matrix (10 classes)')
plt.show()Normalised Confusion Matrix
When classes have different numbers of examples (class imbalance), absolute counts in the confusion matrix can be misleading. A normalised confusion matrix divides each row by the total number of actual examples in that class, converting raw counts to recall rates (proportion correctly identified).
A perfect classifier has 1.0 on every diagonal cell and 0.0 everywhere else. A normalised matrix makes it easy to compare per-class performance regardless of class size.
from sklearn.metrics import confusion_matrix
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
cm = confusion_matrix(y_test, y_pred)
# Normalise by row (actual class total)
cm_norm = cm.astype(float) / cm.sum(axis=1, keepdims=True)
plt.figure(figsize=(9, 7))
sns.heatmap(cm_norm, annot=True, fmt='.2f', cmap='Blues')
plt.ylabel('Actual Class')
plt.xlabel('Predicted Class')
plt.title('Normalised Confusion Matrix (row = recall per class)')
plt.show()
# Per-class recall from diagonal
print('Per-class recall:', np.diag(cm_norm).round(3))ConfusionMatrixDisplay: The Official scikit-learn Tool
Scikit-learn provides ConfusionMatrixDisplay as a high-level tool for plotting confusion matrices without needing seaborn. You can create it directly from the confusion matrix array or by calling from_estimator() which runs prediction internally.
Setting display_labels to your class names makes the plot immediately self-documenting. The display also includes an optional colour bar that helps readers estimate values before reading exact numbers.
from sklearn.metrics import ConfusionMatrixDisplay
from sklearn.datasets import load_breast_cancer
X, y = load_breast_cancer(return_X_y=True)
class_names = load_breast_cancer().target_names # ['malignant', 'benign']
# Method 1: from_estimator (runs predict internally)
disp = ConfusionMatrixDisplay.from_estimator(
model,
X_test_s,
y_test,
display_labels=class_names,
cmap='Blues'
)
disp.ax_.set_title('Confusion Matrix')What the Confusion Matrix Tells You
A well-trained classifier should have most examples on the diagonal. When you see large off-diagonal numbers, diagnose the cause:
- Many FP (false positives): the model is too aggressive — lower the decision threshold or add regularisation.
- Many FN (false negatives): the model misses too many real positives — lower the threshold or improve recall.
- Symmetric errors between class A and B: the two classes are too similar in feature space — add distinguishing features.
- Errors concentrated in one actual class: that class is underrepresented or harder to learn — oversample it or collect more data.
Confusion Matrix to Full Classification Report
Every metric in the classification_report() is derived from the confusion matrix. Understanding the matrix is understanding all metrics at once. From TP, TN, FP, FN you can compute:
- Accuracy = (TP+TN)/N
- Precision = TP/(TP+FP)
- Recall = TP/(TP+FN)
- Specificity = TN/(TN+FP)
- F1 = 2×Precision×Recall/(Precision+Recall)
In the next lesson you will master all these derived metrics and learn when each one is the right choice.
from sklearn.metrics import classification_report, confusion_matrix
import numpy as np
y_true = np.array([1, 0, 1, 1, 0, 0, 1, 0, 1, 0])
y_pred = np.array([1, 0, 1, 0, 1, 0, 1, 0, 0, 0])
cm = confusion_matrix(y_true, y_pred)
TN, FP, FN, TP = cm.ravel()
# Compute all metrics from raw counts
precision = TP / (TP + FP) if (TP + FP) > 0 else 0
recall = TP / (TP + FN) if (TP + FN) > 0 else 0
f1 = 2 * precision * recall / (precision + recall) if (precision + recall) > 0 else 0
print(f'Precision: {precision:.2f}')
print(f'Recall: {recall:.2f}')
print(f'F1-Score: {f1:.2f}')
print('\nscikit-learn report:')
print(classification_report(y_true, y_pred))Quick Check
Test your understanding of Machine Learning with Python concepts from this lesson.
Lesson Recap
In this lesson you learned: the confusion matrix breaks down classifier errors into True Positives, True Negatives, False Positives, and False Negatives, False Negatives (Type II errors) and False Positives (Type I errors) have very different real-world consequences, and all classification metrics — accuracy, precision, recall, F1 — are derived from these four confusion matrix counts. Next up we master precision, recall, and F1-score in depth, and learn the critical trade-off between them that guides real-world classifier evaluation.
Frequently asked questions
Is the “The Confusion Matrix Explained” lesson free?
Yes — the full text of “The Confusion Matrix Explained” is free to read here on the web, and the Machine Learning Academy course includes 4 lessons in total. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the Machine Learning Academy course, upgrade to CoddyKit PRO.
What will I learn in “The Confusion Matrix Explained”?
Learners will build a confusion matrix from predictions and ground truth, then derive true positives, false negatives, and related counts. You practise Machine Learning Academy with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.
Do I need any experience to start Machine Learning Academy?
No prior experience is required. Machine Learning Academy on CoddyKit is structured for beginners through advanced learners; this is — lesson 3 of 4, so you can start here or from the beginning and move at your own pace.
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