Confusion matrix accuracy Equation, Derivation & Rearrangements

Core idea

Overview

Accuracy is the most intuitive performance measure for classification models, representing the ratio of correctly predicted observations to the total number of samples. It combines both true positive and true negative results to provide a broad assessment of how often the classifier is correct across all classes.

When to use: Accuracy is best utilized when the target classes in the dataset are nearly balanced, meaning there is a similar number of samples for each label. It is appropriate when the costs of false positives and false negatives are roughly equal.

Why it matters: It allows stakeholders to quickly grasp the reliability of a system in general terms, such as an OCR engine or a simple sentiment analyzer. High accuracy indicates a model that performs well across the entire distribution, assuming the data is not skewed.

Symbols

Variables

TP = True Positives, TN = True Negatives, FP = False Positives, FN = False Negatives, acc = Accuracy

T P

True Positives

V a r iab l e

T N

True Negatives

V a r iab l e

F P

False Positives

V a r iab l e

F N

False Negatives

V a r iab l e

a cc

Accuracy

V a r iab l e

Walkthrough

Derivation

Confusion Matrix Accuracy

Accuracy measures the proportion of all predictions that were correct: acc = (TP + TN) / (TP + TN + FP + FN). It is reliable only when class distributions are balanced.

The dataset's class distribution is reasonably balanced.
False positives and false negatives carry similar costs.

1

Define the Confusion Matrix Terms

Every prediction falls into one of these four categories. TP and TN are correct; FP and FN are errors.

T P True Positives (correctly predicted positive) T N True Negatives (correctly predicted negative) F P False Positives (negative predicted as positive) F N False Negatives (positive predicted as negative)

2

Sum Correct Predictions

The total number of predictions the classifier got right.

Correct = T P + T N

3

Sum All Predictions

Every sample in the evaluation set, regardless of outcome.

Total = T P + T N + F P + F N

4

Calculate Accuracy

Dividing correct predictions by the total gives accuracy as a value between 0 and 1 (multiply by 100 for %).

a cc = \frac{T P + T N}{T P + T N + F P + F N}

5

Example

A spam filter with TP=45, TN=50, FP=2, FN=3 achieves 95% accuracy.

a cc = \frac{45 + 50}{45 + 50 + 2 + 3} = \frac{95}{100} = 0.95

Note: Accuracy alone can be misleading on imbalanced datasets — consider also precision, recall, and F1-score.

Result

a cc = \frac{45 + 50}{45 + 50 + 2 + 3} = \frac{95}{100} = 0.95

Source: A-Level Data & Computing — Machine Learning

Free formulas

Rearrangements

Solve for $a cc$

Make acc the subject

a cc = \frac{T P + T N}{T P + T N + F P + F N}

acc is already the subject of the formula.

Difficulty: 1/5

Solve for $T P$

Make TP the subject

T P = \frac{a cc ( T N + F P + F N ) - T N}{1 - a cc}

Rearrange the confusion matrix accuracy formula to express True Positives (TP) as the subject in terms of Accuracy (acc), True Negatives (TN), False Positives (FP), and False Negatives (FN).

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line because True Positives appear as a first-degree term, meaning accuracy increases linearly as True Positives rise, provided the other variables remain constant. For a student of Data and Computing, this linear relationship shows that small x-values represent models struggling to identify positive cases, while large x-values indicate a model successfully capturing more of the target class. The most important feature is the y-intercept determined by the negative True Negatives, which highl

Graph type: linear

Why it behaves this way

Intuition

Imagine a 2x2 grid (the confusion matrix) where actual outcomes are rows and predicted outcomes are columns; accuracy is the sum of the diagonal elements (correct predictions)

acc

The proportion of correctly classified instances out of the total number of instances.

A higher value indicates that the model is generally more correct across all predictions.

TP

True Positives: The number of positive instances that were correctly identified by the model.

These are the cases where the model correctly found what it was looking for.

TN

True Negatives: The number of negative instances that were correctly identified by the model.

These are the cases where the model correctly identified what was *not* the target.

FP

False Positives: The number of negative instances that were incorrectly identified as positive by the model.

This represents a 'false alarm' or a Type I error, where the model incorrectly predicted a positive outcome.

FN

False Negatives: The number of positive instances that were incorrectly identified as negative by the model.

This represents a 'miss' or a Type II error, where the model failed to detect a true positive outcome.

Free study cues

Insight

Canonical usage

Accuracy is a dimensionless ratio representing the proportion of correctly classified instances out of the total number of instances, typically expressed as a decimal or percentage.

Common confusion

A common confusion is failing to understand that accuracy, while a numerical value, does not have physical units. It is a proportion and should be interpreted as such, often converted to a percentage for clarity and ease

Dimension note

Accuracy is a dimensionless quantity because it is a ratio of counts (number of correct predictions to total predictions). It represents a proportion and therefore has no physical units.

Unit systems

$T P$ count · True Positives: The number of instances correctly predicted as positive.

$T N$ count · True Negatives: The number of instances correctly predicted as negative.

$F P$ count · False Positives: The number of instances incorrectly predicted as positive (Type I error).

$F N$ count · False Negatives: The number of instances incorrectly predicted as negative (Type II error).

$a cc$ dimensionless · Accuracy: The overall proportion of correct predictions, a value between 0 and 1.

One free problem

Practice Problem

An email spam filter processed 100 messages. It correctly identified 45 as spam and 50 as legitimate. However, it mistakenly marked 2 legitimate emails as spam and failed to catch 3 spam messages. Calculate the accuracy of the filter.

True Positives45

True Negatives50

False Positives2

False Negatives3

Solve for: $a cc$

Hint: Accuracy is the sum of correct predictions (TP and TN) divided by the total number of samples.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Reporting overall model correctness on a test set.

Study smarter

Tips

Always verify class distribution before reporting accuracy as the primary metric.
Incorporate the confusion matrix to see where the specific errors are occurring.
Use accuracy as a baseline to compare different model architectures on the same dataset.

Avoid these traps

Common Mistakes

Ignoring class imbalance.
Using TP only.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Accuracy measures the proportion of all predictions that were correct: acc = (TP + TN) / (TP + TN + FP + FN). It is reliable only when class distributions are balanced.

Accuracy is best utilized when the target classes in the dataset are nearly balanced, meaning there is a similar number of samples for each label. It is appropriate when the costs of false positives and false negatives are roughly equal.

It allows stakeholders to quickly grasp the reliability of a system in general terms, such as an OCR engine or a simple sentiment analyzer. High accuracy indicates a model that performs well across the entire distribution, assuming the data is not skewed.

Ignoring class imbalance. Using TP only.

Reporting overall model correctness on a test set.

Always verify class distribution before reporting accuracy as the primary metric. Incorporate the confusion matrix to see where the specific errors are occurring. Use accuracy as a baseline to compare different model architectures on the same dataset.

References

Sources

Wikipedia: Confusion matrix
An Introduction to Statistical Learning (James, Witten, Hastie, Tibshirani)
Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow by Aurélien Géron
Confusion matrix (Wikipedia article)
Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.).
Géron, A. (2019). Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow (2nd ed.). O'Reilly Media.
A-Level Data & Computing — Machine Learning

Confusion matrix accuracy

Overview

Variables

Derivation

Define the Confusion Matrix Terms

Sum Correct Predictions

Sum All Predictions

Calculate Accuracy

Example

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Precision

Frequently Asked Questions

Sources