What Is a False Negative in a Confusion Matrix?

A false negative is a mistake in which a classifier misses a positive case: the true outcome is positive, but the prediction is negative. In a medical test, that might mean the test reports “no disease” for someone who has the disease.

The key is to identify which class is defined as positive and then compare the prediction with the actual outcome. The matrix’s layout can vary, so read its row and column labels rather than relying on a cell’s position alone.

Locate the false negative

A binary confusion matrix compares actual classes with predicted classes. One common layout puts actual classes in rows and predicted classes in columns:

	Predicted positive	Predicted negative
Actual positive	True positive (TP)	False negative (FN)
Actual negative	False positive (FP)	True negative (TN)

A false negative is therefore in the actual positive, predicted negative cell. Some software reverses the axes, so the false negative may appear in a different position. The definition does not change.

“Positive” does not necessarily mean good, harmful, or favorable. It names the class you want to detect. For a spam filter, spam might be positive; for a quality-control system, a defective item might be positive.

Worked example

Suppose a screening test is used on 100 people. Follow-up establishes that 20 actually have the disease and 80 do not. The test produces these results:

	Test positive	Test negative	Actual total
Disease present	16 (TP)	4 (FN)	20
Disease absent	10 (FP)	70 (TN)	80
Predicted total	26	74	100

The four people in the “disease present, test negative” cell are false negatives. They have the positive condition, but the test predicted the negative class.

The false negative rate is the share of actual positive cases that the test missed:

$$\text{FNR} = \frac{\text{FN}}{\text{TP} + \text{FN}}$$

For these results:

$$\text{FNR} = \frac{4}{16 + 4} = \frac{4}{20} = 0.20 = 20\%$$

So the test misses 20% of people who have the disease. The denominator is all actual positive cases—not all 100 people and not all negative predictions.

False negative rate and related measures

The false negative count is the number of missed positive cases, here 4. The false negative rate is a proportion, here 20%. Be explicit about which one you mean.

The false negative rate is the complement of recall, also called sensitivity or the true positive rate:

$$\text{Recall} = \frac{\text{TP}}{\text{TP} + \text{FN}} = 1 - \text{FNR}$$

In the example, recall is $16/20 = 80\%$. This says the test detects 80% of the people who have the disease.

Do not confuse the false negative rate with the false discovery rate. The false discovery rate asks what fraction of positive predictions are wrong, so its denominator is all predicted positives: $\text{FP}/(\text{TP}+\text{FP})$. The false negative rate instead asks what fraction of actual positives were missed.

Why false negatives matter

The practical cost of a false negative depends on the task. Missing a serious disease, a fraudulent transaction, or a dangerous equipment fault may be costly. In other settings, such as screening a large volume of content, some missed cases might be acceptable if reducing false alarms is the priority.

Many classifiers let you adjust a decision threshold. Lowering the threshold often causes more cases to be classified as positive, which can reduce false negatives while increasing false positives. The right trade-off depends on the consequences of each type of error; there is no universally best threshold.

Common mistakes

• Using the wrong positive class: First establish which outcome the task treats as positive. A matrix does not decide this for you.
• Assuming a fixed cell position: Check the labels for actual and predicted classes; software conventions differ.
• Dividing by the wrong total: For the false negative rate, divide FN by TP + FN, the number of actual positives.
• Calling every negative prediction a false negative: A negative prediction is correct when the actual class is negative. It is a false negative only when the actual class is positive.
• Treating a low count as a low rate: When actual positive cases are rare, even a small number of missed cases can represent a substantial false negative rate.

For a multiclass problem, the same idea applies after choosing one class as the positive class and treating the other classes as negative. You can also compute false negatives separately for each class using a one-versus-rest comparison.