Home » Statistics & Probability Calculators » CI for the Difference of Two Proportions

CI for the Difference of Two Proportions

Home › Statistics & Probability Calculators › CI for the Difference of Two Proportions

Confidence Interval for the Difference of Two Proportions

Use this tool to estimate how much two population proportions differ: the gap in approval between two age groups, in conversion rate between two designs, or in pass rate between two schools. Enter the successes and sizes of both groups and a confidence level.

The interval is (p̂₁ − p̂₂) ± z√[p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂]. Unlike the two-proportion z-test, no pooling is used because the interval does not assume the proportions are equal. The steps show each sample proportion, the difference, the standard error, the critical z, the margin of error and the interval, and a note says whether zero is inside it.

The normal approximation needs roughly 10 successes and 10 failures in each group. For very small groups, exact or score-based intervals are more accurate. The two samples must be independent; for before-and-after proportions on the same people, a paired method such as McNemar’s test is needed instead.

Result and step-by-step solution

95% CI for p₁ − p₂
−0.0104 to 0.1837

(p̂₁ − p̂₂) ± ME = 0.0867 ± 0.0971

  1. Sample proportions
    p̂₁ = 84 ÷ 200 = 0.42; p̂₂ = 60 ÷ 180 = 0.3333
  2. Difference
    p̂₁ − p̂₂ = 0.0867
  3. Standard error (unpooled)
    SE = √[p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂] = √[0.42 × 0.58/200 + 0.3333 × 0.6667/180] = 0.0495
  4. Critical value
    z = 1.96
  5. Margin of error and interval
    ME = 1.96 × 0.0495 = 0.0971; interval = (−0.0104, 0.1837)

The interval contains 0, so the two proportions may be equal.

Frequently asked questions

Why isn't the proportion pooled here?
Pooling assumes p₁ = p₂, which is the null hypothesis of a test. A confidence interval makes no such assumption, so each group’s own proportion is used in the standard error.
How do I interpret a negative interval?
If both ends are negative, group 1’s proportion is lower than group 2’s by between those amounts (in absolute terms).
What sample sizes are needed?
Each group should have at least 10 successes and 10 failures for the normal approximation to be reasonable.
Is this the same as the difference in percentages?
Yes, once multiplied by 100. An interval of 0.05 to 0.15 means group 1 is 5 to 15 percentage points higher.

Related calculators

Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

Scroll to Top