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Statistics

Confidence Interval (95%)

Mean ± 1.96 σ/√n. Free online Confidence Interval (95%). Calculate confidence interval (95%) online — fast, accurate, mobile-friendly, no signup needed.

Lower
48.04
Upper
51.96

Derivation

  1. ├── 01Givenmean = 50, sigma = 10, n = 100
  2. ├── 02FormulaLower: t-1.96 × a / √(n)
  3. ├── 03Substitutet-1.96 × a / √(100)
  4. ├── 04Compute Lower48.04
  5. ├── 05FormulaUpper: t+1.96 × a / √(n)
  6. ├── 06Substitutet+1.96 × a / √(100)
  7. └── 07Compute Upper51.96
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§01What is

Understanding the Confidence Interval (95%)

The Confidence Interval (95%) computes Lower from 3 inputs: mean, std dev, n. Mean ± 1.96 σ/√n.

Statistics is how we make sense of noisy real-world data. Whether you’re analysing survey results, sports scores, or business metrics, a statistics calculator gives you the exact formula-based answer so you can focus on the interpretation. The Confidence Interval (95%) sits in that toolkit — it mean ± 1.96 σ/√n. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Lower = t-1.96 × a / √(n) | Upper = t+1.96 × a / √(n)

Where

mean
Mean
sigma
Std dev
n
n
Lower
Output value
Upper
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Mean = 50, Std dev = 10, n = 100.

  1. 01Start by noting the input — Mean: 50.
  2. 02Start by noting the input — Std dev: 10.
  3. 03Start by noting the input — n: 100.
  4. 04Substitute these values into the formula: Lower = t-1.96 × a / √(n) | Upper = t+1.96 × a / √(n)
  5. 05Compute Lower: the calculator returns 48.04.
  6. 06Compute Upper: the calculator returns 51.96.
  7. 07Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Confidence Interval (95%) Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Mean halved

mean = 25 (from 50)

Keep every other input at its default and halve the mean. See how lower responds.

  1. 01New Mean: 25
  2. 02Baseline Lower: 48.04
  3. 03New Lower: 23.04
  4. 04Lower decreases by 52% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Mean doubled

mean = 100 (from 50)

Keep every other input at its default and double the mean. See how lower responds.

  1. 01New Mean: 100
  2. 02Baseline Lower: 48.04
  3. 03New Lower: 98.04
  4. 04Lower increases by 104.1% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Std dev halved

sigma = 5 (from 10)

Keep every other input at its default and halve the std dev. See how lower responds.

  1. 01New Std dev: 5
  2. 02Baseline Lower: 48.04
  3. 03New Lower: 49.02
  4. 04Lower increases by 2% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Std dev doubled

sigma = 20 (from 10)

Keep every other input at its default and double the std dev. See how lower responds.

  1. 01New Std dev: 20
  2. 02Baseline Lower: 48.04
  3. 03New Lower: 46.08
  4. 04Lower decreases by 4.1% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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