Confidence Interval Calculator (Mean, Proportion & TI-84 Steps)

This free confidence interval calculator finds the interval, margin of error and critical value for a population mean (z-interval when σ is known, t-interval when it is not), a proportion, the difference between two means and the difference between two proportions. You can type summary statistics or paste raw data, pick 80%, 90%, 95%, 98%, 99% or any custom confidence level, and see full step-by-step working, a plain-English interpretation, a shaded t or z curve and the exact TI-84 keystrokes for ZInterval, TInterval and 1-PropZInt. A sample size tab tells you how many observations you need for a target margin of error.

How to Use the Confidence Interval Calculator

  1. Pick the interval type. Mean from summary stats, mean from raw data, proportion, two means, two proportions, or sample size needed.
  2. Choose z or t. If the problem gives the population standard deviation σ, pick “σ known” (z-interval). If you only have the sample standard deviation s, keep “σ unknown” (t-interval). Most real problems use t.
  3. Enter your numbers and confidence level. Click 90%, 95% or 99%, or type any level such as 96.5.
  4. Read the result. You get the interval, the margin of error, the critical value z* or t* with degrees of freedom, and the standard error. Open the step-by-step working and TI-84 keys under the result. Use Copy share link to save or send your exact inputs.

Confidence Interval Formulas

Every confidence interval has the same shape: estimate ± critical value × standard error. The part after ± is the margin of error.

  • Mean, σ known (z-interval): x̄ ± z* · σ/√n
  • Mean, σ unknown (t-interval): x̄ ± t* · s/√n, with df = n − 1
  • One proportion (1-PropZInt): p̂ ± z* · √(p̂(1 − p̂)/n)
  • Two means (2-SampTInt): (x̄₁ − x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂), Welch degrees of freedom
  • Two proportions (2-PropZInt): (p̂₁ − p̂₂) ± z* · √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂)
  • Sample size for a mean: n = (z*·σ/E)², rounded up. For a proportion: n = z*²·p(1 − p)/E²

Worked example: a sample of n = 25 has mean 50 and standard deviation s = 10. With σ unknown, df = 24 and t* for 95% is 2.0639. The standard error is 10/√25 = 2, so the margin of error is 2.0639 × 2 = 4.1278. The 95% confidence interval is (45.872, 54.128). If σ = 10 were known, z* = 1.96 and the interval would be (46.080, 53.920).

Critical Values for Common Confidence Levels

Confidence levelαz*t* (df = 10)t* (df = 30)
80%0.201.28161.37221.3104
90%0.101.64491.81251.6973
95%0.051.96002.22812.0423
98%0.022.32632.76382.4573
99%0.012.57583.16932.7500

The calculator works out t* exactly for any degrees of freedom, including the decimal df from Welch’s two-sample formula, so you never need a printed t-table.

How to Find a Confidence Interval on a TI-84

  1. Press STAT, then the right arrow twice to open TESTS.
  2. Choose 7:ZInterval (σ known), 8:TInterval (σ unknown), A:1-PropZInt (one proportion), 0:2-SampTInt (two means) or B:2-PropZInt (two proportions).
  3. Set Inpt to Stats to type x̄, Sx and n, or to Data if your values are in L1.
  4. Enter the C-Level as .95 (or 95), highlight Calculate and press ENTER.

For the example above, TInterval with x̄ = 50, Sx = 10, n = 25 and C-Level .95 shows (45.872, 54.128). The calculator above writes out these menus with your own numbers and shows what the TI-84 screen should display. You can follow along on our TI-84 Plus CE calculator online.

z-Interval or t-Interval?

Use a z-interval only when the population standard deviation σ is actually given. Use a t-interval whenever you estimate the spread from the sample with s, which is almost always. The t-distribution has heavier tails, so t* is larger than z* and the interval is a little wider. As n grows, t* gets close to z*. To learn more about the normal curve behind z*, see our normal distribution calculator and z-score calculator.

Confidence Interval for a Proportion: Wald, Wilson or Exact

The TI-84’s 1-PropZInt uses the standard (Wald) formula p̂ ± z*·√(p̂(1 − p̂)/n). It works well when there are at least 10 successes and 10 failures. For small samples or proportions near 0 or 1, the calculator also gives the Wilson score interval, which most statisticians now recommend, and the Clopper–Pearson exact interval, which never has less than the stated coverage. The step-by-step working checks the success/failure condition for you.

How to Interpret a Confidence Interval

A 95% confidence interval of (45.87, 54.13) means: we are 95% confident that the population mean is between 45.87 and 54.13. The 95% describes the method. If you took many samples and built an interval from each, about 95% of those intervals would contain the true mean. It does not mean there is a 95% probability that this particular interval is right. For two-sample intervals, if the interval for μ₁ − μ₂ does not contain 0, the difference is statistically significant at α = 1 − confidence level.

What Makes the Interval Wider or Narrower

  • Higher confidence level → wider interval. 99% needs a bigger critical value than 90%.
  • Bigger sample → narrower interval. The standard error shrinks with √n, so four times the data halves the margin of error.
  • More spread → wider interval. A larger standard deviation means more uncertainty. You can compute s from raw data with our standard deviation calculator, or paste the data straight into the “Mean from raw data” tab.

Why This Confidence Interval Calculator Is Different

  • Six tools in one: z and t intervals for a mean, raw data, one proportion (Wald, Wilson and exact), two means (Welch, pooled or z), two proportions, and sample size.
  • Exact t critical values for any degrees of freedom, checked against statistical software to 10+ significant digits.
  • TI-84 keystrokes for ZInterval, TInterval, 1-PropZInt, 2-SampTInt and 2-PropZInt with your own numbers, plus the expected screen.
  • Step-by-step working, a plain-English interpretation, a shaded critical-value curve and an interval number line.
  • Any confidence level, share links, dark mode and a layout that works on phones.

Confidence Interval Calculator FAQ

A confidence interval is a range of plausible values for an unknown population value, such as a mean or a proportion, built from sample data. It is written as estimate ± margin of error, for example 50 ± 4.13.

Find the sample estimate (x̄ or p̂), the standard error (s/√n for a mean) and the critical value for 95% (z* = 1.96, or t* with n − 1 degrees of freedom). Multiply the critical value by the standard error to get the margin of error, then add and subtract it from the estimate.

z* = 1.96 for 95%. Other common values: 1.645 for 90%, 2.326 for 98% and 2.576 for 99%.

Use t whenever the population standard deviation σ is unknown and you use the sample standard deviation s. Use z only when σ is given in the problem, and for proportions.

Press STAT, arrow to TESTS, and pick 7:ZInterval, 8:TInterval or A:1-PropZInt. Enter your statistics or data list and the C-Level, then choose Calculate and press ENTER.

The margin of error is half the width of the interval: critical value × standard error. A 95% interval of (45.87, 54.13) has a margin of error of 4.13.

For a mean, n = (z*·σ/E)². For a proportion, n = z*²·p(1 − p)/E², using p = 0.5 if you have no estimate. Always round up. The Sample size tab does this for you; for example, ±3% at 95% confidence needs 1,068 responses.

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