BYTETOOLS

Confidence Interval Calculator

Build a confidence interval for a mean (z or t) or a proportion. Get the margin of error, lower and upper bounds, and the critical value used, step by step.

45.872
Lower bound
54.128
Upper bound

Working

2.064
Critical t (df 24)
2
Standard error
± 4.128
Margin of error
50
Point estimate

t-interval: x̄ ± t·s/√n with df = 24.

What is the Confidence Interval Calculator?

The ByteTools Confidence Interval Calculator constructs an interval estimate for a population mean or proportion.

  • z-interval, t-interval and proportion (Wald) modes
  • Built-in Student's t table by degrees of freedom
  • Shows critical value, standard error and margin of error separately
  • Any confidence level in z mode; five standard levels in t mode
  • Warns when the proportion interval's assumptions fail
  • 100% private — everything computed in your browser

How to use the Confidence Interval Calculator

  1. 1

    Choose the interval type: mean with known σ (z), mean with sample s (t), or proportion.

  2. 2

    Enter the sample statistic (x̄ or p̂), the spread (σ or s) and the sample size n.

  3. 3

    Pick a confidence level such as 95%.

  4. 4

    Read the critical value, standard error, margin of error and interval bounds.

  5. 5

    Copy the full working with one click.

About the Confidence Interval Calculator

The ByteTools Confidence Interval Calculator constructs an interval estimate for a population mean or proportion. For a mean with known σ it uses the z-interval x̄ ± z·σ/√n; with an unknown σ it switches to the t-interval x̄ ± t·s/√n, pulling the critical value from a built-in Student's t table by degrees of freedom. For a proportion it uses p̂ ± z·√(p̂(1−p̂)/n).

The output shows every ingredient: the critical value, the standard error, the margin of error and the final lower and upper bounds — so you can follow the arithmetic, not just read the answer. Confidence levels of 80%, 90%, 95%, 98% and 99% are supported in t mode, and any custom level in z mode.

All calculation is 100% local to your browser; your data never leaves your device. Note that the proportion interval is the standard Wald interval taught in introductory courses — it becomes unreliable when n·p̂ or n·(1−p̂) is small, and the tool warns you when that happens.

Frequently asked questions

How do you calculate a 95% confidence interval?

Take your sample statistic and add and subtract the margin of error: critical value × standard error. For a mean of 50 with σ = 10 and n = 100, the 95% interval is 50 ± 1.96 × 10/√100 = 50 ± 1.96, i.e. (48.04, 51.96).

When should I use t instead of z?

Use t whenever the population standard deviation is unknown and you are estimating it from the sample — which is most real situations. The t distribution has heavier tails to account for that extra uncertainty; with df = 24 the 95% critical value is 2.064 instead of 1.96. For large n the two converge.

What does a 95% confidence level actually mean?

It means the procedure captures the true parameter in 95% of repeated samples. It does not mean there is a 95% probability the true value lies in this specific interval — the true value is fixed; it is the interval that varies from sample to sample.

How can I make my confidence interval narrower?

Increase the sample size (the width shrinks with √n), reduce variability in the measurement, or accept a lower confidence level. Moving from 99% to 90% confidence narrows the interval but raises the chance it misses the true value.

Why does the tool warn me about small samples for proportions?

The Wald proportion interval relies on the normal approximation, which needs roughly n·p̂ ≥ 10 and n·(1−p̂) ≥ 10 (some texts use 5). Below that the stated coverage can be badly off, and alternatives like the Wilson interval are preferred. The warning tells you when you are in that zone.

Related tools