95% Confidence Interval Calculator

What is a confidence interval?

A confidence interval is a range of values that is likely to contain an unknown population parameter. Rather than estimating the parameter by a single value, an interval estimate gives a range of plausible values along with a probability that the true value falls within that range.

For example, a 95% confidence interval means that if we were to take many samples and calculate a confidence interval from each, approximately 95% of those intervals would contain the true population parameter.

The confidence interval formula

For a population mean, the confidence interval is:

$$\bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}$$

Where:

• x̄ = sample mean
• t = critical value from t-distribution
• s = sample standard deviation
• n = sample size

The margin of error is:

$$E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$$

Confidence levels and critical values

t-distribution (unknown population σ)

Confidence	α	α/2	df = 10	df = 30	df = ∞
90%	0.10	0.05	1.812	1.697	1.645
95%	0.05	0.025	2.228	2.042	1.960
99%	0.01	0.005	3.169	2.750	2.576

z-distribution (known population σ or large n)

Confidence level	z-score
90%	1.645
95%	1.960
99%	2.576

When to use t vs z distribution

Use t-distribution when:

• Population standard deviation (σ) is unknown
• Sample size is small (n < 30)
• Estimating from sample standard deviation (s)

Use z-distribution when:

• Population standard deviation (σ) is known
• Sample size is large (n ≥ 30)
• Working with proportions

As sample size increases, the t-distribution approaches the normal (z) distribution.

Example calculation

A researcher measures the heights of 25 students and finds:

• Sample mean (x̄) = 170 cm
• Sample standard deviation (s) = 8 cm
• Sample size (n) = 25

Calculate the 95% confidence interval:

$$\begin{aligned} df &= n - 1 = 24 \\ t_{0.025, 24} &\approx 2.064 \\ SE &= \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{25}} = 1.6 \\ E &= 2.064 \times 1.6 = 3.30 \\ CI &= 170 \pm 3.30 = (166.7, 173.3) \end{aligned}$$

We are 95% confident the true mean height is between 166.7 cm and 173.3 cm.

Interpreting confidence intervals

Correct interpretation

"We are 95% confident that the interval (a, b) contains the true population parameter."

Common misinterpretations

• ❌ "There's a 95% probability the population mean is in this interval"
• ❌ "95% of the data falls within this interval"
• ❌ "The sample mean will be in this range 95% of the time"

The population mean is fixed—it either is or isn't in the interval. The 95% refers to the long-run success rate of the method.

Factors affecting interval width

Sample size (n)

Larger samples produce narrower intervals:

Sample size	Margin of error (relative)
n = 10	1.00
n = 25	0.63
n = 50	0.45
n = 100	0.32
n = 400	0.16

Variability (s)

Higher variability produces wider intervals. Reducing measurement error or using more homogeneous samples can help.

Confidence level

Higher confidence requires wider intervals:

Confidence	Relative width
90%	0.84
95%	1.00
99%	1.32

Sample size calculation

To achieve a desired margin of error E with confidence level (1-α):

$$n = \left(\frac{z_{\alpha/2} \times s}{E}\right)^2$$

Example

To estimate mean with margin of error ±2, s = 10, at 95% confidence:

$$n = \left(\frac{1.96 \times 10}{2}\right)^2 = 96.04 \approx 97$$

Confidence intervals for proportions

For a proportion p̂ with sample size n:

$$\hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

This requires np̂ ≥ 10 and n(1-p̂) ≥ 10 for the normal approximation to be valid.

One-sided confidence intervals

Sometimes you only care about an upper or lower bound:

Upper bound only

$$\bar{x} + t_{\alpha} \times \frac{s}{\sqrt{n}}$$

Lower bound only

$$\bar{x} - t_{\alpha} \times \frac{s}{\sqrt{n}}$$

Note: Use tα (not tα/2) for one-sided intervals.

Confidence intervals vs hypothesis testing

There's a direct relationship between confidence intervals and hypothesis tests:

• If the 95% CI for μ doesn't contain μ₀, then a two-sided test would reject H₀: μ = μ₀ at α = 0.05
• If the CI contains μ₀, the test would fail to reject

Common applications

Scientific research

• Estimating treatment effects
• Reporting measurement precision
• Comparing groups

Polling and surveys

• Election predictions (margin of error)
• Public opinion estimates
• Market research

Quality control

• Process capability analysis
• Tolerance intervals
• Product specifications

Medical studies

• Drug efficacy ranges
• Diagnostic test accuracy
• Survival analysis

Assumptions

For valid confidence intervals, the following assumptions should be met:

1. Random sampling — Data should be randomly selected from the population
2. Independence — Observations should be independent of each other
3. Normality — For small samples, the population should be approximately normal (less critical for large n due to Central Limit Theorem)
4. No outliers — Extreme values can distort results

Tips for practice

1. Always report CIs with point estimates — They provide more information than significance tests alone
2. Consider the CI width — A statistically significant but very wide CI may not be practically useful
3. Use appropriate confidence level — 95% is standard, but 90% or 99% may be appropriate depending on context
4. Check assumptions — Violations can make CIs invalid or misleading
5. Increase sample size for precision — Larger samples give narrower, more useful intervals