Confidence Interval Calculator

Calculation Type:

Sample Mean (x̄):

Sample Size (n):

Standard Deviation (σ):

Confidence Level (%):

Calculation Results

About Confidence Intervals

A confidence interval provides a range of values that likely contains the true population parameter. It quantifies the uncertainty in our statistical estimates and helps researchers make informed decisions about their data.

How Confidence Intervals Work

When we calculate a 95% confidence interval, we can interpret this as follows: if we were to repeat our sampling process many times and calculate a confidence interval each time, approximately 95% of these intervals would contain the true population parameter.

Example: A survey of 100 UK households found an average monthly energy bill of £120 with a standard deviation of £25. The 95% confidence interval would be approximately £115.10 to £124.90, meaning we can be 95% confident that the true average monthly energy bill for all UK households falls within this range.

Confidence Interval Formulas

For Population Mean (σ known):

CI = x̄ ± z(α/2) × (σ/√n)

Where x̄ is the sample mean, z(α/2) is the critical z-value, σ is the population standard deviation, and n is the sample size.

For Population Proportion:

CI = p̂ ± z(α/2) × √[(p̂(1-p̂))/n]

Where p̂ is the sample proportion, z(α/2) is the critical z-value, and n is the sample size.

Critical Z-Values

Confidence Level	α (Alpha)	α/2	Z-Value
90%	0.10	0.05	1.645
95%	0.05	0.025	1.960
99%	0.01	0.005	2.576
99.9%	0.001	0.0005	3.291

Factors Affecting Confidence Interval Width

Sample Size: Larger samples produce narrower confidence intervals
Confidence Level: Higher confidence levels result in wider intervals
Population Variability: Greater variability leads to wider intervals
Measurement Precision: More precise measurements yield narrower intervals

Assumptions and Limitations

Data should be collected through random sampling
For means: data should be approximately normally distributed or sample size ≥ 30
For proportions: both np̂ and n(1-p̂) should be ≥ 5
Observations should be independent

Important Note: A specific confidence interval either contains the true parameter or it doesn’t. The confidence level refers to the long-run proportion of intervals that would contain the true parameter if the sampling process were repeated many times.

Practical Applications

Market Research: Estimating customer satisfaction rates
Quality Control: Monitoring manufacturing processes
Medical Studies: Evaluating treatment effectiveness
Opinion Polling: Predicting election outcomes
Scientific Research: Quantifying experimental uncertainties

Frequently Asked Questions

What does a 95% confidence interval mean?

A 95% confidence interval means that if you repeated the sampling process 100 times, approximately 95 of the resulting confidence intervals would contain the true population parameter.

How do I choose the right confidence level?

Common choices are 90%, 95%, and 99%. Higher confidence levels provide greater certainty but result in wider intervals. The choice depends on the consequences of being wrong and the precision required for your application.

What if my sample size is small?

For small sample sizes (n < 30) with unknown population standard deviation, use the t-distribution instead of the normal distribution. This calculator assumes either large samples or known population standard deviation.

Can confidence intervals be used for prediction?

Confidence intervals estimate population parameters, not individual observations. For predicting individual values, you would need prediction intervals, which are typically wider than confidence intervals.