A confidence interval turns a single sample estimate into a range, and that range is what honest reporting of a result looks like. Give a mean of 72.4 on its own and you imply a precision your data does not have; give it as 71.4 to 73.4 and you have said something defensible about the population.
This calculator handles the two cases coursework asks for — an interval for a mean and an interval for a proportion — and returns the margin of error and standard error alongside the bounds. It also writes out the interpretation sentence, including the phrasing that loses marks, because misinterpreting the interval is more common in submitted work than miscalculating it.
A 95% confidence interval does not mean there is a 95% probability that the population parameter lies inside your particular interval. The population parameter is a fixed number; it is either in there or it is not. What varies is the interval, which would differ with every new sample you drew.
The correct reading is about the procedure: if you repeated the sampling many times and built an interval each time the same way, about 95% of those intervals would contain the true parameter. In a report, the accepted phrasing is "we are 95% confident that the population mean lies between A and B" — and the calculator gives you that sentence with your numbers in it.
This distinction is examined regularly, and the probability wording is the single most common error in student statistics write-ups. It costs marks even when every calculation above it is correct.
Three things set the width of an interval: the variability in the data, the sample size and the confidence level. You control the last two. Raising confidence from 95% to 99% widens the interval — more certainty of capture, less precision — which is why 99% is not automatically better.
Sample size enters through sqrt(n), and that has a blunt practical consequence: to halve the margin of error you need four times the data, not twice. This is the reason studies plateau at a certain precision, and it is worth stating in a limitations section when your sample is modest.
If an interval for a difference between two groups includes zero, the difference is not statistically significant at that level. Likewise an interval for a ratio or odds ratio that includes one. Reading significance off an interval this way is often quicker than a separate test, and it shows the effect size at the same time.
This calculator uses the z distribution. That is the right choice for a proportion, and for a mean when the sample is reasonably large — the usual rule of thumb is n of at least 30 — or when the population standard deviation is genuinely known. With a small sample and an unknown population standard deviation, the t distribution is required, and its critical value depends on degrees of freedom (n − 1). Check what your module specifies, because the two give different answers on small samples.
For proportions, the normal approximation used here needs enough of both outcomes: at least 10 successes and 10 failures is the standard threshold. The calculator checks this and tells you which side falls short, so you can either report it as a limitation or switch to an exact method such as the Clopper–Pearson interval.
One more condition applies to both: the sample must actually be random and the observations independent. A convenience sample of classmates produces an interval that is arithmetically correct and substantively meaningless, and saying so is what a limitations section is for.
x̄ ± z × (s / sqrt(n)) — the sample mean plus and minus the critical value times the standard error. The calculator shows each step of the substitution.
p̂ ± z × sqrt(p̂(1 − p̂) / n), where p̂ is successes divided by sample size.
No, and writing that will cost you marks. The parameter is fixed; the interval varies by sample. Say you are 95% confident the parameter lies between the bounds.
Use t when the sample is small (broadly n under 30) and the population standard deviation is unknown. This calculator uses z, which is correct for proportions and for larger samples.
Collect more data, or accept a lower confidence level. Because n enters through its square root, cutting the margin of error in half needs four times the sample.
The normal approximation is unreliable. The calculator flags it; report it as a limitation or use an exact method such as Clopper–Pearson.