What does 95% confidence actually mean?

That if you repeated the whole sampling exercise many times, about 95% of the intervals you built this way would contain the true value. It is a statement about the method, not a 95% probability that this particular interval is right.

When should I not use this?

On small samples, where the t-distribution is the correct tool and gives a wider interval. Below about 30 observations the difference matters; this page uses the normal multiplier throughout and will be slightly optimistic there.

Why does a higher confidence level widen the interval?

Because being right more often means claiming less. Demanding 99% instead of 95% raises the multiplier from 1.96 to 2.58, widening the interval by about a third for exactly the same data.

Math

Confidence Interval Calculator

Enter your sample mean, standard deviation and size to see the interval around it.

The formula

standard error = standard deviation ÷ √n

margin = z × standard error

interval = mean ± margin

Worked example

A mean of 72.5 with a standard deviation of 12, from 400 samples, at 95%

12 ÷ √400 = 0.6 standard error
1.95996 × 0.6 = 1.176 margin
72.5 ± 1.176 → 71.32 to 73.68

Where this goes wrong

Expecting a bigger sample to shrink the interval proportionally

The margin falls with the square root of the sample size, not the size itself. Going from 400 samples to 1,600 — four times the work — only halves the margin, from 1.18 to 0.59. This is why survey precision gets expensive fast, and why doubling a sample is often not worth it.

Questions

What does 95% confidence actually mean?: That if you repeated the whole sampling exercise many times, about 95% of the intervals you built this way would contain the true value. It is a statement about the method, not a 95% probability that this particular interval is right.
When should I not use this?: On small samples, where the t-distribution is the correct tool and gives a wider interval. Below about 30 observations the difference matters; this page uses the normal multiplier throughout and will be slightly optimistic there.
Why does a higher confidence level widen the interval?: Because being right more often means claiming less. Demanding 99% instead of 95% raises the multiplier from 1.96 to 2.58, widening the interval by about a third for exactly the same data.