Statistics Reference
Confidence Level vs Confidence Interval
One is a promise about the method; the other is what the method produced from your data. Mixing them up creates the most common misreadings in statistics — “there's a 95% chance the truth is in here” among them. This guide separates the two with one dataset computed at three levels.
Setting vs Output
| Confidence level | Confidence interval | |
|---|---|---|
| What it is | A percentage you choose — the long-run capture rate you demand | A range the data produces — estimate ± margin of error |
| Decided | Before seeing the data | After computing from the sample |
| Belongs to | The procedure (all its repeated uses) | This one sample |
| Example | 95% | (48.04, 51.96) |
The connection runs through the critical value: the level picks it (1.96 for 95%), and the interval is estimate ± critical value × standard error — the anatomy covered in the margin-of-error guide.
One Dataset, Three Levels
Sample: mean 50, standard deviation 10, n = 100 — so the standard error is 10/√100 = 1.
| Level | Critical value | Interval | Long-run miss rate |
|---|---|---|---|
| 90% | 1.645 | (48.36, 51.65) | 1 in 10 |
| 95% | 1.960 | (48.04, 51.96) | 1 in 20 |
| 99% | 2.576 | (47.42, 52.58) | 1 in 100 |
Same data, three different claims. Raising the level never moves the center — it only stretches the range, trading precision for reliability. Reproduce any row with the confidence interval calculator.
The Two Legitimate Ways to Narrow an Interval
- More data. Width scales with 1/√n: quadrupling the sample halves the interval at the same level — the honest route, planned with the sample size calculator.
- Less noise. Better instruments or designs shrink the standard deviation itself.
Lowering the level also narrows the interval — but that is weakening the promise, not strengthening the evidence. A suspiciously tight interval quoted at 80% confidence is the classic form of this trick; always check the level before admiring the width. The margin of error calculator makes the level-width dependence explicit.
Getting the Words Right
Correct
“We are 95% confident the mean lies between 48.04 and 51.96” — shorthand for: the procedure that produced this interval captures the truth 95% of the time.
Incorrect
“95% of the data falls in this interval” (that confuses it with a spread statement) or “the mean has a 95% probability of being in here” (the frequentist framework assigns no probability to a computed interval).
The deeper interpretation questions — what the long-run reading really commits you to, and z vs t choices — are covered in the full confidence intervals guide.
Try the Confidence Interval Calculator
Compute the interval at 90%, 95%, or 99% and watch the level-width trade-off on your own data.
Frequently Asked Questions
What is the difference between confidence level and confidence interval?
The confidence level is a percentage you choose before analyzing - the long-run success rate you demand from the procedure (usually 95%). The confidence interval is the specific range the procedure then computes from your sample (like 48.04 to 51.96). Level is the setting; interval is the output. Two studies can share a level and produce completely different intervals.
Does a 95% confidence level mean my interval has a 95% chance of being right?
Not quite - the 95% belongs to the method, not to your one interval. Across many repeated studies, 95% of the intervals built this way capture the truth. Your specific interval either contains the parameter or it does not; there is no probability left once it is computed. The practical reading: you hold one ticket from a batch where 95% are winners.
What happens to the interval when I raise the confidence level?
It widens - certainty is bought with precision. With the same data (mean 50, SD 10, n = 100), the interval is +/-1.645 at 90%, +/-1.96 at 95%, and +/-2.576 at 99%. The width scales with the critical value, so 99% intervals are about 31% wider than 95% ones. An interval wide enough to be nearly certain can be too wide to be useful.
Which should I report - the level or the interval?
Both, always together: '95% CI: 48.0 to 52.0'. The interval without its level is uninterpretable (48-52 at 80% confidence is a much weaker claim than at 99%), and the level without the interval says nothing about your data. Reporting the sample size alongside completes the picture.
Can I narrow the interval without lowering the confidence level?
Yes - collect more data or reduce noise. Width is (critical value) x (standard error), and the standard error shrinks with the square root of n: quadrupling the sample halves the width at any level. Lowering the level narrows the interval too, but by weakening the claim rather than improving the evidence.
Is a 99% confidence interval always better than a 95% one?
No - it is a different trade, not an upgrade. The 99% interval misses the truth less often but says something vaguer about where it is. Fields settle on 95% as a working balance; use 99% when the cost of a wrong interval is high, and 90% when a tighter range matters more than occasional misses. The choice should be made before seeing the data.