# Narrow confidence interval — higher accuracy?

I have two questions about confidence intervals:

Apparently a narrow confidence interval implies that there is a smaller chance of obtaining an observation within that interval, therefore, our accuracy is higher.

Also a 95% confidence interval is narrower than a 99% confidence interval which is wider.

The 99% confidence interval is more accurate than the 95%.

Can someone give a simple explanation that could help me understand this difference between accuracy and narrowness?

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I think you mean "there is a smaller chance of obtaining an observation outside that interval". Unfortunately, a Confidence Interval may not mean what it appears to mean, due to technical, statistical issues, but in general the narrower the interval (at a given confidence level) the less uncertainty there is about the results. There are many threads on this site discussing what a Confidence Interval means (as opposed to, say, a Credible Interval). We're not even getting into Predictive Intervals... – Wayne Sep 28 '11 at 15:21
@Wayne Why is not the statement be "there is a smaller chance of obtaining an observation within that interval" ? Since narrow interval has a large type 1 error , it is more likely to reject the true null hypothesis , that is , my true null value is not contained in that interval . So , it seems to me a narrow confidence interval implies that there is a smaller chance of obtaining an observation within that interval is correct . Would you please explain me where am I doing the mistake ? – Leaf Jul 23 '15 at 4:16

For a given dataset, increasing the confidence level of a confidence interval will only result in larger intervals (or at least not smaller). That's not about accuracy or precision but rather about how much risk you're willing to take about missing the true value.

If you're comparing confidence intervals for the same sort of parameter from multiple data sets and one is smaller than the other, you could say that the smaller one is more precise. I prefer to talk about precision rather than accuracy in this situation (see this relevant Wikipedia article).

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What is meant by "same sort of parameter" and "multiple data sets" ? Say , a survey on illiteracy and the survey is carried out in different time , 1995, 1998 , etc . Then is the "illiteracy rate" same sort of parameter and do the data sets of 1995, 1998 , etc indicate multiple data sets ? – Leaf Jul 23 '15 at 4:26
For example, a set of confidence intervals, each for the mean of some population. Your example fits, too, I think. – Karl Jul 25 '15 at 3:05
thank you very much . – Leaf Jul 25 '15 at 4:24

The 95% is not numerically attached at all to how confident you are that you've covered the true effect in your experiment. Think about the 95% not as a modifier to confidence but to interval. Perhaps "confidence interval using 95% range calculation" might be more accurate. You are allowed to decide that the interval contains the true value. But you really don't know how likely it is for your experiment.

Q1: Your first query conflates two things and misuses a term. No wonder you're confused. A narrower confidence interval may be more precise but, when calculated the same way, such as the 95% method, they all have the same accuracy. They capture the true value the same proportion of the time.

Also, just because it's narrow doesn't mean you're less likely to encounter a sample that falls within that narrow confidence interval. A narrow confidence interval can be achieved one of three ways. The experimental method or nature of the data could just have very low variance. The confidence interval around the boiling point of tap water at sea level is pretty small, regardless of the sample size. The confidence interval around the average weight of people might be rather large because people are very variable but one can make that confidence interval smaller by just collecting more samples. In that case, as you gain more certainty about where you believe the true value is, by collecting more samples and making a narrower confidence interval, then the probability of encountering an individual in that confidence interval does go down. (OK, it goes down in any case when you increase sample size, but you may not bother collecting the big sample in the boiling water case). Finally, it could be narrow because your sample is unrepresentative. In that case you are actually more likely to have one of the 5% of intervals that does not contain the true value. It's a bit of a paradox regarding CI width and something you should check by knowing the literature and how variable this data typically is.

Further consider that the confidence interval is about trying to estimate the true mean value of the population. If you knew that spot on then you'd be even more precise (and accurate) and not even have a range of estimates. But your probability of encountering an observation with that exact same value would be far lower than finding one within any particular sample based CI.

Q2: A 99% confidence interval is wider than a 95%. Therefore, it's more likely that it will contain the true value. See the distinction above between precise and accurate, you're conflating the two. If I make a confidence interval narrower with lower variability and higher sample size it becomes more precise, the likely values cover a smaller range. If I increase the coverage by using a 99% calculation it becomes more accurate, the true value is more likely to be within the range.

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The variability of the mean (=what you are trying to estimate) in the population is zero. – Nick Sabbe Sep 29 '11 at 7:13

First of all, a CI for a given confidence percentage (e.g.95%) means, for all practical purposes (though technically it is not correct) that you are confident that the true value is in the interval.

If this is interval is "narrow" (note that this can only be regarded in a relative fashion, so, for comparison with what follows, say it is 1 unit wide), it means that there is not much room to play: whichever value you pick in that interval is going to be close to the true value (because the interval is narrow), and you are quite certain of that (95%).

Compare this to a relatively wide 95% CI (to match the example before, say it is 100 units wide): here, you are still 95% certain that the true value will be within this interval, yet that doesn't tell you very much, since there are relatively many values in the interval (about a factor 100 as opposed to 1 - and I ask, again, of purists to ignore the simplification).

Typically, you are going to need a bigger interval when you want to be 99% certain that the true value is in it, than when you only need to be 95% certain (note: this may not be true if the intervals are not nested), so indeed, the more confidence you need, the broader the interval you will need to pick.

On the other hand, you are more certain with the higher confidence interval. So, If I give you 2 intervals of the same width, and I say one is a 95% CI and the other is a 99% CI, I hope you will prefer the 99% one. In this sense, 99% CIs are more accurate: you have less doubt that you will have missed the truth.

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thanks! so then when they say that this new research on neutrinos being faster than light has a very small confidence interval (I guess this means narrow) then that means that they are more likely to be accurate then if it was a wide confidence interval? (disregarding all other aspects) – Dbr Sep 28 '11 at 15:02
Nick, your first statement is wrong. It's not a "technical issue", it's just not correct. The confidence interval is a statement about what would happen in repeated experiments, that they would cover the true value 95% of the time. A statement about the confidence that the true value is within my given range found in my given experiment is not the same as that at all. If you removed the "that" in "that confident" and the parenthetical numerical amount then you'd be closer to the truth. You could just say that it means you believe the true value likely to fall in the interval. – John Sep 28 '11 at 15:17
otherwise, the answer is pretty good... – John Sep 28 '11 at 15:19
@John: I specifically avoided saying that the interval itself is the random variable, though my sentence does not imply it not to be (admittedly, it does suggest so). I know the issues involved, but found them irrelevant for the question. I have never seen a practical situation where the difference mattered either, hence the "for all practical purposes". – Nick Sabbe Sep 28 '11 at 15:38
Haven't encountered the issue? That's like saying the p-value = the probability of the null and then saying that you've never encountered an issue with it. You won't if you stay in the right journals. It's just incorrect to say that you're 95% certain the the true value is in your current range. Treating it as some esoteric matter just means now we'll have (at least) one more person walking around saying, "I'm 95% confident the value is in this range." It would hardly change your answer to correct it. The other issues you skirt could be ignored if you changed that one statement. – John Sep 28 '11 at 17:24