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02.09.2014
Written by Ms.Geethai Narayanasamy, ASQ CSSBB, Quality Management Professional, Chennai, IndiaMost of us would have used these terms and values in our statistical analysis and estimation.
It is the probability that the population parameter value lies between a specified ‘Range’. Confidence interval is always in the same unit as the population parameter or sample statistic. Confidence Intervals are mostly used in hypothesis testing to validate an assumption and in methods like correlation, regression etc, to arrive at intervals for the required confidence level.
To illustrate this, here are the means and confidence intervals for 100 samples of 3 observations from a population with a parametric mean of 5. When you calculate the confidence interval for a single sample, it is tempting to say that "there is a 95% probability that the confidence interval includes the parametric mean." This is technically incorrect, because it implies that if you collected samples with the same confidence interval, sometimes they would include the parametric mean and sometimes they wouldn't. An alternative technique for estimating the confidence limits of a proportion assumes that the sample proportions are normally distributed. This handbook mostly presents "classical" or "frequentist" statistics, in which hypotheses are tested by estimating the probability of getting the observed results by chance, if the null is true (the P value). This approach is valuable if a small deviation from the null hypothesis would be uninteresting, when you're more interested in the size of the effect rather than whether it exists. Using confidence limits this way, as an alternative to frequentist statistics, has many advocates, and it can be a useful approach. Nominal data: If you work with a lot of proportions, it's good to have a rough idea of confidence limits for different sample sizes, so you have an idea of how much data you'll need for a particular comparison.
The descriptive statistics spreadsheet calculates 95% confidence limits of the mean for up to 1000 measurements. This web page calculates confidence intervals of the mean for up to 10,000 measurement observations.
Salvatore Mangiafico's R Companion has sample R programs for confidence limits for both measurement and nominal variables.


The first pair of confidence limits shown is based on the normal approximation; the second pair is the better one, based on the exact binomial calculation. Of the 100 samples, 94 (shown with X for the mean and a thin line for the confidence interval) have the parametric mean within their 95% confidence interval, and 6 (shown with circles and thick lines) have the parametric mean outside the confidence interval. This approximate technique yields symmetrical confidence limits, which for proportions near zero or one are obviously incorrect. An alternative way of doing statistics is to put a confidence interval on a measure of the deviation from the null hypothesis.
Another version of this myth is that if each mean is outside the confidence interval of the other mean, the means are significantly different. A surprising number of papers don't say what their error bars represent, which means that the only information the error bars convey to the reader is that the authors are careless and sloppy.
For proportions near 50%, the confidence intervals are roughly ±30%, 10%, 3%, and 1% for n=10, 100, 1000, and 10,000, respectively.
The confidence intervals for a binomial proportion spreadsheet calculates 95% confidence limits for nominal variables, using both the exact binomial and the normal approximation. Note that if you have more than two values of the nominal variable, the confidence limits will only be calculated for the value whose name is first alphabetically. While the purpose of these two are invariably the same, there is a minor and important difference between these two terms conceptually, which makes them to inevitably devote an article to them.
To explain simply, when a dice is thrown at random the chance of getting ‘3’ in 50 throws varies.
Confidence interval is always expressed in percentage and most of the statistical calculations use a value of 95% or 99%, depending upon the accuracy of data needed. A common rule of thumb says that it is okay to use this approximation as long as npq is greater than 5; my rule of thumb is to only use the normal approximation when the sample size is so large that calculating the exact binomial confidence interval makes smoke come out of your computer.
For example, rather than comparing two means with a two-sample t–test, some statisticians would calculate the confidence interval of the difference in the means.


This is why the "margin of error" in political polls, which typically have a sample size of around 1,000, is usually about 3%.
For example, if the Gus data set included "left," "right," and "both" as values, SAS would only calculate the confidence limits on the proportion of "both." One clumsy way to solve this would be to run the program three times, changing the name of "left" to "aleft," then changing the name of "right" to "aright," to make each one first in one run. It would also be incorrect to say that the confidence limits were 0 and 0.23, because you know the proportion of colorblind men in your population is greater than 0 (your sample had two colorblind men, so you know the population has at least two colorblind men).
Some statisticians don't care about this confusing, pedantic distinction, but others are very picky about it, so it's good to know.
I consider confidence limits for proportions that are based on the normal approximation to be obsolete for most purposes; you should use the confidence interval based on the binomial distribution, unless the sample size is so large that it is computationally impractical. Don't try compare two means by visually comparing their confidence intervals, just use the correct statistical test. On judging the significance of differences by examining overlap between confidence intervals. Unfortunately, more people use the confidence limits based on the normal approximation than use the correct, binomial confidence limits.
For some it might be 99% of the times, and for some other it may be 80% of the times and so on.
Overlapping confidence intervals or standard error intervals: what do they mean in terms of statistical significance?



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