When we calculate a statistic for example, a mean, a variance, a proportion, or a correlation coefficient, there is no reason to expect that such point estimate would be exactly equal to the true population value, even with increasing sample sizes. According to University of Glasgow Department of Statistics, Confidence Interval is defined as: A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.
I appreciate your conceptual illustration of confidence intervals these illustration will be used to used to educate the medical doctors for research purpose. This quick note serves as a supplementnote of my previous Statistical Notes (3): Confidence Intervals for Binomial Proportion Using SAS which I will extend as a SESUG 2015 paper. The farther you try to forecast into the future, the less certain you are -- how can you represent that graphically? In the above session, a binomialc option was used to compute the intervals with a continuity correction(CC). Interval estimation for the difference between independent proportions: comparison of eleven methods. Note that #1 method is the most popular one in textbook, #10 might  be the most wildly used method in industry. Last year I dumped piece of SAS codes to compute confidence intervals for single proportion using 11 methods including 7 presented by one of Prof. If interested, a nice paper on binomial intervals with ties, the paper Confidence intervals for a binomial proportionin the presence of ties,  and the program. This post serves as an implementation using SAS accompany with the first paper on confidence intervals for single proportion(method 1-7).
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Summary Adding confidence intervals to completion rates in usability tests will temper both excessive skepticism and overstated usability findings. In *most* cases I find success in reporting the margin of error or confidence interval in some way. As you point out well, with small sample usability research, it's much easier to show unusability than usability.
To understand Confidence Intervals better, consider this example scenario: Acme Nelson, a leading market research firm conducts a survey among voters in USA asking them whom would they vote if elections were to be held today. If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter.
Basically I added a new Blaker method to my file and found more CIs from SAS PROC FREQ. There are some Bayesian intervals available in SAS  procedures but since I’m not familiar with Bayes, I will leave it blank by far. I’m still trying to figure out why the exact method produces different numbers compared to what’s in Prof. Confidence intervals make testing more efficient by quickly revealing unusable tasks with very small samples. This was especially the case to coexist peaceably with Market Research and Marketing which routinely survey much larger sample sizes. In most Six Sigma projects, there are at least some descriptive statistics calculated from sample data. Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9% (or whatever) confidence intervals for the unknown parameter.
In truth, it cannot be said that such data are the same as the population’s true mean, variance, or proportion value.

In addition to Democrats and Republicans, there is this surprise independent candidate, John Doe who is expected to secure 22% of the vote. There are many situations in which it is preferable instead to express an interval in which we would expect to find the true population value. A confidence interval is an interval, calculated from the sample data, that is very likely to cover the unknown mean, variance, or proportion.
For example, after a process improvement a sampling has shown that its yield has improved from 78% to 83%. If the lower end of the interval is 78% or less, you cannot say with any statistical certainty that there has been a significant improvement to the process. You rush excitedly to tell your manager the results so you can communicate this success to the development team. There is an error of estimation, or margin of error, or standard error, between the sample statistic and the population value of that statistic.
Your manager asks, "OK, this is great with 5 users, but what are the chances that 50 or 1000 will have a 100% completion rate?" You stop, think for a minute and say "pretty good." Fortunately there is a way to be a bit more precise than "pretty good" and you don't need to be a statistician (or bribe the one in your company with jelly-donuts to help you). Not only are they precise, but they make you sound smart when you talk about them. For example, if you are dealing with a sample mean and you do not know the population’s true variance (standard deviation squared) or the sample size is less than 30, than you use the t Distribution confidence interval.
In part one confidence intervals for task completion will be discussed and in part two (coming soon) confidence intervals for task times will be illustrated. If five out of five users complete a task, you can be 95% confident that the completion rate could be as high as 100% but it also could be as low as 48%.
In other words, if you tested another five users, their completion rate will fall somewhere between 48% and 100%. The question every analyst must also ask is, "How much risk are we willing to accept?" That risk is described in two places: The confidence level The width of the confidence interval Both are easy to calculate and present along with the completion rate. It is the percent likelihood statement that accompanies the width of the confidence interval. You would want to lower it to 90% or 85% or raise it to 99% depending on the impact of being wrong. A confidence level of 99% means that 1 out of 100 times your sample completion rate will NOT fall within your confidence intervals. To start with a simple example lets assume you tested 40 users attempting to complete one task. Half of them completed the task and half failed, making the completion rate 50%. Since we're sampling to determine the real mean, the standard error is as the name suggests, the estimate of the error between the true mean of the population and our sample mean. Think of the standard error as the standard deviation of the mean?or the area where the sample mean can "float" as we take multiple samples. The standard error is calculated by dividing the standard deviation by the square root of the sample size.
To obtain the standard deviation we multiply the proportion of people who completed the task (p) times the proportion of people who failed the task (1-p or q). This will return (2.023) which is the critical value for a 95% confidence level with (40-1) degrees of freedom. If you were to continue sampling users to complete a task, 95 times out of 100, the proportion will fall somewhere between 34% and 66%.

More often, the proportion is far from .5 and TWO confidence intervals need to be derived?one for above the proportion and one below the proportion. If this sounds confusing, don't worry, it is and so is the calculation to derive them. The first method is using a technique called the Paulson-Takeuchi approximation and the second uses the incomplete beta function and the F distribution. Don't worry if you've never heard of either one of them, unless you're a total stats geek, you shouldn't have. The important point is that either one of these methods provides accurate asymmetric confidence intervals. Even better, there are calculators on the web that will do the work for you and I've also built an Excel spreadsheet you can download where all you have to do is plug in the values but can still see the workings of the formula. I used the incomplete beta function approach since there is a paper publicly available that shows the formulas.
We know the lower bounds of the interval could be 48% and continue testing to see what happens. Even with a larger than normal sample, the width of the confidence interval is still rather large.
What becomes immediately evident is that it is much easier to proclaim a task completion rate unacceptable than it is to declare it acceptable.
This is a point Jim Lewis made in his 1996 article on Binomial Confidence intervals and it's worth repeating? [Binomial Confidence Intervals] cannot be used with a small sample to prove that a success rate is acceptably high.
With small samples, even if the observed defect percentage is 0 or close to 0 percent, the interval will be wide, so it will include defect percentages that are unacceptable. Therefore, it is relatively easy to prove (requires a small sample) that a product is unacceptable, but it is difficult to prove (requires a large sample) that a product is acceptable (Lewis p.735).
First, binomial confidence intervals are a resource saving tool during formative evaluations. If only two out of five users complete the task, there's less than a 5% chance that the completion rate will ever be above 85%. It will show you and the readers of usability reports the true sense of confidence in a completion rate. Instead of leaving the door wide open for attacks on sample size issues, the door now is partially closed (it's only open as wide as your confidence intervals). Instead of arguing about undefined possibilities, you can discuss probabilities. If the lower limits of the confidence interval are too low, then you can either sample more users or know the unacceptable completion rate is indicative of a problem that needs to be addressed.
In these cases it is necessary to use a z-critical value for a one-sided confidence interval. It is often the case that statistics text or examples you encounter do not make this adjustment for simplicity in instruction. Publication downloaded June 2004 Lewis, James "Binomial Confidence Intervals for Small Sample Usability Studies" in Proceedings of the 1st International Conference on Applied Ergonomics. Istanbul, Turkey, May (1996) Landauer, Thomas K, "Research Methods in Human-Computer Interaction" in The Handbook of Human Computer Interaction M.

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