- Blogs
- Discovery Lean Six Sigma
- What's the Difference between Confidence, Prediction, and Tolerance Intervals?

In statistics, as in life, absolute certainty is rare. That's why statisticians often can't provide a result that is as specific as we might like; instead, they provide the results of an analysis as a range, within which the data suggest the true answer lies.

Most of us are familiar with "confidence intervals," but that's just of several different kinds of intervals we can use to characterize the results of an analysis. Sometimes, confidence intervals are not the best option. Let's look at the characteristics of some different types of intervals, and consider when and where they should be used. Specifically, we'll look at confidence intervals, prediction intervals, and tolerance intervals.

An Overview of Confidence Intervals

A confidence interval refers to a range of values that is likely to contain the value of an unknown population parameter, such as the mean, based on data sampled from that population.

Collected randomly, two samples from a given population are unlikely to have identical confidence intervals. But if the population is sampled again and again, a certain percentage of those confidence intervals will contain the unknown population parameter. The percentage of these confidence intervals that contain this parameter is the confidence level of the interval.

Confidence intervals are most frequently used to express the population mean or standard deviation, but they also can be calculated for proportions, regression coefficients, occurrence rates (Poisson), and for the differences between populations in hypothesis tests.

If we measured the life of a random sample of light bulbs and Minitab calculates 1230 - 1265 hours as the 95% confidence interval, that means we can be 95% confident the mean for the population of bulbs falls between 1230 and 1265 hours.

In relation to the parameter of interest, confidence intervals only assess sampling error—the inherent error in estimating a population characteristic from a sample. Larger sample sizes will decrease the sampling error, and result in smaller (narrower) confidence intervals. If you could sample the entire population, the confidence interval would have a width of 0: there would be no sampling error, since you have obtained the actual parameter for the entire population!

In addition, confidence intervals only provide information about the mean, standard deviation, or whatever your parameter of interest happens to be. It tells you nothing about how the individual values are distributed.

What does that mean in practical terms? It means that the confidence interval has some serious limitations. In this example, we can be 95% confident that the mean of the light bulbs will fall between 1230 and 1265 hours. But that 95% confidence interval does not indicate that 95% of the bulbs will fall in that range. To draw a conclusion like that requires a different type of interval...

An Overview of Prediction Intervals

A prediction interval is a confidence interval for predictions derived from linear and nonlinear regression models. There are two types of prediction intervals.

Confidence interval of the prediction

Given specified settings of the predictors in a model, the confidence interval of the prediction is a range likely to contain the mean response. Like regular confidence intervals, the confidence interval of the prediction represents a range for the mean, not the distribution of individual data points.

With respect to the light bulbs, we could test how different manufacturing techniques (Slow or Quick) and filaments (A or B) affect bulb life. After fitting a model, we can use statistical software to forecast the life of bulbs made using filament A under the Quick method.

If the confidence interval of the prediction is 1400–1450 hours, we can be 95% confident that the *mean *life for bulbs made under those conditions falls within that range. However, this interval doesn't tell us anything about how the lives of *individual *bulbs are distributed.

Prediction interval

A prediction interval is a range that is likely to contain the response value of an individual new observation under specified settings of your predictors.

If Minitab calculates a prediction interval of 1350–1500 hours for a bulb produced under the conditions described above, we can be 95% confident that the lifetime of a new bulb produced with those settings will fall within that range.

You'll note the prediction interval is wider than the confidence interval of the prediction. This will always be true, because additional uncertainty is involved when we want to predict a single response rather than a mean response.

An Overview of Tolerance Intervals

A tolerance interval is a range likely to contain a defined proportion of a population. To calculate tolerance intervals, you must stipulate the proportion of the population and the desired confidence level—the probability that the named proportion is actually included in the interval. This is easier to understand when you look at an example.

Tolerance interval example

To assess how long their bulbs last, the light bulb company samples 100 bulbs randomly and records how long they last in this worksheet.

To use this data to calculate tolerance intervals, go to **Stat > Quality Tools > Tolerance Intervals **in Minitab. (If you don't already have it, download the free 30-day trial of Minitab and follow along!) Under **Data**, choose *Samples in columns*. In the text box, enter *Hours*. Then click **OK**.

The normality test indicates that these data follow the normal distribution, so we can use the Normal interval (1060 1435). The bulb company can be 95% confident that at least 95% of all bulbs will last between 1060 to 1435 hours.

How tolerance intervals compare to confidence intervals

As we mentioned earlier, the width of a confidence interval depends entirely on sampling error. The closer the sample comes to including the entire population, the smaller the width of the confidence interval, until it approaches zero.

But a tolerance interval's width is based not only on sampling error, but also variance in the population. As the sample size approaches the entire population, the sampling error diminishes and the estimated percentiles approach the true population percentiles.

Minitab calculates the data values that correspond to the estimated 2.5th and 97.5th percentiles (97.5 - 2.5 = 95) to determine the interval in which 95% of the population falls. You can get more details about percentiles and population proportions here for more information about percentiles and population proportions.

Of course, because we are using a sample, the percentile estimates will have error. Since we can't say that a tolerance interval truly contains the specified proportion with 100% confidence, tolerance intervals have a confidence level, too.

How tolerance intervals are used

Tolerance intervals are very useful when you want to predict a range of likely outcomes based on sampled data.

In quality improvement, practitioners generally require that a process output (such as the life of a light bulb) falls within spec limits. By comparing client requirements to tolerance limits that cover a specified proportion of the population, tolerance intervals can detect excessive variation. A tolerance interval wider than the client's requirements may indicate that product variation is too high.

Minitab statistical software makes obtaining these intervals easy, regardless of which one you need to use for your data.

Original: http://blog.minitab.com/blog/understanding-statistics/whats-the-difference-between-confidence-prediction-and-tolerance-intervals

By: Eston Martz

Posted: August 22, 2017, 1:58 pm

Dummy user for scooping articles

I'm a dummy user created for scooping great articles in the network for the community.

- July 2018
- June 2018
- May 2018
- April 2018
- March 2018
- February 2018
- January 2018
- December 2017
- November 2017
- October 2017
- September 2017
- August 2017
- July 2017
- June 2017
- May 2017
- April 2017
- March 2017
- February 2017
- January 2017
- December 2016
- November 2016
- October 2016
- September 2016
- August 2016
- July 2016
- June 2016
- May 2016
- April 2016
- March 2016
- February 2016
- January 2016
- December 2015
- November 2015
- October 2015
- September 2015
- August 2015
- July 2015
- June 2015
- May 2015
- April 2015
- March 2015
- February 2015
- January 2015
- December 2014
- November 2014
- October 2014
- August 2014
- July 2014
- June 2014
- April 2014
- March 2014
- February 2014
- January 2014
- December 2013
- November 2013
- October 2013
- September 2013
- August 2013
- July 2013
- June 2013
- May 2013
- April 2013
- March 2013
- February 2013
- January 2013
- November 2012
- October 2012
- September 2012
- August 2012
- July 2012
- June 2012
- May 2012
- March 2012
- February 2012
- November 2011
- October 2011
- September 2011
- August 2011
- July 2011
- June 2011
- May 2011
- April 2011
- February 2011
- January 2011
- December 2010
- November 2010
- October 2010
- September 2010
- August 2010
- July 2010
- June 2010
- April 2010
- March 2010
- February 2010
- December 2009
- November 2009
- August 2009
- June 2009
- March 2009
- November 2008
- October 2008
- July 2008
- May 2008
- April 2008
- March 2008
- February 2008
- June 2007
- February 2007
- August 2005
- February 2002

innovation, Leadership, innovation excellence, Blogartikel, big data, Articles, data management, Data Education, Education Resources For Use & Management of Data, lean manufacturing, & Education, lean, Data Daily | Data News, Quality Insider Article, Twitter Ed, Business, Six Sigma, Management, Management Article, Digitalisierung, systems thinking, lean six sigma, Gastbeiträge, strategy, Lean Management, Big Data News, Operations Article, Smart Data News, Interviews, kaizen, Problem solving, Soft Skills, The Latest, Change, continuous improvement, marketing, Uncategorized, systems view of the world, Organization, Theory of Constraints, quality, Personal, Immobilien, Culture, statistics, agile, MPD, Videos, Sekretariat & Assistenz, Banken