Usability, Customer Experience & Statistics

10 Things to know about Confidence Intervals

Jeff Sauro • March 21, 2012

You don't need a PhD in statistics to understand and use confidence intervals.

Because we almost always sample a fraction of the users from a larger population, there is uncertainty in our estimates.

Confidence intervals are an excellent way of understanding the role of sampling error in the averages and percentages that are ubiquitous in user research.

  1. Confidence intervals tell you the most likely range of the unknown population average or percentage such as the average completion rate, the average level of satisfaction or the percentage of users likely to recommend your product. Note: Confidence intervals don't tell you the likely range of all values, just how much the average value is likely to fluctuate.

  2. Confidence intervals provide both the location and precision of a measure: For example, the graphs below show the 95% confidence intervals around the average System Usability Scale (SUS) score.  The only difference between the graphs is the sample size. Even though they both show the same location (a mean score of 83), the one on the right has a larger sample size (60), making it a more precise estimate of the population SUS score than the sample with only 12. 



  3. Three things impact the width of a confidence interval  
    1. Confidence level: This is the 95% part of the 95% confidence interval and also typically takes values of 90%, 99%, 80% and 85%. Confidence levels are the "advertised coverage" of a confidence interval. If we were to sample from the same user population 100 times, we'd expect the average to fall within the interval 95, 90 etc.,  times out of 100.

    2. Variability: as measured by the standard deviation. Populations (and samples) with more variability generate wider confidence intervals.

    3. Sample Size: Smaller sample sizes generate wider intervals. There is an inverse square root relationship between confidence intervals and sample sizes. If you want to cut your margin of error in half, you need to approximately quadruple your sample size.

  4. The confidence interval is equal to two margins of errors and a margin of error is equal to about 2 standard errors (for 95% confidence). A standard error is the standard deviation divided by the square root of the sample size. The standard error is standard deviation of the sampling distribution of the mean, or in English, it's basically the amount we expect the sample mean to fluctuate for a given sample size due to random sampling error.




  5. For rating scale data (like the SUS, SUPR-Q or SEQ) use the t-confidence interval method. You just need to know the mean, standard deviation, sample size and confidence level or you can use the online calculator to compute an accurate interval at any sample size.

  6. For confidence intervals using task times you should perform a log transformation on the raw values, and then compute the t- interval method. This method corrects for the skew in task time data to generate accurate intervals. Use the online calculator to perform the calculations.

  7. For binary data coded as 1's and 0's (Pass/Fail, Yes/No) use the adjusted wald method which provides accurate intervals at any sample size. It works by "adjusting" the proportion by adding two successes and two failures and generates accurate intervals for any sample size.

  8. Confidence Intervals can be computed on sample sizes as small as two: The intervals will be very wide but there's nothing with the math preventing you form computing them. With small sample sizes you can show that an interface is unusable, but it's harder to show it's usable. For example, if 0 out of 2 people can complete a task, there's only about a 5% chance more than half of all users will.

  9. We are 95% confident in the method of computing confidence intervals, not in any given interval. Any given interval we compute from sample data may or may not contain the population average. Even though 95% of samples will contain the interval we don't know if the one sample we took is one of the 5% that don't contain the average.

  10. You can use the overlap in confidence intervals as a quick way to check for statistical significance. If the intervals do not overlap then you can be at least 95% confident there is a difference (for 95% confidence intervals). If there is a large overlap, then the difference is not significant (at the p <.05 level). The intervals can actually overlap by as much as 25% and still be statistically significant, so when there is some overlap, it's best to conduct the 2-sample t-test and find the p-value. The three graphs below illustrate this.

         
     No Overlap: Significantly different (p <.05)
     Lots of Overlap: Not Significantly different ( p >.05)
     Some Overlap: Can't tell, run the 2-sample t-test (turns out p =.04, go publish!)


We have a chapter dedicated to confidence intervals in our book Quantifying the User Experience (Chapter 3) and in the Companion Book (Chapter 3) which contains step-by-step instructions for computing the interval in R or the Excel statistics package.

I'll be offering a tutorial which will include confidence intervals this fall at the Denver UX Boot Camp.



About Jeff Sauro

Jeff Sauro is the founding principal of MeasuringU, a company providing statistics and usability consulting to Fortune 1000 companies.
He is the author of over 20 journal articles and 5 books on statistics and the user-experience.
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Posted Comments

There are 26 Comments

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September 20, 2016 | Tracey Danny wrote:

Dr Hope is a very genuine spell caster and a very unique and a powerful man of wisdom. Dr Hope brought back joy to my life and to my family again by bringing my husband back to me, by curing my sister's cancer disease and by also curing my cousin's hiv virus with his special powers. Dr Hope is really a gifted, amazing, and a very good man who truly knows how to do his job better and if not for him what would my life turn to? Dr Hope thank you so much and i will never stop talking about your great deeds. Words can not express how much good you have done for me and my family and i believe that even a thousand words won't be enough to describe your good great works. If you are in need of any spiritual help why not run to Dr Hope? just contact Dr Hope on his email: Ultimatesolutionhome55@yahoo.com or you can also contact him on his web site on:rnhttp://ultimatesolutionho5.wixsite.com/dr-hope and he will help you. Contact him now and get your solution. Thank you Dr Hope for you are indeed a great man. 


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September 19, 2016 | JANE CLINTON wrote:

MUST READ HOW I GOT CURED FROM GENITAL WARTS
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DOCTOR OKOSUN CAN AS WELL CURE OTHER DISEASE/VIRUSES. 


September 19, 2016 | JANE CLINTON wrote:

MUST READ HOW I GOT CURED FROM GENITAL WARTSrnGreetings to you all, i am here today on this forum giving a life testimony on how Dr okosun has cured me from Genital Warts, i have been stocked in bondage with this virus for almost 2years now, i have tried different means to get this disease out of my body i also heard there was no cure to the virus, all the possible ways i tried did not work out for me, i do have the faith that i was going to be cured one day, as i was a strong believer in God and also in miracles, One day as i was on the internet i came across some amazing testimonies concerning how Dr okosun has cured different people from various sickness with his Herbal Herbs Medicine, they all advised we contact Dr okosun for any problem, with that i had the courage and i contacted Dr okosun i told him about my Sickness, He told me not to worry that he was going to prepare some Herbal Medicine for me, after some time in communication with Dr okosun he finally prepared for me some herbs which he sent to me and he also gave me prescriptions on how to take them, My good friends after taking Dr okosun Herbs for some weeks i started to experience changes in me and from there, I noticed my Genital warts was no longer in my body, as i have also gone for test, Today i am fit and healthy to live life again, I am so happy for the good work of Dr okosun in my life, Friends if you are having any type of disease/problem kindly email Dr okosun on {Doctorokosunspelltemple@gmail.com} or call/whatsapp him on +2348107584479 God Bless you Dr.rnrnDOCTOR OKOSUN CAN AS WELL CURE OTHER DISEASE/VIRUSES. 


September 17, 2016 | joel sandra wrote:

I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: azaspellcaster@gmail.com or you can whats app his line:+2348107155060 


September 17, 2016 | Tracey Danny wrote:

Dr Hope is a very genuine spell caster and a very unique and a powerful man of wisdom. Dr Hope brought back joy to my life and to my family again by bringing my husband back to me, by curing my sister's cancer disease and by also curing my cousin's hiv virus with his special powers. Dr Hope is really a gifted, amazing, and a very good man who truly knows how to do his job better and if not for him what would my life turn to? Dr Hope thank you so much and i will never stop talking about your great deeds. Words can not express how much good you have done for me and my family and i believe that even a thousand words won't be enough to describe your good great works. If you are in need of any spiritual help why not run to Dr Hope? just contact Dr Hope on his email: Ultimatesolutionhome55@yahoo.com or you can also contact him on his email on:
http://ultimatesolutionho5.wixsite.com/dr-hope and he will help you. Contact him now and get your solution. Thank you Dr Hope for you are indeed a great man. 


September 17, 2016 | Tracey Danny wrote:

Dr Hope is a very genuine spell caster and a very unique and a powerful man of wisdom. Dr Hope brought back joy to my life and to my family again by bringing my husband back to me, by curing my sister's cancer disease and by also curing my cousin's hiv virus with his special powers. Dr Hope is really a gifted, amazing, and a very good man who truly knows how to do his job better and if not for him what would my life turn to? Dr Hope thank you so much and i will never stop talking about your great deeds. Words can not express how much good you have done for me and my family and i believe that even a thousand words won't be enough to describe your good great works. If you are in need of any spiritual help why not run to Dr Hope? just contact Dr Hope on his email: Ultimatesolutionhome55@yahoo.com or you can also contact him on his email on:rnhttp://ultimatesolutionho5.wixsite.com/dr-hope and he will help you. Contact him now and get your solution. Thank you Dr Hope for you are indeed a great man. 


September 17, 2016 | joel sandra wrote:

I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: azaspellcaster@gmail.com or you can whats app his line:+2348107155060 


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September 17, 2016 | nandy wrote:

I have heard of him before, but after watching his documentary “healing spell” I was turned on to the full story rnrnabout Dr osayi. He is a man who has the Cure for AIDS,Cancer, Sickle Cell, Herpes,warts,HPV, ALS Diabetes and every kind of rnrnillness, through a whole foods vegetarian diet and natural herbs, what he calls Electric Foods. He went to herbal school college, rnrnand has thousands of testimonials of healing people with Herpes, HIV AIDS, and HPV several other rnrnillnesses. Lisa “Left Eye” Lopes of TLC was cured of Herpes by him and she was so moved by his knowledge that she went on rnrnto spread the word about him in every way she could. She also opened a Cultural/Healing Center for Children in USA rnrnbefore she passed away tragically. This all inspired me that i had to contact Dr osayi, because i was having Herpesrnrnwhich could not be cure but was only managed, when i contacted Dr. osayi, he sent me some herbs which i took, and now i can tell you, i have be totally cured of HIV, My Name is nandy , I am from the Canada, Please if you have any form rnrnof illness especially STD,Infertility,Herpes etc. you can contact him immediately through his Email AddressrnrnName: nandy lawsonrnrnEmail: drosayispellhome50@gmail.comrnrnPHONE/WHATS APP :+2349057656739 


September 17, 2016 | joel sandra wrote:

I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: azaspellcaster@gmail.com or you can whats app his line:+2348107155060 


September 17, 2016 | mailkproyr wrote:

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September 16, 2016 | tracy morgan wrote:

I NEED A SPELL CASTER HELP URGENTLY TO GET BACK EX LOVER: WIFE, HUSBAND, GIRLFRIEND OR BOYFRIEND? CONTACT (dr.okeytospelltemple@gmail.com) or call he on phone +2349050141440/whats-app.. .he is willing and ever ready to help.okay. This is what i want to tell you all out there,That is thinking that all hope is lost..My name is TRACY MORGAN,I want to testify of a great spell caster known as "Dr OKEYTO" I have never see or met he but he help me cast a spell that bring back my ex this great spell caster cast a spell for me on my ex husband and he returned back to me within 24hours. I will forever be grateful to this great man and his great ancient deity of African. Contact him today for any kind of help...The spell caster is so powerful and after that he helped me with a pregnancy spell and i got pregnant again ..we are now happy been together again and with lovely kids..This spell caster has really changed my life and i will forever be thankful to him..he has helped many friends too with similar problem and they are happy and thankful to him..This Dr Okeyto is indeed the most powerful spell caster that i have ever experienced in life..Am Posting this to the Forum in case there is anyone who has similar problem and still looking for a way out. 


September 15, 2016 | joel sandra wrote:

I never believed in spell casting, but After 6 years of dating, I still imagine how Dr Aza brought my ex lover back to me in just 24 hour. No one could have ever made me believe that there is a real spell caster that really work. am sandra by name,I want to quickly tell the world that there is a real on line spell caster that is powerful and genuine, His name is Dr Aza, He helped me recently to reunite my relationship when my ex lover who left me, When i contacted Dr Aza he cast a love spell for me and my ex lover who said he doesn't have anything to do with me again, he called me and started begging me. he is back now with so much love and caring. today i am glad to let you all know that this spell caster have the powers of bring lovers back. because i am now happy with my lover,and the most surprise,is that our love is very strong,every day is happiness and joy. and there is nothing like been with the man you love.i am so happy my love is back to me with the help of Dr.Aza if you have similar problem i will advice you to contact him ,he is there to help you and put a smile on your face. his email: azaspellcaster@gmail.com or you can whats app his line:+2348107155060 


September 14, 2016 | Christabel wrote:

rnrnrnmy name is Christabel from Australia I want to thank DR acient for what he did for me he brought happiness back into my life. My boyfriend left me and told me is over I was devastated as I loved him so much I decided to contact a spell caster and I met a friend who told me of a great doctor and I decided to contact him and he told me in three days my boyfriend will call me and beg me to ace pt him I thought he was joking in three days everything happened as he said I am so happy now.rnYou can also contact him on his email acientspiritspell@gmail.com for any kind of help. 


September 13, 2016 | Jaharys Archilla wrote:

I’d doubts about miracle spells after buying many spells that never labored and really made the decision never to purchase a spell again. I’ve attempted different casters. A number of them never clarified me once i compensated and were apparent con artists, some really cast a spell but for whatever reason it did not work. I Quickly saw a video online with an individual who was mentioning she’d results with DR, Olorun. Regardless of the accusations I’d her site looked real and so i gave spells a final try. It’s most likely the very best decision I available within my existence since it labored and my boyfriend returned beside me Once I split up with my ex I spent considerable time wanting which i could just turn the time backwards.He assisted me just do that. Actually our relationship seems like the split up never even happened. We’ve never been this happy or passionate, all because of Dr Olorun. If you are searching for a great spell caster your search is over, Dr Olorun exceed other spell-casters, you are able to contact her straight on (Olorunoduduwaspiritualtemple@gmail.com) 


May 6, 2016 | farida yasmin wrote:

i want to study on serum magnesium level of eclamptic patients. in my study place prevalence rate of eclampsia is 2.3%.duration of my sample collection is 06 months. how can i calculate my sample size? please help me to estimate the sample size of my study.rnregardsrndr. shumi 


February 4, 2016 | Aijaz Noonari wrote:

Thanks Jeff Sauro, I got lot of help understanding the confidence interval. I think I may need such sort of help in future too. 


January 29, 2016 | charlotte wrote:

thanks so helpful information  


January 5, 2016 | Ahmed Tahon wrote:

I a candidate in a CIA, I have study unit about statistical sampling, I have understood all issues except the clear relation between the condfidence level and the confidence interval; is it positive or negative relation? and this relation can change in case of descrete variables or continious variables? your answer will be highly appreciated 


November 19, 2014 | Gaha wrote:

Great post! Is it then possible to compute the exact p-value (significance) given the confidence interval? 


March 22, 2012 | Jeff Sauro wrote:

You're exactly right Michael, thanks for articulating that. It is much more likely that the true population mean will fall in the middle of the interval (near the average) than near the ends—it’s a sideways normal curve.  


March 21, 2012 | Michael Zuschlag wrote:

#11 (building on #4). The further you get from the middle of the confidence interval, the less likely the real average/percentage is there. I find some people tend to think of a confidence interval as a uniform probability distribution. I try to remind them that the 95% confidence interval is the _reasonably plausible_ range of values for the average/percentage. The average/percentage is _probably_ (i.e., usually) in a range about half as wide. 


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