Measuring Usability
Quantitative Usability, Statistics & Six Sigma by Jeff Sauro

Interactive Graph of the Standard Normal Curve

Jeff Sauro • December 14, 2007

Hover over the normal curve to display the area and z-score.To enter specific values use the Z-Score to Percentile Calculator or the Percentile to Z-Score Calculator.

Two-Tailed Area Under the Normal Curve

 

The values presented above are computed by adding up all the area under the curve(the shaded area) from the point where the mouse is hovering to its opposite-signed point. For example, by hovering over 1σ the area between -1σ and 1σ is shaded and represents about 68% of the area of the curve. This corresponds to a Z-Score of 1. The area above 1σ and below -1σ is 1 minus the proportion of area covered or about 32%. Contrast the area generated from these Z-score with the area generated below. Add any mean and standard-deviation in the boxes. As an example, a mean of 100 and SD of 16 (similiar to the distribution of IQ scores) has been added to the input boxes. We can then see that 95% of the IQ tests scores should fall between 68 and 132. To enter specific values use the Z-Score to Percentile Calculator.

Download an Excel version of the Normal Curve Graph or take a Crash course in Z-Scores


One-Tailed Area Under the Normal Curve

 

The values presented are computed by adding up all the area under the curve(the shaded area) from negative infinity to the point where the mouse is hovering. For example, by hovering over 1σ about 84% of the area is shaded. This corresponds to a Z-Score of 1. The area above 1σ is 1 minus the proportion of area covered or about 16%. As an example, a mean of 100 and Standard Deviation of 16 (similiar to the distribution of IQ scores) has been added to the input boxes. So for example, if you scored a 132 on an IQ test, you would have an IQ higher than over 97% of the population (a z-score of 2). To enter specific values use the Percentile to Z-Score Calculator.

A Note about the Calculations & Decimal Precision

The values presented in the graphs above are approximations derived from the work of Abramowitz & Stegun. If you need precision to more that 3 decimals you are encouraged to consult multiple published tables of Z-Values.If you need to look up specific values then you will most likely find it easier to use the Z-Score to Percentile Calculator and the Percentile to Z-Score Calculator

About Jeff Sauro

Jeff Sauro is the founding principal of Measuring Usability LLC, a company providing statistics and usability consulting to Fortune 1000 companies.
He is the author of over 20 journal articles and 4 books on statistics and the user-experience.
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Posted Comments

There are 32 Comments

March 21, 2014 | Mary Wilkinson wrote:

Great! Thank you for sharing!  


September 18, 2013 | Emma wrote:

Thank you for these really useful tools! 


July 1, 2013 | Anne Torrisi wrote:

It is what I was looking for to understand my class problem. 


June 17, 2013 | Umar wrote:

Thank you for this. It's really helping me in writing up my report and presentation for class 


May 10, 2013 | AFAA wrote:

nice job ... Thanks  


March 31, 2013 | mary zink wrote:

this is an excellent tool 


October 2, 2012 | Weston wrote:

Excellent! I've been looking for an interactive Z-curve to help my students understand it better. 


September 17, 2012 | wilma wrote:

Thank you so much for this website, it has helped me more than my professor trying to teach us in class! 


July 15, 2012 | anonymous wrote:

Thanks for the interactive graph! I was able to easily get my percentile on an exam I recently took! 


June 14, 2012 | logan Mitchell wrote:

Just did it in class with my favourite teacher. Very Good. Can understand.... 


April 30, 2012 | S Srinivasan wrote:

It is really amazing.
I am able to appreciate that a lot of efforts have gone in to the making of this wonderful stat-tool.
Thanks for the efforts and the excellent presentation and explanation.
This would save a great deal of time and efforts of users.
Regards,
S Srinivasan ('Srini") 


February 29, 2012 | anonymous wrote:

this is a great little tool 


February 17, 2012 | jacob davis wrote:

dont get it 


November 22, 2011 | Dr. A wrote:

fun and creative! Thanks 


November 6, 2011 | Dr.S.Sakthivel wrote:

Article is simple & clear  


November 5, 2011 | Art wrote:

Interactive curve says more than all the words normally used to explain this. 


September 13, 2011 | anonymous wrote:

If you had all the math somewhere (i.e. HOW it takes in a z-score and outputs a Percentile), that would be great! 


July 19, 2011 | Donna Smith wrote:

Thankyou so much!!! With the help of this fantastic web site, I just might pass my Statistics class... 


July 19, 2011 | anonymous wrote:

I am taking Inferential Stats (after 40 years away from the classroom) and have been on another planet. This really helped me understand z-scores. and get back into the classroom. The graphics are excellent teachers. I am a visual learner. 


November 4, 2010 | Six Sigma Student wrote:

Excellent definitions for z-scores and standard deviation. I especially liked the interactive curve. That
pulled it all together for me. I've been studying the Six Sigma workbook for a week and was utterly lost
on z-scores and how they relate to the normal curve graph. Now, it's all clear. Thank you so much.  


December 7, 2009 | srinivasan wrote:

Is 3.4 ppm in 6sigma is applicable to unilateral tolerance , how 


December 7, 2009 | srinivasan wrote:

Is z score and std deviation are same are different 


December 7, 2009 | srinivasan wrote:

I would like to know the z score and ppm calculation for unilateral & bilateral toleraqnce 


November 24, 2009 | ROLY B. BAYO-ANG wrote:

VERY INTERACTIVE SITE 


November 13, 2009 | Christy Muth wrote:

What percentage of architects have Z scores (a) above .10, (b) below .10, (c) above .20, (d) below .20, (e) above 1.10, (f) below 1.10, (g) above -.10, (h) below -.10?
Suppose that the scores of architects on a particular creativity test are normally distributed. What percentage of architects have Z scores (a) above .10, (b) below .10, (c) above .20, (d) below .20, (e) above 1.10, (f) below 1.10, (g) above -.10, (h) below -.10? 


July 20, 2009 | Adell wrote:

I have the results of a final exam for 2 classes. I have the number of students that scored correctly on each question and the percentage. I would like to determine if the students did better on the questions that were "taught" 4th quarter compared to what they had been taught the other 3 quarters. Is a z test appropriate? 


February 16, 2009 | anonymous wrote:

would be much more useful if you also include much larger z-scores 


October 2, 2008 | Jeseme wrote:

How do i compute for the standard score (z-score)of the following data? How will the graph look like?
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March 19, 2008 | Kevin Horton wrote:

Jeff, great website. Do you happen to have a page that calculates 95% CIs for percentiles (i.e. 95th percentile)? Thanks. 


February 22, 2008 | Robert wrote:

What is the difference between z-score and standard deviation? 


January 11, 2008 | Sugato Banerjee wrote:

Beautiful but i wish the graph would cover 6 sigma level too 


January 10, 2008 | Sterling Gardiner wrote:

This is just awsome. . .make stats much easier! Thanks! 


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