ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Standard Normal Distribution Table

Updated on January 8, 2018
tamarawilhite profile image

Tamara Wilhite is a technical writer, industrial engineer, mother of two, and published sci-fi and horror author.


What is a standard normal distribution table? What statistics are used with a standard normal distribution? Can other distributions be converted to a standard normal distribution?

Types of Frequency Distribution Charts

A frequency distribution chart uses a graph to show the incidence of a particular trait. The X axis indicates a variable or value, such as household income or the width of a manufactured part. The Y axis indicates the number of occurrences for the variable. Curves can be negatively skewed or skewed to the left, positively skewed or skewed to the right or bell-shaped.

Some frequency distributions, such as household income, are naturally skewed. Other variables, such as human height, are more symmetrical and evenly distributed. The bell-shaped frequency distribution is called the normal distribution. In a normal distribution, the most common frequency is in the middle and each side is symmetrical.

What If My Distribution Table Has More Than One Peak?

Bi-modal distributions have two peaks. Multi-modal distributions have three or more peaks on the distribution table. If your frequency distribution has two or more peaks, you may have a mixture of two different normal distributions. Input from two different shifts or suppliers create more than one peak; separate the data by source to get the normal distribution table for each sample.

The true curves of a standard normal distribution may become clearer after you have far more data points to gain a clear and much smoother distribution table. If the data still generates more than one peak, you cannot use a standard normal distribution table and related statistics to determine the odds of a value occurring or what percentage of product will fall outside of a selected range such as specification limits.

Six Sigma refers to the range within six standard deviations on either side of the mean of a standard normal distribution.
Six Sigma refers to the range within six standard deviations on either side of the mean of a standard normal distribution. | Source

Statistics Derived from Normal Distributions

In a standard normal distribution, the mean, the median and the mode all occur at the same point, the peak of the bell-shaped curve. Standard deviation is the positive square root of the variance.

For a standard normal distribution, the interval plus and minus one standard deviation from the middle of the mean will include roughly 68% of all occurrences.The range plus and minus two standard deviations from the mean will include roughly 95% of all data points. Plus or minus three standard deviations will include 99.7% of all data points.

The term Six Sigma refers to plus and minus six standard deviations from the mean on a standard normal distribution, and this range will include all values except 37 out of a million data points.

Standard Normal Curves and Probabilities

The probability of a particular occurrence, variable value or range of values can be calculated using the standard normal curve. To determine the probability of an occurrence, the area under the normal curve is set to one. The probability of a value or range of values is now found based upon the area of that section of the standard normal distribution. The area between the mean and one standard deviation on either side of the mean is 0.3413.

The odds of the variable falling within the median and one standard deviation is 34.13%. Converting the variables to a Z-score simplifies the process of determining the probability of a particular value or range of values.


A Z-score is calculated using the equation X – (mean) divided by the standard deviation. The Z-score can then be referenced on a standard normal table to find the probability of that value occurring. The Z-score will tell you the odds of that value or a lower one being found. Z-scores can be positive or negative or even equal zero.

If the mean of a data set is 20 and the standard deviation is 10, let’s calculate the Z-score. For a value of 30, the Z-score is (30-20)/10 or 1.0. For a value of 10, the Z-score is (10-20)/10 or -1.0.

The Z-score allows you to determine the probability of your product and its variation falling outside of the control limits that you set or find the statistical control limits that will contain 90% of your manufactured product. You can use a Z-score to determine how many outliers or exceptions you will see if you set the cut off at a specific level, such as how many people will not have their issue resolved if customer support staff can only spend five minutes on the phone with each. Conversely, the Z-score will tell you at what point 95% of all calls would be resolved, based on your particular standard normal curve.

The Z-score can handle data from any source and any process, converting it to a number that is statistically valid as long as the underlying process is not changing and generates a standard normal distribution. Z-scores can also be used to determine confidence intervals. A greater level of confidence will increase the width of the confidence interval.

Control Limits and Standard Normal Distributions

Statistical process control limits use statistics to set control limits, the lines on a run chart that indicate when a product is out of spec. While a run chart will bounce between the control limits most of the time, it will occasionally generate a data point outside of the specification limits. If all of these data points were put in a frequency distribution chart, you will see a standard normal distribution.

The middle line of the control chart is the median or mean of the standard normal distribution. A run chart with a series of points near the control limit or an increasing number of out of spec data points means that the average or process variability is changing.

Approximating Standard Normal Distributions with Binomial Distributions

Most other types of distributions cannot be converted to standard normal distributions to take advantage of the Z score and other simplified statistics that work with standard normal distributions. Binomial distributions can approximate a normal distribution.

The mean for a binomial distribution is approximated by multiplying the number of trials by the odds of success for a single trial. The standard deviation is the square root of the number of trials multiplied by the total number of successes divided by the number of trials. Using the normal approximation of a binomial distribution allows you to use Z scores to estimate the probability of a particular outcome or set of outcomes.


1. "Statistics for Business and Economics" by David Ray Anderson et al.
2. "The Complete Idiots Guide to Lean Six Sigma" by Neil DeCarlo


Submit a Comment
  • Luis G Asuncion profile image

    Luis G Asuncion 

    2 months ago from City of San Jose Del Monte, Bulacan, Philippines

    Thanks for this article. I love it.


This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

Show Details
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
ClickscoThis is a data management platform studying reader behavior (Privacy Policy)