Home > Articles > Home & Office Computing > Microsoft Applications

The Noncentrality Parameter in the F Distribution

The noncentrality parameter changes the shape of the F distribution in the analysis of variance, when the treatments have an actual effect in the populations. Excel expert Conrad Carlberg, author of Predictive Analytics: Microsoft Excel, shows how to calculate the noncentrality parameter. He also shows how to use it with the probability density function to create Excel charts that display the shape of noncentral F distributions.
Like this article? We recommend

Like this article? We recommend

An F-ratio is the ratio of two variances. When used in the context of the analysis of variance, one variance (the numerator) is based on the variability of the means of sampled groups. The other variance, in the denominator, is based on the variability of individual values within groups.

When the group means differ, the numerator involves a noncentrality parameter that stretches the distribution of the F-ratio, out to the right. This article discusses the meaning, calculation, and symbolic representation of the noncentrality parameter in the literature on ANOVA.

The final article in this series of four papers discusses the relationship of the noncentrality parameter to the calculation of the statistical power of the F-test in an ANOVA. The concept of statistical power is discussed in the first article in this series, and the statistical power of the t-test is discussed in the second article.

Variance Estimates

The rationale for the F-test in ANOVA provides that there are two ways to estimate the variance in the measures of treatment outcome:

  • Between Groups. An estimate that depends exclusively on the differences between group means and the number of observations per group. The estimate is based a rearrangement of the formula for the standard error of the mean.
  • Within Groups. An estimate that depends exclusively on the variance within each group. This estimate does not involve the differences between the group means, but is the average of the within-group variances.

Both figures estimate the same value: the variance of the individual outcome measures. We can form a ratio, termed the F-ratio, of the two variance estimates, dividing the between-groups estimate by the within-groups estimate.

The derivation of the formulas for the variability within groups and the variability between groups is not given here; see Statistical Analysis: Microsoft Excel 2010 (Que Publishing, 2011) for that information. It turns out, though, that:

  • The within groups figure comprises the variance in the population from which the subjects were sampled.
  • The between groups figure comprises the variance in the same population, plus any variance attributable to the differences between the group means.

Central F Distributions

So we wind up with this F-ratio:

    F = (σε2 + σΒ2) / σε2


    σε2 = Estimate of population variance


    σΒ2 = Estimate of variability due to differences between group means

If there are no differences between group means in the population, then σΒ2 is zero and the F-ratio is:

    F = (σε2 + 0) / σε2 = 1.0

When σΒ2 is zero, the ratio follows a central F distribution.

We sample the subjects that make up our treatment groups and control groups from populations: the population of subjects from which we obtain a sample for Group 1, the population of subjects from which we obtain a sample for Group 2, and so on. Those populations would have mean values on the outcome measure if we were able to administer the treatment to a full population. If there is no difference among those population means, we expect the F-ratio to equal 1.0.

Of course, using our sample data, we often calculate an F-ratio that does not equal 1.0, even when the F-ratio comes from a central F distribution. That's because our samples are not perfectly representative of the populations on which they are based. Figure 1 shows the relative frequency of different F-ratios based on samples when there are no differences in the means of the populations.

Figure 1 The distribution of F-ratios, when there are no population differences in group means, is termed the central F distribution.

The distribution of central F-ratios is determined solely by the number of degrees of freedom for the numerator and the number of degrees of freedom for the denominator.

You generally decide that an F-ratio is "statistically significant" if you would observe it by the accident of sampling error, when its population value is 1.0, less than 5% of the time (that is, p < .05), or less than 1% of the time (p < .01), or less than 0.1% of the time (p < .001) and so on. Figure 1 shows the relative likelihood of those accidents of sampling error.

The likelihoods are termed alpha levels. You might decide that you want to limit the mistake of deciding there are differences between means, when there are not, to 5% of the possible experiments like this one that you might carry out. Then you would set alpha to .05.

If your eventual F-ratio turned out to be larger than the F-ratio that cuts off the top 5% of the distribution, you would decide that a true difference in means exists. If no difference in the population means exists, your result would come about only 5% of the time. It is more rational to decide that there is a difference between the population means than it is to decide that a 19-to-1 shot came home.

Noncentral F Distributions

But what if there is a difference in the population means? In that case, the distribution of F ratios does not follow the central F distribution shown in Figure 1. It is instead what's called a noncentral F. Figure 2 shows several noncentral F distributions.

Figure 2 The larger the noncentrality parameter, the more stretched-out the F distribution.

The noncentrality parameter is closely related to the σΒ2 term in the expected value of the F-ratio, shown earlier as:

    F = (σε2 + σΒ2) / σε2

When there are differences between the group means in the population, the term σΒ2 is expected to be greater than zero: It is the variance of the group means. So when that variance, the σΒ2 term in the numerator, is greater than zero, the numerator gets larger, as does the value of the F-ratio, and the distribution stretches out to the right in its chart.

The noncentrality parameter has been defined variously and inconsistently for many years, but the literature on statistics appears to be settling in on both a generally accepted symbol for the parameter (the Greek letter lambda, λ) and on the formula. For example, one generally well-regarded text in its 1968 edition used the Greek character δ to represent the noncentrality parameter and gave this formula:

where j indexes the groups, n is the number of observations per group, and β is the difference between a group mean and the grand mean.

But the same book's 2013 edition gives the character as λ and the equation as:

which is algebraically equivalent except that it is the square of the version in the 1968 edition. With an equal number of observations per group, λ is the ratio of the ANOVA table's Sum of Squares Between to its Mean Square Within.

Other well-regarded books, which are now 30 years old, confuse the noncentrality parameter with a related figure, Φ, which has for decades been used to look up the value of statistical power in charts. For more on how Φ is used (and to get a sense of the difficulty of using those old charts) see, for example, this article excerpt from the Journal of the American Statistical Association.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020