Home > Articles > Home & Office Computing > Microsoft Applications

  • Print
  • + Share This
This chapter is from the book

The Central Limit Theorem

There is a joint feature of the mean and the normal distribution that this book has so far touched on only lightly. That feature is the Central Limit Theorem, a fearsome sounding phenomenon whose effects are actually straightforward. Informally, it goes as in the following fairy tale.

Suppose you are interested in investigating the geographic distribution of vehicle traffic in a large metropolitan area. You have unlimited resources (that's what makes this a fairy tale) and so you send out an entire army of data collectors. Each of your 2,500 data collectors is to observe a different intersection in the city for a sequence of two-minute periods throughout the day, and count and record the number of vehicles that pass through the intersection during that period.

Your data collectors return with a total of 517,000 two-minute vehicle counts. The counts are accurately tabulated (that's more fairy tale, but that's also the end of it) and entered into an Excel worksheet. You create an Excel pivot chart as shown in Figure 7.12 to get a preliminary sense of the scope of the observations.

Figure 7.12

Figure 7.12 To keep things manageable, the number of vehicles is grouped by tens.

In Figure 7.12, different ranges of vehicles are shown as "row labels" in A2:A11. So, for example, there were 48,601 instances of between 0 and 9 vehicles crossing intersections within two-minute periods. Your data collectors recorded another 52,053 instances of between 10 and 19 vehicles crossing intersections within a two-minute period.

Notice that the data follows a uniform, rectangular distribution. Every grouping (for example, 0 to 9, 10 to 19, and so on) contains roughly the same number of observations.

Next, you calculate and chart the mean observation of each of the 2,500 intersections. The result appears in Figure 7.13.

Figure 7.13

Figure 7.13 Charting means converts a rectangular distribution to a normal distribution.

Perhaps you expected the outcome shown in Figure 7.13, perhaps not. Most people don't. The underlying distribution is rectangular. There are as many intersections in your city that are traversed by zero to ten vehicles per two-minute period as there are intersections that attract 90 to 100 vehicles per two-minute period.

But if you take samples from that set of 510,000 observations, calculate the mean of each sample, and plot the results, you get something close to a normal distribution.

And this is termed the Central Limit Theorem. Take samples from a population that is distributed in any way: rectangular, skewed, binomial, bimodal, whatever (it's rectangular in Figure 7.12). Get the mean of each sample and chart a frequency distribution of the means (refer to Figure 7.13). The chart of the means will resemble a normal distribution.

The larger the sample size, the closer the approximation to the normal distribution. The means in Figure 7.13 are based on samples of 100 each. If the samples had contained, say, 200 observations each, the chart would have come even closer to a normal distribution.

Making Things Easier

During the first half of the twentieth century, great reliance was placed on the Central Limit Theorem as a way to calculate probabilities. Suppose you want to investigate the prevalence of left-handedness among golfers. You believe that 10% of the general population is left-handed. You have taken a sample of 1,500 golfers and want to reassure yourself that there isn't some sort of systematic bias in your sample. You count the lefties and find 135. Assuming that 10% of the population is left-handed and that you have a representative sample, what is the probability of selecting 135 or fewer left-handed golfers in a sample of 1,500?

The formula that calculates that exact probability is

or, as you might write the formula using Excel functions:

  • =SUM(COMBIN(1500,ROW(A1:A135))*(0.1^ROW(A1:A135))* (0.9^(1500-ROW(A1:A135))))

(The formula must be array-entered in Excel, using Ctrl+Shift+Enter instead of simply Enter.)

That's formidable, whether you use summation notation or Excel function notation. It would take a long time to calculate its result by hand, in part because you'd have to calculate 1,500 factorial.

When mainframe and mini computers became broadly accessible in the 1970s and 1980s, it became feasible to calculate the exact probability, but unless you had a job as a programmer, you still didn't have the capability on your desktop.

When Excel came along, you could make use of BINOMDIST(), and in Excel 2010 BINOM.DIST(). Here's an example:

  • =BINOM.DIST(135,1500,0.1,TRUE)

Any of those formulas returns the exact binomial probability, 10.48%. (That figure may or may not make you decide that your sample is nonrepresentative; it's a subjective decision.) But even in 1950 there wasn't much computing power available. You had to rely, so I'm told, on slide rules and compilations of mathematical and scientific tables to get the job done and come up with something close to the 10.48% figure.

Alternatively, you could call on the Central Limit Theorem. The first thing to notice is that a dichotomous variable such as handedness—right-handed versus left-handed—has a standard deviation just as any numeric variable has a standard deviation. If you let p stand for one proportion such as 0.1 and (1 - p) stand for the other proportion, 0.9, then the standard deviation of that variable is as follows:

That is, the square root of the product of the two proportions, such that they sum to 1.0. With a sample of some number n of people who possess or lack that characteristic, the standard deviation of that number of people is

and the standard deviation of a distribution of the handedness of 1,500 golfers, assuming 10% lefties and 90% righties, would be

or 11.6.

You know what the number of golfers in your sample who are left-handed should be: 10% of 1,500, or 150. You know the standard deviation, 11.6. And the Central Limit Theorem tells you that the means of many samples follow a normal distribution, given that the samples are large enough. Surely 1,500 is a large sample.

Therefore, you should be able to compare your finding of 135 left-handed golfers with the normal distribution. The observed count of 135, less the mean of 150, divided by the standard deviation of 11.6, results in a z-score of -1.29. Any table that shows areas under the normal curve—and that's any elementary statistics textbook—will tell you that a z-score of -1.29 corresponds to an area, a probability, of 9.84%. In the absence of a statistics textbook, you could use either

  • =NORM.S.DIST(-1.29,TRUE)

or, equivalently

  • =NORM.DIST(135,150,11.6,TRUE)

The result of using the normal distribution is 9.84%. The result of using the exact binomial distribution is 10.48: slightly over half a percent difference.

Making Things Better

The 9.84% figure is called the "normal approximation to the binomial." It was and to some degree remains a popular alternative to using the binomial itself. It used to be popular because calculating the nCr combinations formula was so laborious and error prone. The approximation is still in some use because not everyone who has needed to calculate a binomial probability since the mid-1980s has had access to the appropriate software. And then there's cognitive inertia to contend with.

That slight discrepancy between 9.84% and 10.48% is the sort that statisticians have in past years referred to as "negligible," and perhaps it is. However, other constraints have been placed on the normal approximation method, such as the advice not to use it if either np or n(1-p) is less than 5. Or, depending on the source you read, less than 10. And there has been contentious discussion in the literature about the use of a "correction for continuity," which is meant to deal with the fact that things such as counts of golfers go up by 1 (you can't have 3/4 of a golfer) whereas things such as kilograms and yards are infinitely divisible. So the normal approximation to the binomial, prior to the accessibility of the huge amounts of computing power we now enjoy, was a mixed blessing.

The normal approximation to the binomial hangs its hat on the Central Limit Theorem. Largely because it has become relatively easy to calculate the exact binomial probability, you see normal approximations to the binomial less and less. The same is true of other approximations. The Central Limit Theorem remains a cornerstone of statistical theory, but (as far back as 1970) a nationally renowned statistician wrote that it "does not play the crucial role it once did."

  • + Share This
  • 🔖 Save To Your Account

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