Home > Articles > Home & Office Computing > Microsoft Applications

Statistical Analysis with Excel 2010: Using Excel with the Normal Distribution

Excel offers more worksheet functions that pertain to the normal distribution than to any other. This chapter explains what a normal distribution is and how to use those Excel functions that map it.
This chapter is from the book

About the Normal Distribution

You cannot go through life without encountering the normal distribution, or "bell curve," on an almost daily basis. It's the foundation for grading "on the curve" when you were in elementary and high school. The height and weight of people in your family, in your neighborhood, in your country each follow a normal curve. The number of times a fair coin comes up heads in ten flips follows a normal curve. The title of a contentious and controversial book published in the 1990s. Even that ridiculously abbreviated list is remarkable for a phenomenon that was only starting to be perceived 300 years ago.

The normal distribution occupies a special niche in the theory of statistics and probability, and that's a principal reason Excel offers more worksheet functions that pertain to the normal distribution than to any other, such as the t, the binomial, the Poisson, and so on. Another reason Excel pays so much attention to the normal distribution is that so many variables that interest researchers—in addition to the few just mentioned—follow a normal distribution.

Characteristics of the Normal Distribution

There isn't just one normal distribution, but an infinite number. Despite the fact that there are so many of them, you never encounter one in nature.

Those are not contradictory statements. There is a normal curve—or, if you prefer, normal distribution or bell curve or Gaussian curve—for every number, because the normal curve can have any mean and any standard deviation. A normal curve can have a mean of 100 and a standard deviation of 16, or a mean of 54.3 and a standard deviation of 10. It all depends on the variable you're measuring.

The reason you never see a normal distribution in nature is that nature is messy. You see a huge number of variables whose distributions follow a normal distribution very closely. But the normal distribution is the result of an equation, and can therefore be drawn precisely. If you attempt to emulate a normal curve by charting the number of people whose height is 56", all those whose height is 57", and so on, you will start seeing a distribution that resembles a normal curve when you get to somewhere around 30 people.

As your sample gets into the hundreds, you'll find that the frequency distribution looks pretty normal—not quite, but nearly. As you get into the thousands you'll find your frequency distribution is not visually distinguishable from a normal curve. But if you apply the functions for skewness and kurtosis discussed in this chapter, you'll find that your curve just misses being perfectly normal. You have tiny amounts of sampling error to contend with, for one; for another, your measures won't be perfectly accurate.


A normal distribution is not skewed to the left or the right but is symmetric. A skewed distribution has values whose frequencies bunch up in one tail and stretch out in the other tail.

Skewness and Standard Deviations

The asymmetry in a skewed distribution causes the meaning of a standard deviation to differ from its meaning in a symmetric distribution, such as the normal curve or the t-distribution (see Chapters 8 and 9, for information on the t-distribution). In a symmetric distribution such as the normal, close to 34% of the area under the curve falls between the mean and one standard deviation below the mean. Because the distribution is symmetric, an additional 34% of the area also falls between the mean and one standard deviation above the mean.

But the asymmetry in a skewed distribution causes the equal percentages in a symmetric distribution to become unequal. For example, in a distribution that skews right you might find 45% of the area under the curve between the mean and one standard deviation below the mean; another 25% might be between the mean and one standard deviation above it.

In that case, you still have about 68% of the area under the curve between one standard deviation below and one standard deviation above the mean. But that 68% is split so that its bulk is primarily below the mean.

Visualizing Skewed Distributions

Figure 7.1 shows several distributions with different degrees of skewness.

Figure 7.1

Figure 7.1 A curve is said to be skewed in the direction that it tails off: The log X curve is "skewed left" or "skewed negative."

The normal curve shown in Figure 7.1 (based on a random sample of 5,000 numbers, generated by Excel's Data Analysis add-in) is not the idealized normal curve but a close approximation. Its skewness, calculated by Excel's SKEW() function, is -0.02. That's very close to zero; a purely normal curve has a skewness of exactly 0.

The X2 and log X curves in Figure 7.1 are based on the same X values as form the figure's normal distribution. The X2 curve tails to the right and skews positively at 0.57. The log X curve tails to the left and skews negatively at -0.74. It's generally true that a negative skewness measure indicates a distribution that tails off left, and a positive skewness measure tails off right.

The F curve in Figure 7.1 is based on a true F-distribution with 4 and 100 degrees of freedom. (This book has much more to say about F-distributions beginning in Chapter 10, "Testing Differences Between Means: The Analysis of Variance." An F-distribution is based on the ratio of two variances, each of which has a particular number of degrees of freedom.) F-distributions always skew right. It is included here so that you can compare it with another important distribution, t, which appears in the next section on a curve's kurtosis.

Quantifying Skewness

Several methods are used to calculate the skewness of a set of numbers. Although the values they return are close to one another, no two methods yield exactly the same result. Unfortunately, no real consensus has formed on one method. I mention most of them here so that you'll be aware of the lack of consensus. More researchers report some measure of skewness than was once the case, to help the consumers of that research better understand the nature of the data under study. It's much more effective to report a measure of skewness than to print a chart in a journal and expect the reader to decide how far the distribution departs from the normal. That departure can affect everything from the meaning of correlation coefficients to whether inferential tests have any meaning with the data in question.

For example, one measure of skewness proposed by Karl Pearson (of the Pearson correlation coefficient) is shown here:

  • Skewness = (Mean - Mode) / Standard Deviation

But it's more typical to use the sum of the cubed z-scores in the distribution to calculate its skewness. One such method calculates skewness as follows:

This is simply the average cubed z-score.

Excel uses a variation of that formula in its SKEW() function:

A little thought will show that the Excel function always returns a larger value than the simple average of the cubed z-scores. If the number of values in the distribution is large, the two approaches are nearly equivalent. But for a sample of only five values, Excel's SKEW() function can easily return a value half again as large as the average cubed z-score. See Figure 7.2, where the original values in Column A are simply replicated (twice) in Column E. Notice that the value returned by SKEW() depends on the number of values it evaluates.

Figure 7.2

Figure 7.2 The mean cubed z-score is not affected by the number of values in the distribution.


A distribution might be symmetric but still depart from the normal pattern by being taller or flatter than the true normal curve. This quality is called a curve's kurtosis.

Types of Kurtosis

Several adjectives that further describe the nature of a curve's kurtosis appear almost exclusively in statistics textbooks:

  • A platykurtic curve is flatter and broader than a normal curve. (A platypus is so named because of its broad foot.)
  • A mesokurtic curve occupies a middle ground as to its kurtosis. A normal curve is mesokurtic.
  • A leptokurtic curve is more peaked than a normal curve: Its central area is more slender. This forces more of the curve's area into the tails. Or you can think of it as thicker tails pulling more of the curve's area out of the middle.

The t-distribution (see Chapter 8) is leptokurtic, but the more observations in a sample the more closely the t-distribution resembles the normal curve. Because there is more area in the tails of a t-distribution, special comparisons are needed to use the t-distribution as a way to test the mean of a relatively small sample. Again, Chapters 8 and 9 explore this issue in some detail, but you'll find that the leptokurtic t-distribution also has applications in regression analysis (see Chapter 12).

Figure 7.3 shows a normal curve—at any rate, one with a very small amount of kurtosis, -0.03. It also shows a somewhat leptokurtic curve, with kurtosis equal to -0.80.

Figure 7.3

Figure 7.3 Observations toward the middle of the normal curve move toward the tails in a leptokurtic curve.

Notice that more of the area under the leptokurtic curve is in the tails of the distribution, with less occupying the middle. The t-distribution follows this pattern, and tests of such statistics as means take account of this when, for example, the population standard deviation is unknown and the sample size is small. With more of the area in the tails of the distribution, the critical values needed to reject a null hypothesis are larger than when the distribution is normal. The effect also finds its way into the construction of confidence intervals (discussed later in this chapter).

Quantifying Kurtosis

The rationale to quantify kurtosis is the same as the rationale to quantify skewness: A number is often a more efficient descriptor than a chart. Furthermore, knowing how far a distribution departs from the normal helps the consumer of the research put other reported findings in context.

Excel offers the KURT() worksheet function to calculate the kurtosis in a set of numbers. Unfortunately there is no more consensus regarding a formula for kurtosis than there is for skewness. But the recommended formulas do tend to agree on using some variation on the z-scores raised to the fourth power.

Here's one textbook definition of kurtosis:

In this definition, N is the number of values in the distribution and z represents the associated z-scores: that is, each value less the mean, divided by the standard deviation.

The number 3 is subtracted to set the result equal to 0 for the normal curve. Then, positive values for the kurtosis indicate a leptokurtic distribution whereas negative values indicate a platykurtic distribution. Because the z-scores are raised to an even power, their sum (and therefore their mean) cannot be negative. Subtracting 3 is a convenient way to give platykurtic curves a negative kurtosis. Some versions of the formula do not subtract 3. Those versions would return the value 3 for a normal curve.

Excel's KURT() function is calculated in this fashion, following an approach that's intended to correct bias in the sample's estimation of the population parameter:

The Unit Normal Distribution

One particular version of the normal distribution has special importance. It's called the unit normal or standard normal distribution. Its shape is the same as any normal distribution but its mean is 0 and its standard deviation is 1. That location (the mean of 0) and spread (the standard deviation of 1) makes it a standard, and that's handy.

Because of those two characteristics, you immediately know the cumulative area below any value. In the unit normal distribution, the value 1 is one standard deviation above the mean of 0, and so 84% of the area falls to its left. The value -2 is two standard deviations below the mean of 0, and so 2.275% of the area falls to its left.

On the other hand, suppose that you were working with a distribution that has a mean of 7.63 centimeters and a standard deviation of .124 centimeters—perhaps that represents the diameter of a machine part whose size must be precise. If someone told you that one of the machine parts has a diameter of 7.816, you'd probably have to think for a moment before you realized that's one-and-one-half standard deviations above the mean. But if you're using the unit normal distribution as a yardstick, hearing of a score of 1.5 tells you exactly where that machine part is in the distribution.

So it's quicker and easier to interpret the meaning of a value if you use the unit normal distribution as your framework. Excel has worksheet functions tailored for the normal distribution, and they are easy to use. Excel also has worksheet functions tailored specifically for the unit normal distribution, and they are even easier to use: You don't need to supply the distribution's mean and standard deviation, because they're known. The next section discusses those functions, for both Excel 2010 and earlier versions.

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