Tuesday, June 3, 2014

Overview of the Normal Distribution

Overview of the Normal


This is one of the following eight articles on the normal distribution in Excel

Overview of the Normal Distribution

Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013

Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013

Solving Normal Distribution Problems in Excel 2010 and Excel 2013

Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013

An Important Difference Between the t and Normal Distribution Graphs

The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean

Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

The normal distribution is a very useful and widely-occurring distribution. The normal distribution curve has the well-known bell shape and is symmetrical about a mean. The normal distribution is actually a family of distributions with each unique normal distribution being fully described by the following two parameters: the population mean, µ (Greek letter “mu”), and the population standard deviation, σ (Greek letter “sigma”).

The following Excel-generated image is an example of the PDF (Probability Density Function) of a normal distribution curve whose population mean, µ, equals 10 and population standard deviation, σ, equals 5.

normal distribution, statistics
(Click On Image To See a Larger Version)

Following is image of the PDF of a different normal distribution curve with a population mean, µ, equals 0 and population standard deviation, σ, equals 1. This is a special normal distribution known as the Standard Normal Distribution as shown in the following Excel-generated image:

normal distribution, statistics
(Click On Image To See a Larger Version)

You may notice that the shapes of both normal distribution PDF graphs are the same but the x axis has been shifted and scaled. The entire family of normal distributions is symmetrical about a mean and contains approximately 68 percent of the entire area under the curve within one standard deviation of the mean. Approximately 95 percent of the total area under the curve lies within two standard deviations of the mean, and approximately 99.7 percent of the total curve area within three standard deviations of the mean.

If PDF curves of different normal distributions are created with the mean and standard deviation in the same locations on the horizontal axis, the curves will be in the same place and have exactly the same shape. The two previous images demonstrate this.

If the PDF curves of different normal distributions are placed on the same horizontal and vertical axes, each curve will be shifted left or right so it is symmetrical about its population mean, µ, and its width will be scaled (widened or narrowed) depending on the size of its population standard deviation, σ.

The PDF graph of a normal distribution with a smaller population standard deviation would appear thinner and taller than the PDF graph of a normal distribution with a larger population standard deviation, if both curves were graphed on the same set of horizontal and vertical axes.


Uses of the Normal Distribution

Many real-valued variables are modeled using the normal distribution. An example of normally-distributed natural phenomenon would be the velocities of the molecules in an ideal gas. Test scores for large numbers of people follow the normal distribution. Random variation in output of a machine barring special causes is often normal-distributed.

In other cases it is often a variable’s logarithm that is normal-distributed. Biological sciences provide many such examples such as the length of appendages and blood pressure. A variable whose logarithm is normal-distributed is said to have a log-normal distribution.

An outcome that is the result of a number of small variables that are additive, independent, and similar in magnitude is often normal-distributed. Measurement error of a physical experiment might be one example of such an outcome.

In some fields the changes in the logarithm of a variable are assumed to be normal-distributed. In the financial fields, changes in the logarithms of exchange rates, price indices, and the stock market indices are modeled by the normal distribution.


Useful Normal Approximations of

Other Distributions

Several distributions can be approximated by the normal distribution under certain conditions. When the normal approximation is valid, tools such as normal-distribution-based hypothesis testing and hypothesis testing can be conveniently applied directly to samples from that-Distribution.


Normal Approximation of the Binomial Distribution

The normal approximation of the binomial distribution is of particular importance. Each unique binomial distribution, B(n,p), is described with the following two parameters: n (sample size) and p (probability of each binary sample having a positive outcome of the two possible outcomes). A binomial distribution is approximately normal-distributed with a mean np and variance np(1-p) if n is large an p is not too close to zero. A unique normal distribution, N(µ,σ), is completely described by the following two parameters: µ (population mean) and σ (population standard deviation). When the normal approximation of the binomial distribution is appropriate, normal-distribution-based hypothesis testing and confidence intervals can be performed with binomially-distributed data by applying the following substitution: N(µ,σ) = N(np, SQRT(np(1-p))).

The normal approximation of the binomial distribution enables the convenient analysis of population proportions with normal-distribution-based hypothesis testing and confidence intervals. A well-known example of this type of analysis would be the estimation of the percentage of a population along with a percent margin of error that will vote for or against a particular political candidate. This is known as a confidence interval of a population proportion.

The t-Distribution (sometimes called the Student’s t-Distribution) is closely related to the normal distribution in function and appearance. The t-Distribution is used to analyze normally-distributed data when the sample size is small (n< 30) or the population standard deviation is not known. The t-Distribution is always centered about its mean of zero and closely resembles the standard normal curve. The standard normal curve is the unique normal distribution curve whose mean equals 0 and standard deviation equals 1. The t-Distribution has a lower peak and slightly thicker outer tails than the standard normal distribution, but converges to the exact shape of the standard normal distribution as sample size increases.

In the real world, normally-distributed data are much more frequently analyzed with t-Distribution-based tools than normal-distribution-based tools. The reason is that the t-Distribution more correctly describes the distribution normally-distributed data in the common occurrences of small sample size and unknown population standard deviation. t-Distribution-based tools are also equally appropriate for analysis of large samples of normally-distributed data because the t-Distribution converges to nearly an exact match of the standard normal distribution when sample size exceeds 30.

Normal Approximation of the Poisson Distribution

Of lesser importance is the normal distribution’s approximation of the Poisson and chi-Square distribution under certain conditions. Each unique Poisson distribution curve is completely described by a single parameter λ. When λ is large, data distributed according to the Poisson distribution can be analyzed with normal-distribution-based tools by making the following substitution:

N(µ,σ) = N(λ, SQRT(λ)).

Each unique Chi-Square distribution curve is completely described by the single parameter that is its degrees of freedom, k. When k is large, data distributed according to the Chi-Square distribution can be analyzed with normal-distribution-based tools by making the following substitution:

N(µ,σ) = N(k, SQRT(2k)).

One of the most useful modeling applications of the normal distribution is due to the fact the means of large random samples from a population are approximately normal-distributed no matter how the underlying population is distributed. This property is described by the Central Limit Theorem. Normal-distribution based analysis tools such as hypothesis testing and confidence intervals can therefore be applied to the means of samples taken from populations that are not normal-distributed.


Statistical Tests That Require

Normality of Data or Residuals

Statistical tests that as classified a parametric tests have a requirement that the sample data or the residuals are normal-distributed. For example, linear regression requires that the residuals be normal-distributed. t-tests performed on small samples (n < 30) require that the samples are taken from a normally-distributed population. ANOVA requires normality of the samples being compared.

When normality requirements cannot be met, nonparametric tests can often be substituted for parametric tests. Nonparametric tests do not requirements that data or residuals follow a specific distribution. Nonparametric tests are usually not as powerful as parametric tests.


History of the Normal Distribution

The first indirect mention of the normal distribution is credited to French mathematician and friend of Isaac Newton, Abraham De Moivre. De Moivre developed such repute as a mathematician that Newton often referred questions to him, saying “Go to Mr. de Moivre. He knows these things better than I do.”

In 1738 De Moivre published the second edition of his The Doctrine of Chances that contained a study of binomial coefficients which is considered to the first reference, albeit, indirect, to the normal distribution. That particular book became highly valued among gamblers of the time. The book noted that the distribution of the number of times that a binary event produces a positive outcome, such as a coin toss resulting in heads, becomes a smooth, bell-shaped curve when the number of trials is high enough. This bell-shaped curve approaches the normal curve. DeMoivre was alluding to what we know today as the normal distribution’s approximation of the binomial distribution. DeMoivre did not speak of the normal distribution in terms of a probability density function and therefore does not receive full credit for discovering the normal distribution.

De Moivre, who spent his adult life in London, remained somewhat poor because he was unable to secure a professorship at any local university, partially due to his French origins. He earned a substantial part of his living from tutoring mathematics and being a consultant to gamblers. One day when De Moivre became older, he noticed that he required more sleep every night. He determined that he was sleeping an extra 15 minutes every night. He correctly calculated the date his own death to be November 27, 1754, the date that the total required sleep time would reach 24 hours.

The first true mention of the normal distribution was made by German mathematician Carl Friedrich Gauss in 1809 as a way to rationalize the nonlinear method of least squares. The normal distribution curve is often referred to as the Gaussian curve as a result.

Gauss was one of the greatest mathematicians who ever lived and is sometimes referred to as “the Prince of Mathematicians” and “the greatest mathematician since antiquity.” Gauss was a child prodigy and made some of his groundbreaking mathematical discoveries as a teenager. Gauss was an astonishingly prolific scientist in many fields. Here is a link to a partial list of over 100 scientific topics named after him:


Significant contributions to the normal distribution were made by another of the greatest mathematicians of all-time, Frenchman Pierre Simon LePlace, who was sometimes referred to as the French Newton. LePlace is credited with providing the first proof of one of statistics’ most important tenets related to the normal distribution, the Central Limit Theorem. LePlace has an interesting life and was appointing by Napoleon to be the French Minister of the Interior shortly after Napoleon seized power in a coup. The appointment lasted about six weeks until nepotism took over and the post was given to Napoleon’s brother.

The distribution’s moniker, the “normal distribution,” was made popular by Englishman Karl Pearson, another giant in the field of mathematics. Karl Pearson is credited with establishing the discipline of mathematical statistics and founded the world’s first university statistics department in 1911 at the University College of London. Many of topics covered in all basic statistics courses such a p Values and correlation are the direct result of Karl Pearson’s work.

Some of Pearson’s publishings, particularly his book The Grammar of Science, provided a number of themes that Einstein would weave into several of his most well-known theories of relativity. Pearson postulated that a person traveling faster than the speed of light would experience time being reversed. Pearson also discussed the concept that the operation of physical laws depended on the relative position of the observer. These are central themes in Einstein’s relativity theories.


Properties of the Normal


This graph of the standard normal distribution’s PDF is once again presented to assist in the understanding of each listed property of the normal distribution.

normal distribution, statistics
(Click On Image To See a Larger Version)

The normal distribution’s probability density function, f(x), has the following properties:

1) It is symmetric about its population mean µ. Half of the values of a normally-distributed population will be less than (to the left of) the population mean and the other half of the population’s value will be greater than (to the right of) the population mean.

2) Its mode and median are equal to the population mean µ.

3) It is unimodal. This means that it has only one peak, i.e., only one point that is a local maximum.

4) The total area under the normal distribution’s PDF is equal to 1.

5) Each unique normal distribution curve is entirely defined by its two parameters population mean µ and population standard deviation σ.

6) The density of the normal distribution’s PDF is the highest at its mean and always decreases as distance from the mean increases.

7) 50 percent of values of a normally-distributed population are less than the population mean and 50 percent of the values are greater than the mean.

8) Approximately 68 percent of the total area under the PDF curve resides within one σ from the mean, approximately 95 percent of the total resides within two σs, and approximately 99.7 percent of the total area resides within three σs from the mean. This is sometimes known as the Empirical Rule or the 68-95-99.7 Rule.

9) f(x) is infinitely differentiable.

10) The first derivative of f(x) is positive for all x < µ and negative for all x > µ.

11) The second derivative of f(x) has two inflection points which are located one population standard deviation above and below the population mean. These inflection points are located at x = µ ± σ. An inflection point occurs at the point that the 2nd derivative equals zero and changes sign as x continues.

12) It is log-concave. A function f(x) is log-concave if its natural log, ln[f(x)], is concave. A log-concave function does not have multiple separate maxima and its tails are not “too thick.” Other well-known distributions that are log-concave include the following:

- exponential distribution

- uniform distribution over any convex set

- logistic distribution

- gamma distribution if its shape parameter is >=1

- Chi-Square distribution if the number of degrees of freedom >=2

- beta distribution if both shape parameters are >=1

- Weibull distribution if the shape parameter is >=1

The following well-known distributions are non-log-concave for all parameters:

- t-Distribution

- log-normal distribution

- F-distribution


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