# The Normal Distribution The Log Normal Distribution Geometric

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The Normal Distribution & The Log Normal Distribution & Geometric Mean • Level: Intermediate • Version No: 1 • Version Date: June 2013 Rehena Sultana, M. Sc (Statistics) Associate Biostatistician Duke-NUS Medical School

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Financial Disclosures • No Disclosures

Learning Objectives • Normal Distribution – Properties – Standard Normal Distribution – The Central Limit Theorem • Log Normal distribution – Properties – Geometric mean

The Normal Distribution

Outline Normal distribution • Properties • The standard normal distribution • Application • Central Limit Theorem

Definition: Normal Distribution • Discovered by De-Moivre in 1733. • A continuous probability distribution. – Continuous variable uncountable number of values. – Probability distribution realization of data

Density Definition: Normal Distribution Height of Singaporeans (cm)

Properties: Normal Distribution • Bell shaped. • Uni-modal. • Symmetrical. • The mean, median, and mode are equal.

Properties: Normal Distribution (con’t) • Asymptotic to X – axis. • The amount of variation in the random variable determines the height and spread of the normal distribution. • If the random variable X follows normal distribution with mean and variance 2, then we write X ~ N( , 2). – and 2 are the parameters of the distribution.

Properties: Normal Distribution (con’t) • The probability density of X~N(µ, σ2) can be written as and = 3. 14 • If x ~ N(µ 1, 12) and y ~ N(µ 2, 22) and they are independent to each other, then x + y ~ N(µ 1 + µ 2, 12 + 22).

Properties: Normal Distribution (con’t) N(100, 64) N(100, 100) N(130, 100) N(100, 100)

Properties: Normal Distribution (con’t) the f o % 68 er d n u a are mits i l ± µ the f o 95% er d n u area o f the ± 2 der 9µ 9 % usn a t i e r m i al 3 ± µ limits Figure: Probability density function of normal distribution

Standard Normal Distribution: Definition • If X ~ N(µ, σ2), then x any value in horizontal axis µ mean of the normal distribution σ sd of the normal distribution z scaled value. Z follows standard normal distribution. General notation for standard distribution is Z~N(0, 1).

Applications of normal distribution in Clinical Research (examples) • Height / Weight • IQ scores • Body temperature • Diastolic / systolic blood pressure • Repeated measurements of same quantity

Central Limit Theorem (CLT) If Xi (i = 1, . . , n) are independent random variables with mean µi and σi 2, then under central limit theorem, the random variable Sn= X 1+. . +Xn is asymptotically normal For the purpose of applying the central limit theorem, we will consider a sample size to be large when n > 30.

Application of CLT: Example A random sample of n = 64 observations are drawn from a population of mean µ = 15 and standard deviation = 4. As sample size is large enough we can easily apply CLT. Hence we can write

Application of CLT: Example http: //www. astroml. org/book_figures/chapter 3/fig_central_limit. html

Lognormal Distribution & Geometric Mean

Outline • Log normal distribution – Property – Application • Geometric mean

Application of Log normal distribution in Clinical Research (examples) • Latency period of diseases like chicken pox, bacterial food poisoning, amoebic dysentery • Survival times of cancer diagnosis • Age of onset of disease e. g. Alzheimer

Definition: Log Normal Distribution • A continuous probability distribution. • If X is normal distribution, then y = ex follows log normal distribution. • General Notation: X ~ LN(µ, σ2) • Density function of X when X ~ LN(µ, 2) is: , x > 0

Properties: Log Normal Distribution • Mean: • Variance: • If Y ~ N(µ, σ2), then ey ~ LN(µ, σ2). • If Y ~ LN(µ, σ2), then log(X) ~ N(µ, σ2) • The product of two log normal distribution is also lognormal.

Properties: Log Normal Distribution • Lognormal distribution is positively skewed. • Mode > Median > Mean • Most of the properties of the lognormal distribution can be derived by transforming a corresponding normal distribution and vice versa

Geometric Mean • Geometric mean of data set x 1, x 2, …, xn is • Log of the geometric mean is

Properties: Geometric Mean • As the log of the geometric mean is an average, we can apply CLT (under same assumption) • Geometric mean ≤ mean (arithmetic mean). • It gives comparatively more weight to small items. • It is not affected much by fluctuations of sampling.

Application of Geometric Mean • Rate of population growth. • Growth rate of infection rate. • Interest rate in bank. • To construct index numbers.

Geometric Mean: Example Suppose that in a population of interest, the prevalence of a disease rose 2% one year, then fell 1% in next year, then rose 2%, then rose 1%; since these factors act multiplicatively it makes sense to consider the geometric mean. (1. 02 x 0. 99 x 1. 02 x 1. 01)1/4 = 1. 01 Hence overall increased disease prevalence over these 4 years was 1%.

Summary • Lognormal distribution is always a better model for original data • Normal distribution is very common because of symmetry nature. • People like to use additive property compared to multiplicative property, so it makes sense to use geometric mean. • Defining the normal limits of a clinical measurement is not straightforward and requires clinical thinking.

Key References • Harvey Motulsky. Intuitive Biostatistics: A nonmathematical guide to statistical thinking (2 nd ed. ): Oxford • Aitchison J, Brown JAC. 1957. The Log-normal Distribution. Cambridge (UK): Cambridge University Press. • Wlodzimierz Bryc. Normal Distribution characterizations with applications: Lecture Notes in Statistics 1995, Vol 100 (Revised June 7, 2005)

Acknowledgements • Special Thanks to Ms. Shruti Shah for helping to amend our slides.

Thank you

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