![Binomial distribution
Assumptions : Each trial has only two possible outcomes.
Probability of success or failure remains
constant from trail to trail.
In n trials, success denoted by r, and failure by n – r;
probability ( r successes out of n trials ) :
p(r) = [n! / r!(n – r)!] . pr qn-r
6](https://image.slidesharecdn.com/qaunitv-140929001053-phpapp02/85/Qaunitv-6-320.jpg)
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- 2. Overview • Types • Importance • Properties • Area under normal distribution • Standard normal curve • Z transformation • Calculating the areas • Application 2
- 3. Introduction • Probability : likelihood of occurrence of an event • Probability distribution : is a mathematical representation of the probabilities associated with the values of random variable • Random variable : variable whose value is determined by outcome of a random experiment 3
- 4. Types I. Discrete probability distributions : Binomial distribution Poisson distribution II. Continuous probability distributions : Normal distribution 4
- 5. Binomial distribution • Bernoulli distribution • Bernoulli process/trial : is one where an experiment can result only in one or two mutually exclusive outcomes. • Binomial distribution : distribution of probabilities where there are only two possible outcomes for each trail of an experiment. 5
- 6. Binomial distribution Assumptions : Each trial has only two possible outcomes. Probability of success or failure remains constant from trail to trail. In n trials, success denoted by r, and failure by n – r; probability ( r successes out of n trials ) : p(r) = [n! / r!(n – r)!] . pr qn-r 6
- 7. Poisson distribution • Limiting the distribution of binomial distribution where number of trials, n, is very large and the probability, p, of success for every trial is very small. • ‘np’ is the fixed number which is called poisson distribution. • This distribution studies the probabilities of rare events which are common in science. 7
- 8. Poisson distribution • Ex : no of defective articles produced by a high quality machine • Probability of r successes : p(r) = e-m mr / r! where r = 0, 1, 2,….. N successes e = 2.7183 (constant) 8
- 9. Normal distribution • First discovered by De Moivre • Also by Laplace and Guass • The Normal distribution is also known as the Gaussian Distribution and the curve is also known as the Gaussian Curve. • Named after German Mathematician Astronomer Carl Frederich Gauss. 9
- 10. Normal distribution • It is defined as a continuous frequency distribution of infinite range (can take any values not just integers as in the case of binomial and Poisson distribution). • Curve with important statistical properties, having a smooth curve with symmetrical distributed items on both sides of the peak, is called normal distribution curve. 10
- 11. Importance • In biological analyses • Sample size too large-normal distribution serves a good approximation of discrete distribution. • Making references regarding the value of population mean from sample mean. 11
- 12. Properties • Bell shaped and symmetrical • Single peak-unimodal • Mean lies at the centre • Mean, median and mode are all equal • The total area under the curve the same as any other probability distribution is 1 or (100%) 12
- 13. • Height declines at either side of the peak • Height max at the mean • Area on both sides is equal to each other • Curve is asymptotic to the base on either side • Probabilities for the normal random variable are given by areas under the curve. 13
- 14. 14 Normal curves are defined by two parameters : • Mean : measure of location • Standard deviation : measure of spread Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. μ
- 15. Variations with mean and standard deviation 15 Same mean with different standard deviation
- 16. Same standard deviation with different mean Different mean values with different standard deviation 16
- 17. Area under normal distribution • ± 1s covers 68.27% area; 34.14% area lie on either side of the mean 17
- 18. • ± 2s covers 95.45% area; 47.73% area lie on either side of the mean 18
- 19. • ± 3s covers 99.73% area; 49.87% area lie on either side of the mean 19
- 20. Area under normal distribution 20 No matter what and are, the area between - and + is about 68%; the area between -2 and +2 is about 95%; the area between -3 and +3 is about 99.7%. Almost all values fall within 3 standard deviations.
- 21. Standard normal curve Curve with zero mean and unit standard deviation 21
- 22. Z transformation • Normal distribution : z = x – / s (sample) z = x – μ / σ (population) 22
- 23. Table of cumulative areas under standard normal cuvre z = x-μ/σ Area from -∞ to Z -2.576 0.005 -2.326 0.01 -1.96 0.025 -1.645 0.05 -1.58 0.057 -1.28 0.10 -1 0.16 -0.5 0.31 0 0.50 23
- 24. Table of cumulative areas under standard normal cuvre z = x-μ/σ Area from -∞ to Z 0.50 0.69 1.00 0.84 1.28 0.90 1.58 0.943 1.645 0.95 1.96 0.975 2.326 0.99 2.567 0.995 24
- 25. Calculating the areas • Probability of choosing a tablet at random that weighs between 190 and 210 mg ? 25
- 26. Calculating the areas • To calculate the area between 190 and 210 : a. Area between -∞ and 210 : z = x-μ/σ = 210-200/10 = 1 From the table area between -∞ and 210 is 0.84. 26
- 27. b. Area between -∞ and 190 is 0.16 z = 190-200/10 = -1 27
- 28. c. Area between 190 and 210 is 0.68 = (area between -∞ and 210) – (area between -∞ and 190 is 0.16) • Probability of choosing a tablet at random between 190 and 210 mg is 0.68 or 68% 28
- 29. Practical Application • According to USP, for tablets of weight 100mg : not more than 2 tablets – may deviate by more than 10% (in a batch of 20 tablets) no tablet must differ by more than 20% • For passing this test, 98% of the tablets must weight within 10% of the mean 29
- 30. Practical Application • So, 1000 tablets from a batch of 30,00,000 tablets are weighed and mean and standard deviation are calculated with following formulae's : • Mean = 101.2, standard deviation = 3.92 30
- 31. Practical Application • Within 10% of mean (101.2) = 91.1 and 111.3 • We should find out what probability of tablets is between 91.1 and 111.3? (If 98% or more, USP requirements are met) • For this, a normal distribution is constructed and areas are found out. 31
- 32. a) Area between -∞ and 111.3 z = x-μ/σ = 111.3 – 101.2 / 3.92 = 2.58 From table, area is 0.995 b) Area between -∞ and 91.1 z = 91.1 – 101.2 / 3.92 = -2.57 From table, area is 0.005 32
- 33. c) Area between 91.1 and 111.3 = 0.995 – 0.005 = 0.99 Thus probability of tablets is between 91.1 and 111.3 is 0.99 or 99% Hence passes the test 33
- 34. References • Khan and Khanum, Fundamentals of biostatistics, 3rd edition, Ukaaz Publicaions, pg no. 181 • Leon Lachman and Herbert A. Lieberman, The theory and practice of Industrial Pharmacy, 2009 edition, pg no. 246 • Sanford Bolton, Charles Bon, Pharmaceutical Statistics: Practical and clinical applications, 4th edition, volume 135, pg no. 54 34
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Editor's Notes
- #5: Measured more and more precisely acc to sensitivity of the measuring instruments. ex wts of tablet, bp.