BINOMIAL Distribution M Sc II Semester What is

BINOMIAL Distribution M. Sc II Semester

What is binomial distribution • A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. • The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice).

Theory simplifies The Bernoulli Distribution • The binomial distribution is closely related to the Bernoulli distribution. According to Washington State University, “If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a binomial Distribution. On the other hand, the Bernoulli distribution is the Binomial distribution with n=1. ” • A Bernouilli distribution is a set of Bernouilli trials. Each Bernouilli trial has one possible outcome, chosen from S, success, or F, failure. In each trial, the probability of success, P(S)=p, is the same. The probability of failure is just 1 minus the probability of success: P(F) = 1 -p. (Remember that “ 1” is the total probability of an event occurring…probability is always between zero and 1). Finally, all Bernouilli trials are independent from each other and the probability of success doesn’t change from trial to trial, even if you have information about the other trials’ outcomes.

Possible outcome is two • Tossing a Coin: • Did we get Heads (H) or / Tails (T) We say the probability of the coin landing H is ½ And the probability of the coin landing T is ½

Variable criteria • The binomial distribution describes the behavior of a count variable X if the following conditions apply: • 1: The number of observations n is fixed. • 2: Each observation is independent. • 3: Each observation represents one of two outcomes ("success" or "failure"). • 4: The probability of "success" p is the same for each outcome.

Criteria for Binomial Distribution • Binomial distributions must also meet the following three criteria: • The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%. • Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial. • The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another.

Binomial formula

Measure of dispersion • The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations. • Dispersion is the state of getting dispersed or spread. Statistical dispersion means the extent to which a numerical data is likely to vary about an average value. In other words, dispersion helps to understand the distribution of the data.

Characteristics of Measures of Dispersion • A measure of dispersion should be rigidly defined • It must be easy to calculate and understand • Not affected much by the fluctuations of observations • Based on all observations

Classification of Measures of Dispersion The measure of dispersion is categorized as: • (i) An absolute measure of dispersion: • The measures which express the scattering of observation in terms of distances i. e. , range, quartile deviation. • The measure which expresses the variations in terms of the average of deviations of observations like mean deviation and standard deviation. • (ii) A relative measure of dispersion: • We use a relative measure of dispersion for comparing distributions of two or more data set and for unit free comparison. They are the coefficient of range, the coefficient of mean deviation, the coefficient of quartile deviation, the coefficient of variation, and the coefficient of standard deviation.

Range • A range is the most common and easily understandable measure of dispersion. It is the difference between two extreme observations of the data set. If X max and X min are the two extreme observations then • Range = X max – X min • Merits of Range • It is the simplest of the measure of dispersion • Easy to calculate • Easy to understand • Independent of change of origin • Demerits of Range • It is based on two extreme observations. Hence, get affected by fluctuations • A range is not a reliable measure of dispersion • Dependent on change of scale

Standard Deviation • A standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. It is denoted by a Greek letter sigma, σ. It is also referred to as root mean square deviation. The standard deviation is given as

Variance

Mean Deviation

Merits & Demerits • Merits of Standard Deviation • Squaring the deviations overcomes the drawback of ignoring signs in mean deviations • Suitable for further mathematical treatment • Least affected by the fluctuation of the observations • The standard deviation is zero if all the observations are constant • Independent of change of origin • Demerits of Standard Deviation • Not easy to calculate • Difficult to understand for a layman • Dependent on the change of scale