Essay on Statistical Analysis (762 Words)

Here is your Essay on Statistical Analysis !

In statistical analysis, one has to deal also with twin problems —”Estimation” and “Testing of Hypothesis”. Estimation is the method of estimating the characteristic value of any character (variable) which has a population of values. !n previous sections, we have observed that as the size of the sample (from a population) increases the distribution of the character is close to a symmetrical unimodal curve.

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This shape of the curve is important as it is found that in nature the distribution of many of the characters follow this trend. So, a general functional form of this curve has been derived. This form should naturally be based on characteristic values of the population. These characteristic values are the parameters of the algebraic equation to the curve.

The algebraic equation for the curve has been derived by considering the nature of the variable, viz., discrete or continuous and the probability of occurrence of this random variable. These considerations gave the way to different probability distribution, the fundamental three probability distri­butions known as ‘Normal’, ‘Binomial’ and ‘Poisson’.

Binomial distribution:

This probability distri­bution has been formulated by James Bernoulli. This is the distri­bution of a discrete random variable. A random variable X is de­fined as follows:

X is said to be a random variable if takes values x₁, x₂ …….x_n with probabilities p₁, p₂,….. p_n respectively such that p₁+p₂+……+p_n = 1.

As the name of the distribution itself indicates, this distri­bution is for the variable or event which can be in two alternative forms. These can be considered as success and failure of the event.

For instance, in tossing a coin we can consider the coin showing up the face E₁ as success and the showing up of the face E₂ as failure. Here the probability of success, P (E₁) and the pro­bability of failure, P (E₂) are both equal to 1/2. But if we consider the throwing of a die which is unbiassed, considering the appea­rance of ace (the face numbered one) at the top and the appearance of any other face at the top as failure, the probability of success P (F₁) = 1/6, and the probability of failure, P (ƒ) =P (F₂) + P (F₃) + P (F₄) + P (F₅) + P (F₆) = 5/6. Similarly, if the male birth is consi­dered success, P (M) = 1/2, then the female birth is the alternative form (failure) with the probability P (F) = 1/2.

Normal distribution:

This is the most important distribution in biometry because; most of the characters in agricul­ture, biology and genetics follow this distribution. The algebraic form of this distribution was first derived in 1733 by De Moivre (1667-1754) but this was later rediscovered and developed by Gauss (1809) and by Leplace (1812). For these reasons the normal distribution is also called the De Moivre, Gaussian or Leplace distribution.

Further, it is also known as Gaussian law of error because Gauss found that the errors of observations on any character follow the normal distribution. The term ‘normal’ is not used to mean the normal trend of any character nor it stands for the norm of the character whose distribution is under considera­tion.

In fact, the normal distribution is the distribution of random variable which varies continuously. The shape of the curve of the normal distribution is unimodal, symmetrical and the ends of the curve tail off asymptotically to the base. Graphically, the total arc under the curve is unity.

The algebraic equation to the curve is

P (X) = 1/σ√2π e – (x – µ)² / 2σ²; µ = Mean, σ = S. D.

When the total number of observations (total frequency) on a character is N, if the distribution follows normal, then, the fre­quency with which Xi of the character X occurs is,

F (Xi) = N/σ√2π – e – (Xi – µ)² /2σ²

Where µ and σ are called parameters.

Poisson distribution:

This is the distribution of a discrete (discontinuous) random variable whose probability of occurrence is very small but the mean and variance of the distri­bution are equal and finite. Thus, the Poisson distribution is the distribution of a rare event with a finite mean and variance. The equality of mean and variance is an important property of Poisson distribution. The Poisson distribution is named after its inventor French mathematician S. D. Poisson, who described it in (1837).

If X follows a Poisson distribution, the probability that X takes the value X_i is

P (X_i) = mX_ie^-m / X_i

Here, m is the only parameter that is estimated by the mean of the distribution.