MCom I Semester Statistical Analysis Theoretical Frequency Distribution Study Material notes ( part 2)

MCom I Semester Statistical Analysis Theoretical Frequency Distribution Study Material notes ( part 2) : Normal Distribution Relation Between Binomial Poisson and Normal Distribution properties Characteristics of Normal Distribution Importance and  Applications of Normal Distribution mathematical Relationships of Normal Curve  Application of the Normal Distribution

MCom I Semester Statistical Analysis Theoretical Frequency Distribution Study Material notes

Normal Distribution

The distributions discussed so far. viz. Binomial distribution and distribution are discrete probability distributions, since the variables and distributions, since the variables under study were discrete random variables. Now we confine the discussion to continuous problem distributions which arise when the underlying variable is a continuous one.

Normal probability distribution or commonly called the normal is one of the most important continuous theoretical distributions in statistics. graph of this distribution is called Normal Curve, a bell-shaped curve extending in both the directions, arriving nearer and nearer to the horizontal axis but never touches it.

The normal distribution was first discovered by English Mathematician De-Moiré (1667-1754) in 1733 who obtained the mathematical equation for this distribution while dealing with problems arising in the game of chance. It was later rediscovered and applied in sciences, both natural and social by the French mathematician Laplace (1749-1827). The normal distribution is often referred to as the Gaussian distribution in honor of Karl Friedrich Gauss (1777-1855), who also derived its equation from the study of errors in repeated measurements of the same quantity.

Relation between Binomial, Poisson and Normal Distribution

The three distributions, namely, Binomial, Poisson and Normal are very closely related to each other. As explained earlier when n is large while the probability p of the occurrence of an event is close to zero so that q = (1 – p) the binomial distribution is very closely approximated by the Poisson distribution with m = np.

Since there is a relation between the Binomial and Normal distributions, it follows that there is also a relation between the Poisson and Normal distributions. In fact, it can be proved that the Poisson distribution approaches a normal distribution with standardized variable as m increases to infinity. Thus, normal distribution

Vm may also be regarded as a limiting case of Poisson distribution as the parameter mo.

Conditions for Normality : The following four conditions must prevail among the factors affecting individual events that make up a given population, if the distribution is to be normal :

(1) The forces affecting events must be independent of one another.

(2) The casual forces are numerous and of approximately equal weight.

(3) The operation of the casual forces must be such that deviations above the population mean are balanced as to magnitude and number by deviations below the mean. This is the condition of symmetry.

(4) These forces must be the same over the universe from which the observations are drawn. Though their incidence will vary from event. This states the condition of homogeneity.

Frequency Distribution

Properties (Characteristics) of Normal Distribution

The following are the important properties of the normal curve and the normal distribution:

(1) Shape: It is perfectly symmetrical about the mean (1) and is bell shaped The distribution of the frequencies on either side of the maximum ordinate of the curve is exactly the same. The number of cases above the mean value and below the mean value are equal. If the curve is folded along its vertical axis, the two halves would coincide. The height of the normal curve is maximum at the mean value This ordinate divides the curve into two equal parts.

(2) Continuous Distribution : As distinguished from Binomial and Poisson distributions where the variable is discrete, the variable distributed according to the normal curve is a continuous one. Hence, normal distribution is a continuous theoretical distribution.

(3) Unmoral Distribution: Since there is only one maximum point, the normal curve is unmoral, i.e., it has only one mode.

(4) Equality of Central Values ie., X = M = Z : The value of mean, median and mode, will coincide because the distribution is symmetrical and single peaked. Mean = Median = Mode

(5) Asymptotic to the Base Line : There is one maximum point of the normal curve which occurs at the mean. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touches it, i.e., the curve is asymptotic to the base on either side. Hence, its range is unlimited or infinite in both directions.

(6) Main Parameters: The mean (u) and standard deviation (o) are the main parameters of the Normal distribution and with the help of these parameters we can construct the whole distribution.

(7) Points of Inflexion : The point at which the curve changes its direction is called point of inflexion. Points of inflexion of the normal curve are at x= X(14) oi.e., they are equidistant from mean at a distance of o and are given by :

The normal curve is concave near the mean value, while near + 30 it is convex to the horizontal axis.

(8) Constants : The following are the descriptive measures of the normal distribution:

Constants of the Normal Distribution The mean of the normal distribution is X The standard deviation of the normal distribution is a

Importance and Applications of Normal Distribution

It is said “… if a statistician could select but one distribution to work wim during his life time, he would almost surely select the normal distribution…. modern statisticians and applied economists could perhaps get along without compare I would be erceedingly dificult to do without the normal distribution. This storm indicates the importance of normal distribution in the theory of statistics. following points will further highlight the importance of normal distributio

(1) By virtue of the “Central Limit Theorem” the distribution of the means

samples taken from any population, which need not be normal, tends towards the normal distribution if sample size n becomes large.

(2) As n becomes large, the normal distribution gives a good approximation of many discrete distributions, such as binomial, Poisson, etc., where exact discrete probability is difficult or impossible to obtain correctly.

(3) In theoretical as well as in applied statistics, there are many problems which can be solved under the assumption of normal distribution with satisfactory results.

(4) The normal distribution has extensive uses in the sampling theory. Any statistic based on a large sample generally follows normal distribution. Hence, it helps us to estimate parameters from statistic and to find the confidence limits of parameters.

(5) The normal distribution has a wide use in testing statistical hypothesis and test of significance in which the assumption is that the population from which the sample have been drawn is in normal distribution.

(6) It is very useful in statistical quality control where the control limits are set by using this distribution.

(7) The mathematical properties of the normal distribution make it popular and comparatively easy to manipulate.

Frequency Distribution

Mathematical Relationships of Normal Curve

(1) Measures of Central Tendency: The value of mean, median and mode, will coincide because the distribution is symmetrical and single peaked.

Mean = Median = Mode

(2) Equidistance of Quartiles : Since the curve is symmetrical, the first and the third quartiles are equidistant from the median, i.e..

Q3 – Median = Median – Q,

(3) Quartile Deviation : Quartile Deviation = Probable Error

Q.D. = 0.67450

e, + Q.D. = Q – Q.D. = M = X = Z

(4) Mean Deviation : Mean deviation is 0.7979 or 4/5 of the standard deviation.

(5) Probable Error and Mean Deviation: The probable error is 0.845 of Mean Deviation.

(6) Ordinate Relationship : The ordinate at the mean is the highest ordinate. The height of the ordinate at a distance of one standard deviation from the mean IS 60.653% of the height of the mean ordinate and the heights of other ordinates at various sigma distance from the mean are also in a fixed relationship with the height of the mean ordinate.

The following table shows the proportion of height of other ordinates to maximum ordinate:

Frequency Distribution

(7) Area Relationship : Of all the important relationships of the normal curve the most important relationship is the area relationship. Since ordinate at a given sigma distance from the mean has always the same relationship with the mean ordinate, it follows that the area of the curve enclosed between the mean ordinate and an ordinate at a certain sigma distance from the mean would always be the same proportion of the total area of the curve. In other words, in any normal curve, an exact percentage of observations in the distribution falls within ranges established by the standard deviation in conjunction with the mean. The standard deviation distributes the area under the normal curve as given below:

Finding Area Under the Normal Curve : In order to compute the probability of a random variable lying between two specified values, we need to know the area under the normal curve between two specified values. The equation of the normal distribution involves two parameters, the mean (u or X) and standard deviation (o). Since u and o can assume an infinite number of values, it is impossible to tabulate the areas under the curve for different values of u (X) and o. But it is possible to convert these different normal distributions into one standardised form. Hence, it becomes necessary to explain the ‘Standard Normal Distribution and ‘Standard Normal Variety’

Standard Normal Distribution : A normal distribution with u (X) = 0 and standard deviation = 1 is called the standard normal distribution or unit normal distribution. Such a normal curve with zero mean and unit standard deviation is called as the standard normal curve. Under this curve, the area 99.73% is covered by + 30.

Note : Standard Normal Distribution is applicable to any distribution that is normal in form regardless of the particular mean, standard deviation and number of observations in the distribution.

Standard Normal Variety : The variable z is termed as the standard normal variate and plays a very important role in finding the area.

For convenience, it is useful to transform a normally distributed variable into such a form that a single table of areas under the normal curve would be applicable regardless of the units of the original data. We need to know the area between the mean and a point above the mean some specified distance measured in standard deviations. Because the distance will vary with the situation, it is treated as a variable

Frequency Distribution