Normal Distribution

The random variables that can take any value on a continuum. It has a infinite or uncountable infinite number of values on its sample space. Such random variables are continuous distributions. Many types of data are continuous in nature because the observations are obtained by measurements such as time, distance, or temperature, heights, weight etc. In continuous variables, the successive observations may differ by infinitesimal amounts. Any random variable whose values are measured as a continuous variable it is counted. The probabilistic model of a continuous variable involves the selection of a curve usually smooth called the probability distribution or probably density function. The distributions may assume a variety of shapes. It is interesting to note that very large numbers of random of random variables observed in nature possess a curve which is approximately bell shaped and is commonly referred as normal probability distribution. This occupies a central position in statistical inference theory and is widely applied in practical situations.

- Large Number of phenomenons tends to follow a normal distribution pattern. Numerous variables tend to follow a pattern of variation that is similar to the normal distribution. Natural traits such as heights, weights, I.Q. of children, test scores in an exam, life of electric lamp, breaking strength of a steel rope, errors of measurement, life of an automobile tyre etc. can be approximated through normal distribution.
- Sampling distribution of many sampling statistics such as mean has an approximately normal distribution. Many distributions in business, economic, sociology and other social sciences do not resemble the normal distribution but the distribution of sample means in each case, usually is normal as long as the sample size is large. This property has a wide application in sampling estimation and testing of hypothesis.
- Other types of sampling distribution can be approximated by normal function. In the previous chapter, we observed that binomial, and Poisson distribution approach to normal distribution when the value of ‘n’ is very large. It is too laborious to work out the probabilities for a binomial random variable or Poisson variable when ‘n’ is very large. In these conditions, normal distribution can be used as approximation to binomial distribution.

The shape of the normal curve is bell shaped. The curve has a single peak thus it is unimodel. The mean of a normally distributed population lies at the centre of its curve. Because of the symmetry, the median (Positional average) a mode (concentration of frequency) also lies at the centre. Thus in normal distribution mean, median and mode all coincide. The two tails of the curve extend indefinitely and never touches the X –axis.

There are only two parameters of normal distribution i.e. mean (μ) and variance (σ2). The mean determines the location of the curve and variance determines the shape of the curve. Given variance, a change in mean will shift the curve as a whole along the x-axis. The following figure depicts the three normal curves, having same variance but different means.

In this sub-section, we will study how a particular normal distribution is transformed into a standard normal distribution. This special distribution has a mean 0 and a standard deviation 1 and is written as N (0, 1).

- Standard normal distribution is the distribution of another normal variable called Z-scores, which is defined as,
**z = (X-μ)/σ**

The Z – score is the difference of an observation X from the mean (μ) expressed in term of standard deviation (σ). It is called Z-scores because random variable take on many different units of measures. Since we use only one table which should be standard unit and is represented by 7 this table gives the value or area of probability between variable and mean.

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