**Â THE NORMAL DISTRIBUTION**

The Normal Distribution is the most important distribution for a continuous random variable. Many naturally occurring phenomena have a distribution that is normal, or approximately normal. Some example are:

â€¢ Physical attributes of a population such as height and weight

â€¢ Crop yields

â€¢ Test scores taken from a large population.

Once a normal model has been established, we can use it to make predictions about a distribution.

**Characteristics** :

- The data can be considered to be continuous, that is, the data can take any real value (not just integer values).
- The data are symmetrically distributed about the mean. Data are just as likely to be below the mean as above
- The probability of getting a result in a certain interval will decrease as the distance the interval is from the mean

increases. - The results are in a bell-shaped distribution, as seen below.
- Â The standard deviation is the distance from the mean to the points of inflection.
- Â The area under the curve is equal to one.
- Notation: Xâˆ¼N(Âµ,Ïƒ
^{2})Â Â N= mean variance

For a normal distribution with meanÂ Âµ and standard deviation Ïƒ, the proportional breakdown of where the random variable could lie is given below.

**Standard Normal Distribution**

In a Standard Normal Distribution the mean= 0 and standard deviation= 1.

**Xâˆ¼N(Âµ,Ïƒ)**

For a normal distribution other than Xâˆ¼N(0,1) ,we must perform a transformation on the variables into a z-score where z =\(\frac{x-Âµ}{Ïƒ}\)

If each data value of a normally distributed random variable x is transformed into a z-score, the result will be the standard normal distribution.

Use the Standard Normal Table to find the cumulative area under the standard normal curve.

**Inverse Normal Distribution**

1.Calculating inverse normal distribution is much like calculating the normal distribution. The first thing you should do is press the ‘2ND’ button on your calculator and then press the ‘VARS’ button which has the second function (the blue text above the button) of ‘DISTR’. Once you have done this, you should see the screen shown to the right (or below if you are on a mobile device).

2. Calculating inverse normal distribution is much like calculating the normal distribution. The first thing you should do is press the ‘2ND’ button on your calculator and then press the ‘VARS’ button which has the second function (the blue text above the button) of ‘DISTR’. Once you have done this, you should see the screen shown to the right (or below if you are on a mobile device).

Many quantities in life are distributed symmetrically about the mean and can be described with the mathematical model known as the **bell curve **(normal distribution)

A population that follows the normal distribution can be described by its:

â€¢ Mean, Âµ, and standard deviation, Ïƒ.

â€¢ Mode= median = mean because of its symmetric shape.

â€¢ The larger the value of cr, the more dispersion of the data about the mean.

â€¢ The smaller the value of Ïƒ, the more the data cluster about the mean.

For a normal distribution, X;

- About 68% of the data values of X will lie within the range Âµ- Ïƒ and Âµ+ Ïƒ
- About 95% of the data values of X will lie _within the rangeÂµ- 2Ïƒ andÂµ+ 2Ïƒ.
- About 99. 7% of the data values of X will lie within the range Âµ- 3Ïƒ- andÂµ +3Ïƒ.There is no easy formula for calculating areas under the normal distribution curve. Instead we approximate using z-scores, where z =\(\frac{x-Âµ}{Ïƒ}\). A z-score indicates the number of standard deviations a value lies from the mean. A z-score also converts a particular normal distribution to a standard normal distribution, so z-scores can be used with the areas under the normal distribution curve to find probabilities. The notation for a normal distribution is
**Xâˆ¼N(Âµ,Ïƒ**. The Standard Normal Distribution is written as^{2})**Xâˆ¼N(0,1**) and is the distribution of z-scores of a normally distributed variable with mean = 0 and standard deviation = 1.^{2})

The process of reducing a normal distribution to the standard normal distribution Xâˆ¼N(0,1^{2}) is called standardizing.

In the following example, the value x = 6.2 has been standardized to N(O, 1) using z-scores.

Recall, a negative z-score indicated that the value lies below the mean. The value also has the same position on the standard normal distribution (on the right) as the normal distribution (on the left).

Areas under the normal curve can be found on the graphing calculator using the normal(function from the DISTR menu. The syntax for this function is: normal (lower bound, upper bound, mean, standard deviation)