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:
• Physical attributes of a population such as height and weight
• Crop yields
• 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 :
For a normal distribution with mean µ and standard deviation σ, 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.
X∼N(µ,σ)
For a normal distribution other than X∼N(0,1) ,we must perform a transformation on the variables into a z-score where z =\(\frac{x-µ}{σ}\)
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:
• Mean, µ, and standard deviation, σ.
• Mode= median = mean because of its symmetric shape.
• The larger the value of cr, the more dispersion of the data about the mean.
• The smaller the value of σ, the more the data cluster about the mean.
For a normal distribution, X;
The process of reducing a normal distribution to the standard normal distribution X∼N(0,12) 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)