2.2.2.1 Normal Distribution

Normal Distribution is a distribution that is symmetric about the mean, showing that the probabilities near the mean are more than probabilities farther from the mean. The distribution is also known as the Bell Curve or the Gaussian Distribution. A lot of real-world scenarios follow the pattern of normal distribution, making it the most common distribution amongst all. Fig 2.11 shows a typical normal distribution curve.

        

The distribution has some key characteristics:

• The mean, mode, and median are all equal
• The total area under the curve (highlighted in red) is equal to 1
• The curve is symmetric around the mean ()
• Half of the observations lie to the left of the center and half the observations are to the right
• Normal Distribution follows the Empirical Rule:

• ~68% of the area of the curve lies within 1 standard deviation of the mean (are under  -   and  +  )
• ~95% of the area of the curve lies within 2 standard deviations of the mean (are under  -   and  +  )
• ~99.7% of the area of the curve lies within 3 standard deviations of the mean (are under  -   and  +  )

The area under the two tails of the curve is significantly less (~0.15% each)

Few scenarios where Normal Distribution can be used to find probabilities are, height of people, human blood pressure, flight timings, errors in measurements.

The shape of the bell curve (i.e., the height of every value of ) can be calculated through the following formula:

        

Where:

•  is the standard deviation
•  is the Mean
•  π is 3.14159
•  is the Euler's constant, equal to approximately 2.71828

Although we don't need to memorize this formula as we are using Z-score & Z-tables (discussed in the next section) for calculations, it's worth noting that  and  have a significant impact on the shape and probabilities of the curve. Let's see how these two parameters impact the bell curve:

Impact of Mean(): The Mean defines the peak location of the distribution. Thus, decreasing the mean to a negative value, shifts the curve towards left while increasing it to a positive value, shifts the curve towards the right, on the x-axis.

        

Impact of Standard Deviation(): The standard deviation defines the width of the distribution. Thus, increasing the standard deviation, increases the area under the tails and reduces the peak height while decreasing it, decreases the area under the tails and increases the peak height.

        

The Standard Normal Distribution

In statistics, when analyzing a diverse range of data values (e.g., weights ranging from 50 pounds to 400 pounds), it becomes a lot more easier if we standardise these values.

The Standard Normal Distribution standardizes values by setting the mean to 0 and the standard deviation to 1. With a mean of 0 and a standard deviation of 1, the formula for the standard normal distribution is given as:

        

However, we rarely need this formula to find the area under the bell curve, as Z-score along with the Z-table can do that for us.