Explain the concept of normal distribution. Explain divergence from normality
The Normal Distribution is defined by the probability density function for a continuous random variable in a system. Let us say, f(x) is the probability density function and X is the random variable. Hence, it defines a function which is integrated between the range or interval (x to x + dx), giving the probability of random variable X, by considering the values between x and x+dx.
f(x) ≥ 0 ∀ x ϵ (−∞,+∞)
Normal Distribution Formula
The probability density function of normal or gaussian distribution is given by;
- x is the variable
- μ is the mean
- σ is the standard deviation
A distribution is normal when the Mean, Median and Mode coin side together and there is a perfect balance between the right and left halves of the figure. But when the Mean, Median and Mode fall at different points in the distribution, and the center of gravity is shifted to one side it is said to be skewed. In a normal distribution the mean equals the Median-Means.
Mean—Median = 0. So the skewness is ‘0’. Collins Dictionary of Statistics defines the skewness as “a distribution not having equal probabilities above and below the mean.” So in fact greater the gap between the mean and the median, greater is the skewness.
When in a distribution the scores are massed at the high end of the scale i.e. to the right end and are spread out more gradually towards the left side at that time the distribution is said to be Negatively Skewed.
In a negatively skewed distribution the Median is greater than the Mean. So when the skewness is negative the mean lies to the left of the Median. Similarly when in a distribution the scores are massed at the low end of the scale i.e. to the left end and are spread out more gradually to the right side at that time the distribution is said to be Positively Skewed.
In a positively skewed distribution the Median is less than the mean. So when the skewness is positive the mean lies to the right of the Median. Skewness can be computed in different ways.
Out of these methods the following two methods are most widely used:
a. Person’s Measure of Skewness:
In this method we can compute skewness from a frequency distribution.
SK= 3 (Mean-Median)/σ
Where Sk = Skewness
σ = Standard deviation
b. Measure of Skewness in terms of percentiles:
In this method we can compute skewness from percentiles.
where Sk = Skewness
P90 = 90th percentile
P10 = 10th percentile
P50 = 50th percentile, or Median.