To draw inference about data, the basic way is to use the parameters of the distribution.
Let us consider a scenario where we want to understand the weights of an equipment produced in manufacturing company. The historical data says that the mean weight will be 90 Kg with standard deviation = 4 Kg and the weights follow Gaussian distribution.
With this information, we can conclude that 95% of the equipment weight will lie between 82 Kg and 98 Kg, because in Gaussian distribution , we know the rule 68–95–99.7 which says that P(µ-2σ ≤ X ≤ µ + 2σ) = 95%.
What if the distribution is not Gaussian i.e the data comes from unknown distribution. In this case, Chebyshevs Inequality can be used
P( µ - kσ < X < µ + kσ) > 1-1/k²
Using the above inequality , if we want to find what percentage of equipments have the weights between 82 kg and 98 Kg.
µ-2σ = 82, µ = 90 , µ+2σ = 98
P( 82 < X < 98) > 1 - (1/2²)
P( 82 < X < 98) > 0.75
So the probability that equipments weights lie between 82 Kg and 98 Kg is greater than 0.75.
i.e Atleast 75 % of the Equipments weights will like between 82 Kg and 98 Kg.
Though it cannot give exact percentage of the equipment weight, Chebyshev’s inequality gives the minimum value for which the range of weights will fall for the data whose distribution is unknown.
