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# How to Find Maximum of Normal Distribution Using Kolmogorov Smirnov Test - Tutorial

## How to Find Maximum of Normal Distribution - Formula, Example

##### Definition:

Kolmogorov Smirnov test (K-S test) is a statistical analysis, used to evaluate probability of normal distribution for mean variables.

#### Formula:

Where, D = Maximum Value of Normal Distribution, N = Number of Statistic Data, F = Kolmogorov-Smirnov Index.
##### Example:

Calculate maximum of normal distribution mean variable for 6 rows from date set of population {120,123,110,100,200,210} using Kolmogorov Smirnov test method.

RowDataSet
1120
2123
3110
4100
5200
6210
##### Solution:

Substitute the given values in the formula to solve maximum of normal distribution,

##### Step1:

First, calculate the value of Mean(μ)

Mean (μ) = ∑ X / n = ∑ (120+123+110+100+200+210) / 6 = 863 / 6 = 143.833

##### Step2:

Standard Deviation (σ) = √∑(X-μ)2 / n-1 = √∑(120-143.8)2 + (123-143.8)2 + (110-143.8)2 + (100-143.8)2 + (200-143.8)2 + (210-143.8)2 / 5 = √2320.166 = 48.168

##### Step3:

F0(100) = Pμ=143.833, σ=14.85 (100) Pμ=143.833, σ=14.85 (100) = Normal[(100 - 143.833) / 48.168] = 0.181409 Fn(100) = 1/n = 1/6 = 0.167 Fn-1(100) = (i-1)/n = (1-1)/6 = 0

##### Step4:

Follow the procedure of Step 3, upto N th value.

##### Step5:

The following table shows all the values

RowDataSetF0 Fn Fn-1D+=Fn-F0D-=F0-Fn-1
11000.1814090.1666670-0.0147420.181409
21100.2412150.3333330.1666670.0921180.074548
31200.3103720.50.3333330.189628-0.022961
41230.3326840.6666670.50.333983-0.167316
52000.8782040.8333330.666667-0.0448710.211537
62100.91522710.8333330.0847730.081894
##### Max Value
0.333983 0.211537
KS statistic value : 0.333983