Don't understand Kolmogorov-Smirnov test

Question

I am stumped doing a journal club for class. I am looking at a scientific article in which they used the "Kolmogorov-Smirnov test" and I have no idea what that is. I'm trying to decipher if it was an appropriate test but I just am not getting it. Any help would be greatly appreciated!

Answer 1

NONPARAMETRIC TESTING IS USED WHEN ONE OR MORE OF THE ASSUMPTIONS OF THE PREVIOUS TESTS HAVE NOT BEEN MET. USUALLY THE ASSUMPTION IN QUESTION IS THE NORMALITY OF THE DISTRIBUTION (DISTRIBUTION OF THE DATA IS UNKNOWN OR THE SAMPLE SIZE IS SMALL). NONPARAMETRIC TESTS ARE OFTEN BASED ON COUNTING TECHNIQUES THAT ARE EASIER TO CALCULATE AND MAY BE USED FOR ORDINAL DATA AS WELL AS FOR QUANTITATIVE DATA. THESE TESTS ARE NOT EFFICIENT IF THE DISTRIBUTION IS KNOWN OR IF THE SAMPLE SIZE IS LARGE ENOUGH FOR A PARAMETRIC TEST.

FOR A NONPARAMETRIC TEST OF GOODNESS-OF-FIT, THE KOLMOGOROV-SMIRNOV TEST COMPARES CUMULATIVE PROBABILITIES OF THE DATA TO A HYPOTHESIZED DISTRIBUTION.

1. ARRANGE DATA FROM SMALLEST VALUE TO LARGEST VALUE.
2. THE PROPORTION OF DATA BELOW EACH VALUE IS COMPARED WITH CUMULATIVE PROBABILITY BELOW THAT VALUE FROM THE HYPOTHESIZED DISTRIBUTION.
3. THE TEST STATISTIC IS THE MAXIMUM DIFFERENCE FOUND IN STEP 2 ABOVE, WHICH CAN BE COMPARED TO THE CRITICAL VALUE TAKEN FROM A "TABLE OF KOLMOGOROV-SMIRNOV CRITICAL VALUES FOR VARIOUS SIGNIFICANCE LEVELS".

EXAMPLE: A store owner wants to determine at the 5% significance level whether sales are normally distributed with mean of 10 units and standard deviation of 3 units. Sales for a week are observed of 2, 8, 4, 18, 9, 11, and 13 units. The small sample size rules out the use of the Chi-Square-Goodness-Of-Fit Test, but the nonparametric Kolmogorov-Smirnov Test may be used to test between H₀: Normally Distributed With Mean = 10 And Standard Deviation = 3 and H₁: Not Normally Distributed With Mean = 10 And Standard Deviation = 3.

ORDERED DATA VALUES---------2-----------4-------------8--------------9------------11------------13------------18

PROPORTION BELOW, %------14.29----- 28.57------ 42.86------- 57.14------ 71.43------- 85.71-------- 100

NORMAL

CUMULATIVE

PROBABILITY, %------------------- 0.38------- 2.27------ 25.24------- 36.94------ 63.05------- 84.13------- 99.61

DIFFERENCE, %------------------ 13.91----- 26.3------- 17.62-------- 20.20------- 8.38--------- 1.58--------- 0.39

The maximum difference is 26.30% (0.2630) which is less than the table's critical value of 0.486. Therefore, accept the null hypothesis that sales are normally distributed with a mean of 10 and a standard deviation of 3.