- Statistics Tutorial
- Home
- Adjusted R-Squared
- Analysis of Variance
- Arithmetic Mean
- Arithmetic Median
- Arithmetic Mode
- Arithmetic Range
- Bar Graph
- Best Point Estimation
- Beta Distribution
- Binomial Distribution
- Black-Scholes model
- Boxplots
- Central limit theorem
- Chebyshev's Theorem
- Chi-squared Distribution
- Chi Squared table
- Circular Permutation
- Cluster sampling
- Cohen's kappa coefficient
- Combination
- Combination with replacement
- Comparing plots
- Continuous Uniform Distribution
- Continuous Series Arithmetic Mean
- Continuous Series Arithmetic Median
- Continuous Series Arithmetic Mode
- Cumulative Frequency
- Co-efficient of Variation
- Correlation Co-efficient
- Cumulative plots
- Cumulative Poisson Distribution
- Data collection
- Data collection - Questionaire Designing
- Data collection - Observation
- Data collection - Case Study Method
- Data Patterns
- Deciles Statistics
- Discrete Series Arithmetic Mean
- Discrete Series Arithmetic Median
- Discrete Series Arithmetic Mode
- Dot Plot
- Exponential distribution
- F distribution
- F Test Table
- Factorial
- Frequency Distribution
- Gamma Distribution
- Geometric Mean
- Geometric Probability Distribution
- Goodness of Fit
- Grand Mean
- Gumbel Distribution
- Harmonic Mean
- Harmonic Number
- Harmonic Resonance Frequency
- Histograms
- Hypergeometric Distribution
- Hypothesis testing
- Individual Series Arithmetic Mean
- Individual Series Arithmetic Median
- Individual Series Arithmetic Mode
- Interval Estimation
- Inverse Gamma Distribution
- Kolmogorov Smirnov Test
- Kurtosis
- Laplace Distribution
- Linear regression
- Log Gamma Distribution
- Logistic Regression
- Mcnemar Test
- Mean Deviation
- Means Difference
- Multinomial Distribution
- Negative Binomial Distribution
- Normal Distribution
- Odd and Even Permutation
- One Proportion Z Test
- Outlier Function
- Permutation
- Permutation with Replacement
- Pie Chart
- Poisson Distribution
- Pooled Variance (r)
- Power Calculator
- Probability
- Probability Additive Theorem
- Probability Multiplecative Theorem
- Probability Bayes Theorem
- Probability Density Function
- Process Capability (Cp) & Process Performance (Pp)
- Process Sigma
- Quadratic Regression Equation
- Qualitative Data Vs Quantitative Data
- Quartile Deviation
- Range Rule of Thumb
- Rayleigh Distribution
- Regression Intercept Confidence Interval
- Relative Standard Deviation
- Reliability Coefficient
- Required Sample Size
- Residual analysis
- Residual sum of squares
- Root Mean Square
- Sample planning
- Sampling methods
- Scatterplots
- Shannon Wiener Diversity Index
- Signal to Noise Ratio
- Simple random sampling
- Skewness
- Standard Deviation
- Standard Error ( SE )
- Standard normal table
- Statistical Significance
- Statistics Formulas
- Statistics Notation
- Stem and Leaf Plot
- Stratified sampling
- Student T Test
- Sum of Square
- T-Distribution Table
- Ti 83 Exponential Regression
- Transformations
- Trimmed Mean
- Type I & II Error
- Variance
- Venn Diagram
- Weak Law of Large Numbers
- Z table
- Statistics Useful Resources
- Statistics - Discussion

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# Statistics - Kolmogorov Smirnov Test

This test is used in situations where a comparison has to be made between an observed sample distribution and theoretical distribution.

## K-S One Sample Test

This test is used as a test of goodness of fit and is ideal when the size of the sample is small. It compares the cumulative distribution function for a variable with a specified distribution. The null hypothesis assumes no difference between the observed and theoretical distribution and the value of test statistic 'D' is calculated as:

## Formula

$D = Maximum |F_o(X)-F_r(X)|$

Where −

${F_o(X)}$ = Observed cumulative frequency distribution of a random sample of n observations.

and ${F_o(X) = \frac{k}{n}}$ = (No.of observations ≤ X)/(Total no.of observations).

${F_r(X)}$ = The theoretical frequency distribution.

The critical value of ${D}$ is found from the K-S table values for one sample test.

**Acceptance Criteria:** If calculated value is less than critical value accept null hypothesis.

**Rejection Criteria:** If calculated value is greater than table value reject null hypothesis.

### Example

**Problem Statement:**

In a study done from various streams of a college 60 students, with equal number of students drawn from each stream, are we interviewed and their intention to join the Drama Club of college was noted.

B.Sc. | B.A. | B.Com | M.A. | M.Com | |
---|---|---|---|---|---|

No. in each class | 5 | 9 | 11 | 16 | 19 |

It was expected that 12 students from each class would join the Drama Club. Using the K-S test to find if there is any difference among student classes with regard to their intention of joining the Drama Club.

**Solution:**

${H_o}$: There is no difference among students of different streams with respect to their intention of joining the drama club.

We develop the cumulative frequencies for observed and theoretical distributions.

Streams | No. of students interested in joining | ${F_O(X)}$ | ${F_T(X)}$ | ${|F_O(X)-F_T(X)|}$ | |
---|---|---|---|---|---|

Observed (O) | Theoretical (T) | ||||

B.Sc. | 5 | 12 | 5/60 | 12/60 | 7/60 |

B.A. | 9 | 12 | 14/60 | 24/60 | 10/60 |

B.COM. | 11 | 12 | 25/60 | 36/60 | 11/60 |

M.A. | 16 | 12 | 41/60 | 48/60 | 7/60 |

M.COM. | 19 | 12 | 60/40 | 60/60 | 60/60 |

Total | n=60 | ||||

Test statistic ${|D|}$ is calculated as:

The table value of D at 5% significance level is given by

Since the calculated value is greater than the critical value, hence we reject the null hypothesis and conclude that there is a difference among students of different streams in their intention of joining the Club.

## K-S Two Sample Test

When instead of one, there are two independent samples then K-S two sample test can be used to test the agreement between two cumulative distributions. The null hypothesis states that there is no difference between the two distributions. The D-statistic is calculated in the same manner as the K-S One Sample Test.

## Formula

${D = Maximum |{F_n}_1(X)-{F_n}_2(X)|}$

Where −

${n_1}$ = Observations from first sample.

${n_2}$ = Observations from second sample.

It has been seen that when the cumulative distributions show large maximum deviation ${|D|}$ it is indicating towards a difference between the two sample distributions.

The critical value of D for samples where ${n_1 = n_2}$ and is ≤ 40, the K-S table for two sample case is used. When ${n_1}$ and/or ${n_2}$ > 40 then the K-S table for large samples of two sample test should be used. The null hypothesis is accepted if the calculated value is less than the table value and vice-versa.

Thus use of any of these nonparametric tests helps a researcher to test the significance of his results when the characteristics of the target population are unknown or no assumptions had been made about them.