Difference between Parametric and Non Parametric T: Overview, Questions, Preparation

Parametric and non-parametric tests play a vital role in statistics. Parametric tests are widely used, and results can be obtained in less time. Non-parametric tests are conducted with no assumptions.

Parametric Test

Parametric tests use the available statistical distributions to determine the result. These are tests that provide generalisations for producing the records related to the mean of the original population.

Non-Parametric Test

The non-parametric test does not use any available statistical distributions to determine the result. These are the tests that provide results based on median and not on population distribution.

Difference between parametric and non-parametric tests.

1. Assumptions are made in parametric tests, but not in the case of non-parametric tests.
2. The mean is used in parametric tests, while the median is used in the case of non-parametric tests.
3. The parametric test uses Pearson correlation, while the non-parametric test uses Spearman correlation.
4. The population data is required for parametric, while it is not required for non-parametric tests.
5. The parametric test uses normal probabilistic distribution, while the non-paramedic test uses arbitrary probabilistic distribution.
6. The parametric test is used to determine interval data, whereas a non-parametric test is used to determine nominal data.
7. The parametric test is applicable for variables, whereas a non-parametric test is applicable for both variables and attributes.
8. Examples of the parametric test are the T-test and z-test, where for the non-parametric test are Kruskal-Wallis and Mann-Whitney test.

The chapter ‘Statistics’ describes the parametric and non-parametric tests, different modes of dispersion, range, analysis of frequency deviation, variance, standard deviation, and mean deviation. 

1. The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, and 14. Find the other two observations?

Solution.

Let the two unknown observations be x and y.
= (2+4+10+12+14+x+y)/7 = 8
= x + y = 14 ——— (1)
= (1/n) _i=1∑⁷(X - X)² =  16
= (/7)* [(-6)² + (-4)² + (2)² + (4)² + (6)² + x² + y² - 2 * 8* (x+y) + 2 * (8)²]
= x² + y² = 100 ——— (2)
From (1),
= x² + y² + 2xy = 196 ——— (3)
 From (2) and (3),
= 2xy = 96 ——— (4)
Subtracting (4) from (2) 
= x2 + y2 -2xy = 100 - 96
= x - y = -+2 ———(5)
From (1) and (5)
When x-y=2, x= 8 and y=6
When x-y=-2, x=6 and y= 8

2. The mean and SD of 100 observations were found to be 20 and 3, respectively. Later it was found that three observations were wrong and were recorded as 21, 12, and 18. Find the mean and SD by omitting these three observations.

Solution.

n= 100, 
Incorrect mean, (x̄) = 20
= x̄ = (∑x_i/n)
= 20 = (1/100)* _i=1∑¹⁰⁰xi
= 2000= incorrect sum of observations
Correct sum of observations= 2000 - 21 - 12 - 18 = 1940
Correct mean= (1940/ 100-3) = 20
SD = Square root of { (_i=1∑ⁿ x_i² /n) - (i=1∑n xi /n)²}
3² = (∑ x_i²/100) - (20)²
Incorrect ∑ xi2 = 100* (9 + 400) = 40900
= 39694
Correct SD = Square root {(Correct ∑ xi2/n) - (correct mean)2
Correct SD = 3.036

3. Given that x̄ is the mean and σ2 is the variance on n observations. Prove that the mean and variance of the observations are ax̄ and a2σ2. (a is not equal to 0)
Solution.

σ² = 1/n * _i=1∑ⁿ yi (xi- x̄)2 ——— (1)
y_i = ax_i,  
y= 1/n * _i=1∑ⁿ yi = _i=1∑ⁿ axi = (a/n) * _i=1∑ⁿ xi = ax̄ 
So, the mean of observation is ax̄.
Putting the value of xi and x̄ in (1),
= a²σ² = 1/n * _i=1∑ⁿ (yi - y)²

Q: What is a parametric test?

Q: What is a non-parametric test?

Q: Name some examples in parametric and non-parametric tests?

Q: State some differences in parametric and non-parametric tests?

Q: In which test do you require the population data?