# Evaluation of the Effects of Coal Jigging by Means of Kruskal–Wallis and Friedman Tests

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- Each population has to represent a normal distribution;
- -
- The samples must be independent;
- -
- Samples collected from each population must be random and simple;
- -
- Variances in populations must be equal.

## 2. Materials and Methods

#### 2.1. Experiments

^{2}, with an amplitude of 110 mm.

^{3}/h. Industrial experiments were performed at the set operation parameters and representative samples were obtained for further analyses. Then, each of the separation products was subjected to laboratory float–sink and size analyses. The float–sink analysis was performed in zinc chloride solutions with densities of 1.3, 1.4, 1.5, 1.6, 1.7, 1.8 and 2.0 t/m

^{3}. The separated density fractions of all samples were screened using sieves with the mesh sizes of 2.0, 3.15, 5.0, 6.3, 8.0, 10.0, 12.5, 16.0 and 20.0 mm. As a result, size–density fractions were obtained and classified according to the particle size and density of the material simultaneously.

- v—system capacity: 200, 300 and 400 t/h;
- u—amount of hutch water: 35, 50 and 70 m
^{3}/h.

#### 2.2. Methodology

_{i}, by a certain amount of hatch water, u, and system capacity, v, a vector which coordinates mean yield of the material originating from i-th particle size fraction in individual particle density fraction;

- y
_{i}—center of i-th particle size fraction (according to particle size); - x
_{ij}—share percentage of yield in i-th particle size fraction and j-th density fraction; - u—amount of hutch water (u∈{35, 50, 70});
- v—capacity (v∈{200, 300, 400});
- i = 1, 2,…, 9; j = 1, 2,…, 8.

^{2}distribution [71].

_{ij}(x) distribution functions of random variables, which assumed the values of the output volume (in %) in a given density fraction at fixed values of u and v, where i = 1, 2, 3; and j = 1, 2, 3. The H

_{0}hypothesis was checked, represented by

_{1}that these distributions are not equal.

_{1}= n

_{2}= n

_{3}= 8, n = n

_{1}+ n

_{2}+ n

_{3}= 24; R

_{i}is the sum of the ranks assigned to the scores in the i-th trial. The measurement results for all cases are numbered from the lowest value to the highest.

**H**

_{0}:**H**

_{1}:_{i}represents sum of ranks in the same particle size fraction and k shows number of samples (k = 3).

## 3. Results and Discussion

^{3}/h (Figure 2c). In experiments with the capacity of 200 t/h and the smallest amount of additional water, the yield was improved together with the increase in particle size and reached a value above 60% for the coarsest particles of 16.0–20.0 mm (Figure 2a), while for 50 m

^{3}/h of additional water, the yield above 50% could be observed for particles above 10 mm (Figure 2b). For higher yields of concentrate, the highest values were detected in the coarsest fractions of 12.0–16.0 mm (Figure 2a) and 16.0–20.0 mm (Figure 2b). By analyzing the quality of concentrates, an increase in ash in the finest fraction < 2.0 mm (Figure 2e,f) and in the coarsest fraction > 16.0 mm could be seen in the case of the lowest capacity (Figure 2d). In the remaining cases of variable capacity and the amount of additional water, the ash grade in the concentrate remained at a constant level.

#### 3.1. Kruskal–Wallis Test for Yield Depending on the Amount of Hutch Water and System Capacity

_{0}hypothesis, it could be assumed that the yield in individual particle size fractions, regardless of the amount of hutch water added and the system capacity, was subjected to the same distributions.

#### 3.2. Friedman’s Test for the Yield Depending on the Hutch Water Amount and Capacity

^{3}/h, size fraction (8.00–10.00 mm) at u = 70 m

^{3}/h and fractions (8.00–10.00 mm) at v = 200 ton/h, the test values exceeded the critical value χ

^{2}, which indicated that in these cases the H

_{0}hypothesis should be rejected.

#### 3.3. Friedman Test for Two Subgroups of Coal Density Fractions

^{3}) and (1.50–2.20 g/cm

^{3}). The test results are presented in Table 3 and the remaining outcomes are positioned in Appendix A (Table A11, Table A12, Table A13, Table A14 and Table A15).

_{11}, R

_{12}and R

_{13}are rank sums including density fractions (1.20–1.70 g/cm

^{3}) and R

_{21}, R

_{22}and R

_{23}are rank sums including density fractions (1.50–2.20 g/cm

^{3}); ${\chi}_{1}^{2}$—test value for the density range (1.20–1.70 g/cm

^{3}),${\chi}_{2}^{2}$—test value for the density range (1.50–2.20 g/cm

^{3}).

^{3}/h, then in group I only in one case the H

_{0}hypothesis should be rejected. In group II, H

_{0}should be rejected in four cases. If u = 50 m

^{3}/h, the hypothesis was rejected in group I in two cases and in group II in six cases. If, on the other hand, u = 70 m

^{3}/h, then in group I the H

_{0}hypothesis was not rejected even once, while in group II it was rejected in three cases.

_{0}was not rejected in group I in any case, and in group II it was rejected in one case. If v = 300 t/h, then in group I the null hypothesis should be rejected in three cases, and in group II it should also be rejected in three cases. If v = 400 t/h, then in group I the H

_{0}hypothesis was not rejected even once, and in group II it was rejected in two cases.

#### 3.4. The Friedman Test for Ash Grade

^{2}= 5.991 (for the significance level α = 0.05), so there are no grounds for the rejection of hypothesis H

_{0}. It can therefore be concluded that the dependence of the ash grade on both the amount of hutch water and the capacity of the system is low.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The Kruskal–Wallis test results for yield (u = 50); R

_{i}= sum of ranks for v = 200 t/h (R

_{1}), 300 t/h (R

_{2}) and 400 t/h (R

_{3}).

Particle Size Fraction [mm] | v | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 70.0 | 119.0 | 111.0 | 3.220 |

2.00–3.15 | 71.0 | 125.0 | 104.0 | 3.730 |

3.15–5.00 | 71.0 | 124.0 | 105.0 | 3.600 |

5.00–6.30 | 71.0 | 124.0 | 105.0 | 3.600 |

6.30–8.00 | 70.0 | 122.0 | 108.0 | 3.620 |

8.00–10.00 | 75.0 | 110.0 | 115.0 | 2.375 |

10.00–12.00 | 79.0 | 103.0 | 118.0 | 1.935 |

12.00–16.00 | 85.0 | 97.0 | 118.0 | 1.395 |

16.00–20.00 | 78.0 | 99.0 | 123.0 | 2.535 |

**Table A2.**The Kruskal–Wallis test results for yield (u = 70); R

_{i}= sum of ranks for v = 200 t/h (R

_{1}), 300 t/h (R

_{2}) and 400 t/h (R

_{3}).

Particle Size Fraction [mm] | v | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 112.0 | 85.0 | 103.0 | 0.945 |

2.00–3.15 | 92.0 | 98.0 | 110.0 | 0.420 |

3.15–5.00 | 92.5 | 95.0 | 112.5 | 0.593 |

5.00–6.30 | 84.0 | 108.0 | 108.0 | 0.960 |

6.30–8.00 | 90.0 | 94.0 | 116.0 | 0.980 |

8.00–10.00 | 80.0 | 100.0 | 120.0 | 2.000 |

10.00–12.00 | 85.5 | 93.0 | 121.5 | 1.803 |

12.00–16.00 | 88.0 | 102.0 | 110.0 | 0.620 |

16.00–20.00 | 84.0 | 104.5 | 111.5 | 1.020 |

**Table A3.**The Kruskal–Wallis test results for yield (v = 200); R

_{i}= sum of ranks for u = 35 m

^{3}/h (R

_{1}), 50 m

^{3}/h (R

_{2}) and 70 m

^{3}/h (R

_{3}).

Particle Size Fraction [mm] | u | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 108.0 | 78.0 | 114.0 | 1.860 |

2.00–3.15 | 112.0 | 84.0 | 104.0 | 3.000 |

3.15–5.00 | 117.0 | 83.0 | 100.0 | 1.445 |

5.00–6.30 | 119.5 | 79.5 | 101.0 | 2.005 |

6.30–8.00 | 122.0 | 82.0 | 96.0 | 2.060 |

8.00–10.00 | 124.0 | 95.0 | 81.0 | 2.405 |

10.00–12.00 | 116.0 | 88.0 | 95.0 | 0.560 |

12.00–16.00 | 126.0 | 86.0 | 88.0 | 2.540 |

16.00–20.00 | 115.0 | 90.0 | 95.0 | 0.875 |

**Table A4.**The Kruskal–Wallis test results for yield (v = 300); R

_{i}= sum of ranks for u = 35 m

^{3}/h (R

_{1}), 50 m

^{3}/h (R

_{2}) and 70 m

^{3}/h (R

_{3}).

Particle Size Fraction [mm] | u | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 101.0 | 128.0 | 71.0 | 4.065 |

2.00–3.15 | 101.0 | 127.0 | 72.0 | 3.785 |

3.15–5.00 | 102.0 | 124.0 | 74.0 | 3.140 |

5.00–6.30 | 103.0 | 118.0 | 79.0 | 1.935 |

6.30–8.00 | 113.0 | 113.0 | 74.0 | 2.535 |

8.00–10.00 | 116.0 | 106.0 | 78.0 | 1.940 |

10.00–12.00 | 120.0 | 95.0 | 85.0 | 0.625 |

12.00–16.00 | 120.0 | 97.0 | 83.0 | 1.745 |

16.00–20.00 | 120.0 | 96.0 | 84.0 | 1.680 |

**Table A5.**The Kruskal–Wallis test results for yield (v = 400); R

_{i}= sum of ranks for u = 35 m

^{3}/h (R

_{1}), 50 m

^{3}/h (R

_{2}) and 70 m

^{3}/h (R

_{3}).

Particle Size Fraction [mm] | u | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 107.0 | 111.0 | 82.0 | 1.235 |

2.00–3.15 | 104.0 | 112.0 | 84.0 | 1.604 |

3.15–5.00 | 99.0 | 111.0 | 90.0 | 0.555 |

5.00–6.30 | 116.0 | 100.0 | 84.0 | 1.280 |

6.30–8.00 | 103.0 | 96.0 | 101.0 | 0.065 |

8.00–10.00 | 109.0 | 104.0 | 87.0 | 0.665 |

10.00–12.00 | 101.0 | 102.0 | 97.0 | 0.035 |

12.00–16.00 | 123.0 | 97.0 | 80.0 | 2.345 |

16.00–20.00 | 103.0 | 115.0 | 82.0 | 1.395 |

Particle Size Fraction [mm] | v | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 13.0 | 17.0 | 18.0 | 1.750 |

2.00–3.15 | 12.0 | 17.0 | 19.0 | 3.250 |

3.15–5.00 | 13.0 | 17.0 | 18.0 | 1.750 |

5.00–6.30 | 12.0 | 17.0 | 19.0 | 3.250 |

6.30–8.00 | 12.0 | 17.0 | 19.0 | 3.250 |

8.00–10.00 | 12.0 | 16.0 | 20.0 | 4.000 |

10.00–12.00 | 15.0 | 13.0 | 20.0 | 3.250 |

12.00–16.00 | 13.0 | 14.0 | 21.0 | 4.750 |

16.00–20.00 | 14.0 | 14.0 | 20.0 | 4.000 |

Particle Size Fraction [mm] | v | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 20.0 | 13.0 | 15.0 | 3.250 |

2.00–3.15 | 14.0 | 15.0 | 19.0 | 1.750 |

3.15–5.00 | 13.5 | 14.0 | 20.0 | 3.060 |

5.00–6.30 | 14.0 | 14.0 | 20.0 | 3.000 |

6.30–8.00 | 11.0 | 16.0 | 21.0 | 3.000 |

8.00–10.00 | 12.0 | 15.0 | 21.0 | 6.250 |

10.00–12.00 | 11.0 | 18.0 | 19.0 | 5.250 |

12.00–16.00 | 11.5 | 16.5 | 19.0 | 4.750 |

16.00–20.00 | 11.5 | 16.5 | 20.0 | 4.560 |

Particle Size Fraction [mm] | u | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 18.0 | 11.0 | 19.0 | 4.750 |

2.00–3.15 | 18.0 | 12.0 | 18.0 | 3.000 |

3.15–5.00 | 19.0 | 13.0 | 16.0 | 2.250 |

5.00–6.30 | 20.0 | 11.0 | 17.0 | 5.250 |

6.30–8.00 | 20.0 | 12.0 | 16.0 | 4.000 |

8.00–10.00 | 22.0 | 15.0 | 11.0 | 7.750 |

10.00–12.00 | 35.0 | 20.0 | 12.0 | 4.000 |

12.00–16.00 | 21.0 | 13.0 | 14.0 | 4.750 |

16.00–20.00 | 19.0 | 14.0 | 15.0 | 1.750 |

Particle Size Fraction [mm] | u | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 19.0 | 17.0 | 12.0 | 3.250 |

2.00–3.15 | 18.0 | 18.0 | 12.0 | 3.000 |

3.15–5.00 | 18.0 | 17.0 | 13.0 | 1.750 |

5.00–6.30 | 17.0 | 16.0 | 15.0 | 0.250 |

6.30–8.00 | 19.0 | 16.0 | 13.0 | 2.250 |

8.00–10.00 | 19.0 | 15.0 | 14.0 | 1.750 |

10.00–12.00 | 21.0 | 13.0 | 14.0 | 4.750 |

12.00–16.00 | 20.0 | 12.0 | 16.0 | 4.000 |

16.00–20.00 | 20.0 | 12.0 | 16.0 | 4.000 |

Particle Size Fraction [mm] | u | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 13.0 | 20.0 | 15.0 | 3.250 |

2.00–3.15 | 16.0 | 18.0 | 14.0 | 1.000 |

3.15–5.00 | 14.0 | 18.0 | 16.0 | 1.000 |

5.00–6.30 | 18.0 | 16.0 | 14.0 | 1.000 |

6.30–8.00 | 16.0 | 13.0 | 19.0 | 2.250 |

8.00–10.00 | 17.0 | 15.0 | 16.0 | 0.250 |

10.00–12.00 | 15.0 | 17.0 | 16.0 | 0.250 |

12.00–16.00 | 20.0 | 15.0 | 13.0 | 3.250 |

16.00–20.00 | 16.0 | 19.0 | 13.0 | 2.250 |

Particle Size Fraction [mm] | V | v | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 10.0 | 8.0 | 12.0 | 5.0 | 14.0 | 11.0 | 1.600 | 8.400 |

2.00–3.15 | 9.0 | 8.0 | 13.0 | 5.0 | 14.0 | 11.0 | 2.800 | 8.400 |

3.15–5.00 | 10.0 | 8.0 | 12.0 | 5.0 | 14.0 | 11.0 | 1.600 | 8.400 |

5.00–6.30 | 9.0 | 8.0 | 13.0 | 5.0 | 14.0 | 11.0 | 2.800 | 8.400 |

6.30–8.00 | 9.0 | 8.0 | 13.0 | 5.0 | 14.0 | 11.0 | 2.800 | 8.400 |

8.00–10.00 | 9.0 | 7.0 | 14.0 | 5.0 | 13.0 | 12.0 | 5.200 | 7.600 |

10.00–12.00 | 12.0 | 5.0 | 13.0 | 7.0 | 10.0 | 13.0 | 7.600 | 3.600 |

12.00–16.00 | 10.0 | 6.0 | 14.0 | 6.0 | 11.0 | 13.0 | 6.400 | 3.600 |

16.00–20.00 | 10.0 | 7.0 | 13.0 | 6.0 | 11.0 | 13.0 | 3.600 | 5.200 |

Particle Size Fraction [mm] | v | v | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 12.0 | 10.0 | 8.0 | 14.0 | 7.0 | 9.0 | 1.600 | 5.200 |

2.00–3.15 | 11.0 | 19.0 | 10.0 | 7.0 | 9.0 | 14.0 | 0.400 | 5.200 |

3.15–5.00 | 10.0 | 9.0 | 11.0 | 7.5 | 8.5 | 14.0 | 0.400 | 4.900 |

5.00–6.30 | 10.0 | 9.0 | 11.0 | 7.0 | 8.0 | 15.0 | 0.400 | 7.600 |

6.30–8.00 | 10.0 | 9.0 | 11.0 | 8.0 | 8.0 | 14.0 | 0.400 | 4.800 |

8.00–10.00 | 7.0 | 10.0 | 13.0 | 6.0 | 10.0 | 14.0 | 3.600 | 6.400 |

10.00–12.00 | 8.0 | 10.0 | 12.0 | 6.0 | 10.0 | 14.0 | 1.600 | 6.400 |

12.00–16.00 | 7.5 | 11.5 | 11.0 | 6.0 | 11.0 | 13.0 | 1.900 | 5.200 |

16.00–20.00 | 7.0 | 12.0 | 11.0 | 6.5 | 9.5 | 14.0 | 5.200 | 5.700 |

Particle Size Fraction [mm] | u | u | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 11.0 | 7.0 | 12.0 | 12.0 | 6.0 | 12.0 | 2.800 | 4.800 |

2.00–3.15 | 10.0 | 8.0 | 12.0 | 12.0 | 7.0 | 11.0 | 1.600 | 2.800 |

3.15–5.00 | 10.0 | 8.0 | 12.0 | 14.0 | 7.0 | 9.0 | 1.600 | 5.200 |

5.00–6.30 | 12.0 | 7.0 | 11.0 | 14.0 | 6.0 | 10.0 | 2.800 | 6.400 |

6.30–8.00 | 11.0 | 8.0 | 11.0 | 15.0 | 6.0 | 9.0 | 1.200 | 8.400 |

8.00–10.00 | 13.0 | 9.0 | 8.0 | 15.0 | 9.0 | 6.0 | 2.800 | 8.400 |

10.00–12.00 | 12.0 | 7.0 | 11.0 | 14.0 | 8.0 | 9.0 | 2.800 | 8.400 |

12.00–16.00 | 12.0 | 8.0 | 10.0 | 15.0 | 7.0 | 8.0 | 1.600 | 7.600 |

16.00–20.00 | 12.0 | 8.0 | 10.0 | 13.0 | 9.0 | 8.0 | 1.600 | 2.800 |

_{11}, R

_{21}mean the sum of ranks in individual subgroups for u = 35 m

^{3}/h; R

_{12}, R

_{22}are the sum of ranks in individual subgroups for u = 50 m

^{3}/h and R

_{13}, R

_{23}mean the sum of ranks in individual subgroups for u = 70 m

^{3}/h.

Particle size Fraction [mm] | u | u | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 13.0 | 8.0 | 9.0 | 11.0 | 14.0 | 5.0 | 2.800 | 8.400 |

2.00–3.15 | 12.0 | 9.0 | 9.0 | 11.0 | 14.0 | 5.0 | 1.200 | 8.400 |

3.15–5.00 | 12.0 | 8.0 | 10.0 | 5.0 | 12.0 | 13.0 | 1.600 | 7.600 |

5.00–6.30 | 11.0 | 7.0 | 12.0 | 10.0 | 13.0 | 7.0 | 2.800 | 3.600 |

6.30–8.00 | 13.0 | 7.0 | 10.0 | 11.0 | 13.0 | 6.0 | 3.600 | 5.200 |

8.00–10.00 | 13.0 | 6.0 | 11.0 | 12.0 | 12.0 | 6.0 | 5.200 | 4.800 |

10.00–12.00 | 14.0 | 5.0 | 11.0 | 13.0 | 10.0 | 7.0 | 8.400 | 3.600 |

12.00–16.00 | 13.0 | 5.0 | 12.0 | 13.0 | 9.0 | 8.0 | 7.600 | 2.800 |

16.00–20.00 | 13.0 | 5.0 | 12.0 | 13.0 | 9.0 | 8.0 | 7.600 | 4.400 |

Particle Size Fraction [mm] | u | u | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 6.0 | 12.0 | 12.0 | 10.0 | 14.0 | 6.0 | 4.800 | 6.400 |

2.00–3.15 | 8.0 | 11.0 | 11.0 | 12.0 | 12.0 | 6.0 | 1.200 | 4.800 |

3.15–5.00 | 8.0 | 10.0 | 12.0 | 9.0 | 12.0 | 9.0 | 1.600 | 1.600 |

5.00–6.30 | 9.0 | 11.0 | 10.0 | 13.0 | 9.0 | 8.0 | 0.400 | 1.400 |

6.30–8.00 | 8.0 | 9.0 | 13.0 | 11.0 | 7.0 | 12.0 | 2.800 | 2.800 |

8.00–10.00 | 10.0 | 10.0 | 10.0 | 13.0 | 9.0 | 8.0 | 0.000 | 2.800 |

10.00–12.00 | 7.0 | 11.0 | 12.0 | 11.0 | 10.0 | 9.0 | 2.800 | 0.400 |

12.00–16.00 | 11.0 | 10.0 | 9.0 | 15.0 | 9.0 | 6.0 | 0.400 | 8.400 |

16.00–20.00 | 9.0 | 12.0 | 9.0 | 11.0 | 12.0 | 7.0 | 1.200 | 2.800 |

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**Figure 2.**Yield of concentrate and ash grade in the concentrate according to the size fraction: (

**a**) yield of concentrate - hutch water 35 m

^{3}/h, (

**b**) yield of concentrate - hutch water 50 m

^{3}/h, (

**c**) yield of concentrate - hutch water 70 m

^{3}/h, (

**d**) ash grade - hutch water 35 m

^{3}/h, (

**e**) ash grade - hutch water 50 m

^{3}/h, (

**f**) ash grade - hutch water 70 m

^{3}/h.

**Table 1.**The Kruskal–Wallis test results for yield (u = 35); R

_{i}= sum of ranks for v = 200 t/h (R

_{1}), 300 t/h (R

_{2}) and 400 t/h (R

_{3}).

Particle Size Fraction [mm] | v | H | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 120.0 | 92.0 | 97.0 | 1.145 |

2.00–3.15 | 88.0 | 112.0 | 100.0 | 0.720 |

3.15–5.00 | 102.0 | 114.0 | 84.0 | 1.140 |

5.00–6.30 | 87.0 | 111.0 | 102.0 | 0.735 |

6.30–8.00 | 86.0 | 120.0 | 94.0 | 1.580 |

8.00–10.00 | 78.0 | 120.0 | 102.0 | 2.220 |

10.00–12.00 | 88.0 | 114.5 | 97.5 | 0.900 |

12.00–16.00 | 77.0 | 112.0 | 111.0 | 0.985 |

16.00–20.00 | 79.0 | 129.0 | 92.0 | 3.365 |

Particle Size Fraction [mm] | v | χ^{2} | ||
---|---|---|---|---|

R_{1} | R_{2} | R_{3} | ||

0.00–2.00 | 12.0 | 18.0 | 18.0 | 3.000 |

2.00–3.15 | 17.0 | 18.0 | 13.0 | 1.750 |

3.15–5.00 | 14.0 | 21.0 | 13.0 | 4.750 |

5.00–6.30 | 16.0 | 17.0 | 15.0 | 0.250 |

6.30–8.00 | 14.0 | 17.0 | 17.0 | 0.750 |

8.00–10.00 | 12.0 | 21.0 | 15.0 | 5.250 |

10.00–12.00 | 14.0 | 19.0 | 15.0 | 1.780 |

12.00–16.00 | 12.0 | 19.0 | 17.0 | 3.250 |

16.00–20.00 | 13.0 | 22.0 | 13.0 | 6.750 |

Particle Size Fraction [mm] | v | v | ${\mathit{\chi}}_{1}^{2}$ | ${\mathit{\chi}}_{2}^{2}$ | ||||
---|---|---|---|---|---|---|---|---|

R_{11} | R_{12} | R_{13} | R_{21} | R_{22} | R_{23} | |||

0.00–2.00 | 9.0 | 12.0 | 9.0 | 6.0 | 11.0 | 13.0 | 1.200 | 5.200 |

2.00–3.15 | 12.0 | 12.0 | 6.0 | 10.0 | 11.0 | 9.0 | 4.800 | 0.400 |

3.15–5.00 | 11.0 | 13.0 | 6.0 | 7.0 | 14.0 | 9.0 | 5.200 | 5.200 |

5.00–6.30 | 13.0 | 9.0 | 8.0 | 8.0 | 12.0 | 10.0 | 2.800 | 1.600 |

6.30–8.00 | 11.0 | 8.0 | 11.0 | 5.0 | 14.0 | 11.0 | 1.200 | 8.400 |

8.00–10.00 | 9.00 | 12.0 | 9.0 | 5.0 | 15.0 | 10.0 | 1.200 | 10.000 |

10.00–12.00 | 11.0 | 12.0 | 7.0 | 5.0 | 13.0 | 12.0 | 2.800 | 7.600 |

12.00–16.00 | 9.0 | 12.0 | 9.0 | 8.0 | 15.0 | 7.0 | 2.800 | 7.600 |

16.00–20.00 | 8.0 | 13.0 | 9.0 | 8.0 | 15.0 | 7.0 | 2.800 | 7.600 |

**Table 4.**The Friedman test results for the ash grade at a fixed u value (the amount of hutch water).

Particle Size Fraction [mm] | u = 35 m^{3}/h | u = 50 m^{3}/h | u = 70 m^{3}/h |
---|---|---|---|

χ^{2} | χ^{2} | χ^{2} | |

0.00–2.00 | 1.160 | 1.500 | 2.670 |

2.00–3.15 | 0.500 | 2.170 | 0.500 |

3.15–5.00 | 2.170 | 2.670 | 1.170 |

5.00–6.30 | 3.170 | 0.170 | 1.500 |

6.30–8.00 | 2.050 | 5.500 | 2.170 |

8.00–10.00 | 0.670 | 0.170 | 1.500 |

10.00–12.00 | 0.670 | 0.670 | 0.670 |

12.00–16.00 | 0.670 | 0.500 | 2.000 |

16.00–20.00 | 1.500 | 4.500 | 2.000 |

Particle Size Fraction [mm] | v = 200 t/h | v = 300 t/h | v = 400 t/h |
---|---|---|---|

χ^{2} | χ^{2} | χ^{2} | |

0.00–2.00 | 2.670 | 0.670 | 1.500 |

2.00–3.15 | 0.500 | 2.670 | 0.670 |

3.15–5.00 | 0.670 | 2.000 | 4.670 |

5.00–6.30 | 1.500 | 2.000 | 1.170 |

6.30–8.00 | 2.170 | 2.000 | 0.670 |

8.00–10.00 | 1.500 | 2.000 | 0.670 |

10.00–12.00 | 0.670 | 0.670 | 0.670 |

12.00–16.00 | 2.000 | 0.670 | 2.670 |

16.00–20.00 | 0.125 | 0.670 | 4.670 |

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**MDPI and ACS Style**

Niedoba, T.; Surowiak, A.; Hassanzadeh, A.; Khoshdast, H.
Evaluation of the Effects of Coal Jigging by Means of Kruskal–Wallis and Friedman Tests. *Energies* **2023**, *16*, 1600.
https://doi.org/10.3390/en16041600

**AMA Style**

Niedoba T, Surowiak A, Hassanzadeh A, Khoshdast H.
Evaluation of the Effects of Coal Jigging by Means of Kruskal–Wallis and Friedman Tests. *Energies*. 2023; 16(4):1600.
https://doi.org/10.3390/en16041600

**Chicago/Turabian Style**

Niedoba, Tomasz, Agnieszka Surowiak, Ahmad Hassanzadeh, and Hamid Khoshdast.
2023. "Evaluation of the Effects of Coal Jigging by Means of Kruskal–Wallis and Friedman Tests" *Energies* 16, no. 4: 1600.
https://doi.org/10.3390/en16041600