New two-way discrete frequency table with application to English Premier League data

Figures & data

Table 1. Typical bivariate discrete frequency table

$Y$	$X$	$x_1$	$x_2$	$\cdots$	$x_{m_1}$
$y_1$		$f_{11}$	$f_{12}$	$\cdots$	$f_{1m_1}$
$y_2$		$f_{21}$	$f_{22}$	$\cdots$	$f_{2m_1}$
$⋮$		$⋮$	$⋮$	$⋮$	$⋮$
$y_{m_2}$		$f_{m_{2}1}$	$f_{m_{2}2}$	$\cdots$	$f_{m_2{m}_1}$

Table 2. General $g$-element bivariate discrete frequency table

X Y	Class	$x_1,\cdots ,x_{n_{11}}$	$x_{n_{12}-g+1},\cdots ,x_{n_{12}}$	$\cdots$	$x_{n_{c_1-2}+1},\cdots ,x_{n_{c_1}-1}$	$x_{n_{c_1-1}+1},\cdots ,x_{n_{c_1}}$
Class	$x_{mo}$$y_{mo}$	$x_{m{o}_1}$	$x_{m{o}_2}$	$\cdots$	$x_{m{o}_{c_1-1}}$	$x_{m{o}_{c_1}}$
$y_1,\cdots ,y_{n_{2}1}$	$y_{m{o}_1}$	$f_{11}$	$f_{12}$	$\cdots$	$f_{1(c_1-1)}$	$f_{1c_1}$
$y_{(n_{2}1)+1},\cdots ,y_{(n_{2}1)+(n_{2}2)}$	$y_{m{o}_2}$	$f_{21}$	$f_{22}$	$\cdots$	$f_{2(c_1-1)}$	$f_{2c_1}$
$⋮$	$⋮$	$⋮$	$⋮$	$\cdots$	$⋮$	$⋮$
$y_{n_{c_2-2}+1},\cdots ,y_{n_{c_2}-1}$	$y_{m{o}_{c_2-1}}$	$f_{(c_2-1)1}$	$f_{(c_2-1)2}$	$\cdots$	$f_{(c_2-1)(c_1-1)}$	$f_{(c_2-1)c_1}$
$y_{n_{c_2-1}+1},\cdots ,y_{n_{c_2}}$	$y_{m{o}_{c_2}}$	$f_{c_{2}1}$	$f_{c_{2}2}$	$\cdots$	$f_{c_2(c_1-1)}$	$f_{c_2{c}_1}$

Table 3. Bi-element bivariate discrete frequency table with both $m_1$ and $m_2$ even

X Y	Class	$x_1,x_2$	$x_3,x_4$	$\cdots$	$x_{m_1-3},x_{m_1-2}$		$x_{m_1-1},x_{m_1}$
Class	$x_{mo}$ $y_{mo}$	$x_{m{o}_1}$	$x_{m{o}_2}$	$\cdots$		$x_{m{o}_{c_1-1}}$	$x_{m{o}_{c_1}}$
$y_1,y_2$	$y_{m{o}_1}$	$f_{11}$	$f_{12}$	$\cdots$	$f_{1(c_1-1)}$		$f_{1c_1}$
$y_3,y_4$	$y_{m{o}_2}$	$f_{21}$	$f_{22}$	$\cdots$	$f_{2(c_1-1)}$		$f_{2c_1}$
$⋮$	$⋮$	$⋮$	$⋮$	$\cdots$	$⋮$		$⋮$
$y_{m_2-3},y_{m_2-2}$	$y_{m{o}_{c_2}-1}$	$f_{(c_2-1)1}$	$f_{(c_2-1)2}$	$\cdots$	$f_{(c_2-1)(c_1-1)}$		$f_{(c_2-1)c_1}$
$y_{m_2-1},y_{m_2}$	$y_{m{o}_{c_2}}$	$f_{c_{2}1}$	$f_{c_{2}2}$	$\cdots$	$f_{c_2(c_1-1)}$		$f_{c_2{c}_1}$

Table 4. Bi-element bivariate discrete frequency table with $m_1$ even and $m_2$ odd

X Y	Class	$x_1,x_2$	$x_3,x_4$	$\cdots$	$x_{m_1-3},x_{m_1-2}$	$x_{m_1-1},x_{m_1}$
Class	$x_{mo}$$y_{mo}$	$x_{m{o}_1}$	$x_{m{o}_2}$	$\cdots$	$x_{m{o}_{c_1-1}}$	$x_{m{o}_{c_1}}$
$y_1,y_2$	$y_{m{o}_1}$	$f_{11}$	$f_{12}$	$\cdots$	$f_{1(c_1-1)}$	$f_{1c_1}$
$y_3,y_4$	$y_{m{o}_2}$	$f_{21}$	$f_{22}$	$\cdots$	$f_{2(c_1-1)}$	$f_{2c_1}$
$⋮$	$⋮$	$⋮$	$⋮$	$\cdots$	$⋮$	$⋮$
$y_{m_2-2},y_{m_2-1}$	$y_{m{o}_{c_2}-1}$	$f_{(c_2-1)1}$	$f_{(c_2-1)2}$	$\cdots$	$f_{(c_2-1)(c_1-1)}$	$f_{(c_2-1)c_1}$
$y_{m_2}$	$y_{m_2}$	$f_{c_{2}1}$	$f_{c_{2}2}$	$\cdots$	$f_{c_2(c_1-1)}$	$f_{c_2{c}_1}$

Table 5. Bi-element bivariate discrete frequency table with $m_1$ odd but $m_2$ even

X Y	Class	$x_1,x_2$	$x_3,x_4$	$\cdots$	$x_{m_1-2},x_{m_1-1}$	$x_{m_1}$
Class	$x_{mo}$$y_{mo}$	$x_{m{o}_1}$	$x_{m{o}_2}$	$\cdots$	$x_{m{o}_{c_1}-1}$	$x_{m_1}$
$y_1,y_2$	$y_{m{o}_1}$	$f_{11}$	$f_{12}$	$\cdots$	$f_{1(c_1-1)}$	$f_{1c_1}$
$y_3,y_4$	$y_{m{o}_2}$	$f_{21}$	$f_{22}$	$\cdots$	$f_{2(c_1-1)}$	$f_{2c_1}$
	$⋮$	$⋮$	$⋮$	$\cdots$	$⋮$	$⋮$
$y_{m_2-3},y_{m_2-2}$	$y_{m{o}_{c_2}-1}$	$f_{(c_2-1)1}$	$f_{(c_2-1)2}$	$\cdots$	$f_{(c_2-1)(c_1-1)}$	$f_{(c_2-1)c_1}$
$y_{m_2-1},y_{m_2}$	$y_{m{o}_{c_2}}$	$f_{c_{2}1}$	$f_{c_{2}2}$	$\cdots$	$f_{c_2(c_1-1)}$	$f_{c_2{c}_1}$

Table 6. Bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate binomial distribution with parameters $n=20$, ${\pi }_x=0.5$, ${\pi }_y=0.5$

$X$$Y$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
3	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0
5	0	1	0	6	4	4	1	0	0	0	0	0	0	0	0	0
6	0	0	2	1	11	13	5	1	1	0	0	0	0	0	0	0
7	0	0	0	1	18	24	20	12	1	0	0	0	0	0	0	0
8	0	0	0	0	4	20	56	27	14	2	0	0	0	0	0	0
9	0	0	0	0	1	10	34	64	51	18	4	0	0	0	0	0
10	0	0	0	0	0	3	10	38	60	39	16	4	1	0	0	0
11	0	0	0	0	0	0	5	13	28	60	36	8	2	0	0	0
12	0	0	0	0	0	0	2	2	8	23	48	15	7	1	0	0
13	0	0	0	0	0	0	0	2	5	14	22	31	8	1	1	0
14	0	0	0	0	0	0	0	0	0	1	8	12	15	1	1	0
15	0	0	0	0	0	0	0	0	0	0	0	1	2	3	2	0
16	0	0	0	0	0	0	0	0	0	0	0	1	0	1	2	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1

Table 7. Bi-element bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate binomial distribution with parameters $n=20$, ${\pi }_x=0.5$, ${\pi }_y=0.5$

$X$$Y$	Class interval	2,3	4,5	6,7	8,9	10,11	12,13	14,15	16,17
Class interval	$x_{mo}$$y_{mo}$	3	5	7	9	10	12	14	16
3,4	3	2	2	1	0	0	0	0	0
5,6	6	1	9	32	7	1	0	0	0
7,8	8	0	1	66	115	17	0	0	0
9,10	9	0	0	14	146	168	24	1	0
11,12	11	0	0	0	22	119	107	10	0
13,14	13	0	0	0	2	20	73	25	2
15,16	15	0	0	0	0	0	2	6	4
17	17	0	0	0	0	0	0	0	1

Table 8. Bivariate discrete frequency table constructed using a sample of size 100,000 simulated from the bivariate binomial distribution with parameters $n=50$, ${\pi }_x=0.5$, ${\pi }_y=0.5$

$X$$Y$	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37
15	0	0	0	1	1	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
16	0	0	1	3	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
17	1	1	2	0	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
18	0	0	0	3	4	4	2	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
19	0	0	1	3	8	6	5	5	3	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0
20	0	0	0	2	2	9	12	9	8	3	1	2	0	0	0	0	0	0	0	0	0	0	0	0
21	0	0	0	0	2	7	12	11	13	10	6	2	0	0	0	0	0	0	0	0	0	0	0	0
22	0	1	0	0	0	4	3	9	18	24	11	8	2	1	0	0	0	0	0	0	0	0	0	0
23	0	0	0	0	0	0	4	12	20	17	19	7	6	3	2	0	0	0	0	0	0	0	0	0
24	0	0	0	0	1	1	1	8	10	22	22	22	17	6	3	3	0	1	0	0	0	0	0	0
25	0	0	0	0	0	0	1	0	6	13	28	27	19	19	6	4	0	0	0	0	0	0	0	0
26	0	0	0	0	0	0	0	1	2	2	11	15	24	15	14	4	3	0	1	0	0	0	0	0
27	0	0	0	0	0	0	0	0	3	3	8	10	26	23	17	7	4	1	0	0	0	0	0	0
28	0	0	0	0	0	0	0	0	0	0	3	7	13	14	8	11	7	1	2	0	0	0	0	0
29	0	0	0	0	0	0	0	0	0	0	1	2	4	12	11	13	5	3	1	0	0	0	0	0
30	0	0	0	0	0	0	0	0	0	0	0	2	1	3	12	10	7	6	3	0	1	0	0	0
31	0	0	0	0	0	0	0	0	0	0	0	0	2	1	3	4	6	6	1	0	1	1	0	0
32	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	5	2	0	2	1	2	0	0
33	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2	2	2	0	0	0	1
34	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	1	0	0	3	1	1	0
35	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	0	0	0	0
36	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
37	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
38	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0

Table 9. Tri-element bivariate discrete frequency table constructed using a sample of size 100,000 simulated from the bivariate binomial distribution with parameters $n=50$, ${\pi }_x=0.5$, ${\pi }_y=0.5$

$X$$Y$	Class interval	14,15,16	17,18,19	20,21,22	23,24,25	26,27,28	29,30,31	32,33,34	35,36,37
Class interval	$x_{mo}$$y_{mo}$	16	19	22	24	26	29	32	35
15,16,17	17	5	8	3	1	0	0	0	0
18,19,20	20	1	41	45	8	0	0	0	0
21,22,23	23	1	13	102	104	14	0	0	0
24,25,26	25	0	2	29	162	123	15	1	0
27,28,29	27	0	0	3	34	128	52	3	0
30,31,32	30	0	0	0	2	23	46	9	3
33,34,35	33,34	0	0	0	0	1	4	8	3
36,37,38	36,37,38	0	0	0	0	0	0	0	3

Table 10. Bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=2.5$, ${\mu }_2=3.5$, and ${\mu }_3=2.5$

$X$$Y$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	15
0	0	1	2	1	0	0	1	0	0	0	0	0	0	0	0
1	2	3	4	6	6	6	1	0	2	0	1	0	0	0	0
2	0	2	5	15	18	18	10	6	2	1	1	2	0	0	0
3	0	4	12	20	24	30	22	12	7	2	0	0	1	0	0
4	0	2	12	21	31	37	30	22	14	8	11	4	1	1	0
5	0	1	6	17	19	23	40	27	19	10	4	1	3	0	0
6	0	0	4	10	22	28	20	31	29	14	6	1	3	0	0
7	0	1	2	3	9	12	18	17	11	12	4	3	1	0	0
8	0	0	0	2	4	6	14	10	10	12	6	3	1	0	1
9	0	0	0	0	2	3	4	4	4	8	3	1	0	2	0
10	0	0	0	0	0	3	0	3	2	1	4	2	1	1	0
11	0	0	0	0	0	1	0	0	1	3	0	1	0	0	0
12	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
13	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0

Table 11. Bi-element bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=2.5$, ${\mu }_2=3.5$, and ${\mu }_3=2.5$

$X$$Y$	Class interval	0,1	2,3	4,5	6,7	8,9	10,11	12,13	15
Class interval	$x_{mo}$$y_{mo}$	1	3	5	6	8	10	12	15
0,1	1	6	13	12	2	2	1	0	0
2,3	3	6	52	90	50	12	3	1	0
4,5	4	3	56	110	119	51	20	5	0
6,7	6	1	19	71	86	66	14	4	0
8,9	8	0	2	15	32	34	13	3	1
10,11	10	0	0	4	3	7	7	2	0
12,13	12,13	0	0	0	0	0	0	2	0

Table 12. Bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=6.5$, ${\mu }_2=5.5$, and ${\mu }_3=4.5$

$X$$Y$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
2	1	0	0	0	0	2	1	0	2	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	1	1	1	0	1	1	2	0	1	0	0	0	0	0	0	0	0
5	0	0	2	5	2	1	2	1	2	2	0	1	2	0	0	0	0	0	0	0
6	1	1	2	3	6	6	6	4	5	3	6	1	0	0	0	0	0	0	0	0
7	0	0	3	4	1	15	9	10	5	2	4	2	1	1	0	0	0	0	0	0
8	0	2	6	6	3	13	11	10	13	4	6	6	4	1	0	1	1	0	0	0
9	0	0	3	4	12	6	5	27	13	7	16	5	4	1	3	1	2	0	0	0
10	0	0	3	3	8	9	19	15	19	16	9	8	4	3	3	0	1	0	0	0
11	0	0	2	2	4	8	14	13	16	19	8	10	8	4	3	2	0	0	0	0
12	0	2	2	2	5	10	15	15	14	18	13	9	7	4	3	1	1	1	0	0
13	0	0	0	0	4	2	6	16	11	12	10	11	3	4	4	0	1	2	0	0
14	0	0	0	0	3	8	8	10	7	15	11	12	4	2	2	2	2	0	0	1
15	0	0	0	1	0	1	7	5	9	5	9	1	4	2	0	1	1	1	1	0
16	0	0	0	0	0	1	1	3	6	8	5	0	3	6	1	0	2	0	0	0
17	0	0	1	1	0	1	1	0	2	1	3	4	4	2	2	1	1	0	0	1
18	0	0	0	0	0	0	1	0	1	0	0	1	2	2	0	1	0	0	0	0
19	0	0	0	0	0	0	0	0	1	2	1	4	1	1	2	1	1	0	0	0
20	0	0	0	0	0	0	0	0	1	0	1	0	0	1	1	1	0	0	0	0
21	0	0	0	0	0	0	0	0	0	0	0	2	0	0	0	0	0	0	0	0
24	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0

Table 13. Tri-element bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=6.5$, ${\mu }_2=5.5$, and ${\mu }_3=4.5$

$X$$Y$	Class interval	2,3,4	5,6,7	8,9,10	11,12,13	14,15,16	17,18,19	20,21
Class interval	$x_{mo}$$y_{mo}$	4	7	9	11	14	17,18	21
2,3,4	4	1	6	6	3	0	0	0
5,6,7	7	9	43	44	21	4	0	0
8,9,10	10	14	64	132	77	23	6	0
11,12,13	12	6	37	120	110	40	8	0
14,15,16	14	0	14	56	66	24	9	2
17,18,19	17	1	2	6	16	16	5	1
20,21,24	20	0	0	1	3	2	2	0

Table 14. Bivariate discrete frequency table constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=6.5$, ${\mu }_2=5.5$, and ${\mu }_3=4.5$

$X$$Y$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
3	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	2	0	1	0	0	1	0	0	2	0	0	0	0	0	0	0	0	0	0
5	0	0	1	2	4	4	2	1	0	0	2	1	0	0	0	0	0	0	0	0	0	0
6	0	2	1	6	4	4	10	4	7	4	2	3	0	0	0	0	0	0	0	0	0	0
7	1	0	1	5	6	6	8	8	6	10	5	1	1	1	0	0	0	0	0	0	0	0
8	0	1	0	6	9	11	10	20	14	11	7	6	4	1	1	0	0	0	0	0	0	0
9	0	2	3	7	8	16	13	21	14	11	9	4	3	1	0	0	0	0	0	0	0	0
10	0	0	6	4	12	14	19	16	18	12	8	8	4	4	2	0	3	0	0	0	0	0
11	0	0	1	3	7	11	12	11	18	10	11	9	1	6	0	2	0	0	0	0	0	0
12	0	0	3	3	4	8	17	7	7	12	12	7	6	7	1	2	0	0	2	0	0	0
13	0	0	0	1	2	2	7	13	8	10	15	8	7	6	2	0	2	0	0	1	1	0
14	0	0	0	1	1	5	7	7	10	14	7	12	9	8	3	4	0	2	0	0	0	0
15	0	0	0	0	1	1	4	4	7	10	8	6	6	3	3	5	2	0	0	1	0	1
16	0	0	0	0	0	0	3	0	4	13	4	8	2	2	3	1	0	0	0	0	0	0
17	0	0	0	0	1	1	1	1	2	3	4	4	2	1	3	0	0	0	0	0	0	0
18	0	0	0	0	0	0	1	1	0	2	0	1	3	2	0	1	2	0	0	0	0	0
19	0	0	1	0	0	0	1	0	0	2	0	1	1	0	0	1	0	0	0	0	0	0
20	0	0	0	0	0	0	0	0	1	0	1	0	1	2	0	0	0	0	0	0	0	0
21	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0

Table 15. Tri-element bivariate discrete frequency table, where the first class is having a different number of elements, constructed using a sample of size 1000 simulated from the bivariate poisson distribution with parameters ${\mu }_1=6.5$, ${\mu }_2=5.5$, and ${\mu }_3=4.5$

$X$$Y$	Class interval	2	3,4,5	6,7,8	9,10, 11	12,13,14	15,16,17	18,19,20	21,22,23
Class interval	$x_{mo}$ $y_{mo}$	2	5	8	11	12	15	18	21
3	3	0	0	2	0	0	0	0	0
4,5,6	6	0	14	29	17	10	0	0	0
7,8,9	9	1	25	87	115	40	4	0	0
10,11,12	10	0	20	104	111	66	24	5	0
13,14,15	14	0	2	30	83	87	34	6	4
16,17,18	16	0	0	7	26	28	12	2	0
19,20,21	19	0	1	1	3	4	4	0	0

Table 16. Bivariate discrete frequency table constructed using data on the number of wins and clean sheets for English Premier League clubs from season 2006/2007 to 2017/2018

$X$$Y$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	24
1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0	1	0	0	0	0	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	1	1	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	0	1	1	5	1	1	1	0	0	1	0	0	0	0	0	0	0	0	0	0
8	0	3	0	0	2	2	2	2	1	1	1	0	0	0	0	0	0	0	0	0	0	0
9	0	0	0	4	0	3	3	1	2	1	2	0	0	0	0	0	0	0	0	0	0	0
10	0	1	0	2	1	1	4	2	6	2	1	0	0	0	0	0	0	0	0	0	0	0
11	0	0	0	1	2	6	6	5	4	1	1	1	4	0	1	0	0	0	0	0	0	0
12	1	1	0	1	1	2	1	8	3	2	2	2	2	0	0	0	0	0	0	0	0	0
13	0	0	0	0	0	1	1	3	2	1	1	1	0	0	0	0	0	0	0	0	0	0
14	0	0	0	0	0	0	1	0	2	1	3	1	0	1	0	0	0	0	0	0	0	0
15	0	0	0	0	0	0	2	1	2	1	1	0	1	1	0	0	0	0	0	0	0	0
16	0	0	0	0	0	0	1	1	0	4	1	2	0	0	2	0	0	0	0	0	0	0
17	0	0	0	0	1	0	0	0	0	0	0	2	1	1	0	1	0	0	0	0	0	0
18	0	0	0	0	0	0	0	0	1	1	1	0	1	1	0	2	0	0	0	0	0	0
19	0	0	0	0	0	0	0	1	0	0	1	4	1	1	1	0	1	0	0	0	0	0
20	0	0	0	0	0	0	0	0	0	1	0	0	1	0	1	0	1	0	1	0	0	0
21	0	0	0	0	0	0	0	1	0	0	0	2	2	2	1	1	2	0	0	0	0	0
22	0	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0	0	0	0	0	0	0
23	0	0	0	0	0	0	0	0	0	0	1	1	1	2	1	0	1	0	0	0	0	0
24	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	1	0	0	0	0	1	0
25	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0
26	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	2	0	0	0	0	0	0
27	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	1	0	1	0	0
28	0	0	0	0	0	0	0	0	0	0	0	1	0	0	1	1	0	0	1	0	0	1
30	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
32	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0

Table 17. Tri-Element bivariate discrete frequency table constructed using data on the number of wins and clean sheets for English Premier League clubs from season 2006/2007 to 2017/2018

$X$$Y$	Class interval	2,3,4	5,6,7	8,9,10	11,12,13	14,15,16	17,18,19	20,21,22	24
Class interval	$x_{mo}$$y_{mo}$	3	7	9	13	14	17,18	20	24
1,3,4	1,3,4	1	2	0	0	0	0	0	0
5,6,7	7	1	10	5	2	0	0	0	0
8,9,10	10	4	15	23	8	0	0	0	0
11,12,13	11	2	14	33	12	7	0	0	0
14,15,16	16	0	0	10	14	5	0	0	0
17,18,19	19	0	1	2	9	7	4	0	0
20,21,22	21	0	0	1	5	8	4	1	0
23,24,25	23	0	0	0	2	6	4	4	0
26,27,28	28	0	0	1	1	2	5	2	1
30,32	30,32	0	0	0	0	1	1	0	0