Illustrations and Examples

In this subsection we list some numerical examples, which are taken from the published literature, in order to shortly illustrate the introduced notions. The numerical tables should suggest the computing of exercises.

(i)

(David, 1977[5], S. 35): 10 values of ore samples (Fe) in % are given:

1	55.8
2	54.8
3	56.5
4	56.0
5	57.5

6	54.7
7	55.3
8	56.3
9	55.9
10	56.3

Find $/bar{z}, /; s$ and an approximate 95% confidence interval for the mean $m$.

Solution: $/bar{z}=55.91/% Fe, s=.835/% Fe$, 95% confidence interval = [55.4,56.4].

(ii)

(Koch and Link, 1970-71[13], S. 30): Frequency distribution of 224 phosphate samples in %.

	Mean of	Abs.	Rel.	Cum.	Rel. Cum.
Interval	Interval	Frequ.	Frequ.	Frequ.	Frequ.
$[/% P_{2}O_{5}]$	$c_{j}$	$h_{j}$	$f_{j}[/%]$		[%]	$h_{j}c_{j}$	$h_{j}c_{j}^{2}$
14-16	15	1	0.45	1	0.45	15	225
16-18	17	1	0.45	2	0.90	17	289
18-20	19	8	3.57	10	4.47	152	2,888
20-22	21	21	9.37	31	13.84	441	9,261
22-24	23	44	19.64	75	33.48	1,012	23,276
24-26	25	54	24.12	129	57.60	1,350	33,750
26-28	27	56	25.00	185	82.60	1,512	40,824
28-30	29	30	13.39	215	95.99	870	25,230
30-32	31	7	3.12	222	99.11	217	6,727
32-34	33	2	0.89	224	100.00	66	2,178

units <1.0cm,1.0cm> x from 0 to 6.0, y from 0.0 to 6.0 right ticks in long withvalues / at 1 2 3 4 5 / /

top ticks in long withvalues / at .4 1.8 3.2 4.6 / /

left label Frequ-encies$h_{j}$ ticks in long withvalues 0 10 20 30 40 50 60 / at 0 1 2 3 4 5 6 / /

bottom label Analyses ( % $P_{2}O_{5}$ ) ticks length <0cm> withvalues 15 20 25 30 35 / at .4 1.8 3.2 4.6 6.0 / /

span <0.07cm> .12 0.0 .68 0.1 1.24 0.1 1.8 0.8 2.36 2.1 2.92 4.4 3.48 5.4 4.04 5.6 4.6 3.0 5.16 0.75 5.72 0.2 /

Computations:
$m /simeq /bar{z}= /sum h_{j}c_{j}//sum h_{j}$
$/sum h_{j}c_{j}=5652$
$/sum h_{j}=224$
$/bar{z}=25.23$
$s^{2}=/frac{SS}{/sum h_{j}-1}=/frac{/sum hc^{2}-(/sum hc)^{2}/ /sum h}{/sum h-1}$
$/sum hc^{2}=144.648$
$(/sum hc)^{2}=31.945.104$
$(/sum hc)^{2}/n=142.612$
SS = 2.036
$s^{2}=9.13$
s = 3.02
v = 0.119 .

(iii)

(David, 1977[5]):

units <0.9cm,0.9cm> x from 0 to 7.0, y from 0.0 to 6.2 right / top / left / bottom / <0.2cm> [0.2,0.4] from 0.5 0.5 to 6.5 0.5 <0.2cm> [0.2,0.4] from 0.5 3.5 to 6.5 3.5 $50' /enspace /mbox{BLOCKS}$ at 4.5 5 $10' /enspace /mbox{SAMPLES}$ at 4.5 1.8 $0.0$ at 0.7 3.3 $0.30$ at 1.7 3.3 $0.60$ at 2.7 3.3 $0.90$ at 3.7 3.3 $1.20$ at 4.7 3.3 $/% /enspace Cu$ at 5.9 3.3 $0.0$ at 0.7 0.3 $0.30$ at 1.7 0.3 $0.60$ at 2.7 0.3 $0.90$ at 3.7 0.3 $1.20$ at 4.7 0.3 $/% /enspace Cu$ at 5.9 0.3 0.5 3.5 1.0 4.5 1.5 5.8 2.0 4.5 2.5 3.8 3.0 3.7 3.5 3.6 / 0.5 0.5 1.0 1.3 1.5 1.8 2.0 1.6 2.5 1.2 3.0 0.9 3.5 0.8 4.0 0.7 4.5 0.6 5.0 0.6 5.5 0.52 /

Histograms of grades of 10'-samples compared with 50'-blocks in a Porphyre Copper deposit.

units <1.0cm,1.0cm> x from 0.0 to 5.5, y from 0.0 to 3.9 <0.2cm> [0.2,0.4] from 5.1 0.0 to 5.5 0.0

left label F
r
e
q
e
n
c
y ticks short unlabeled at 0.29 0.58 0.87 1.46 1.75 2.04 2.63 2.92 3.21 / long withvalues 0 5 10 15 / at 0.0 1.17 2.34 3.51 / / bottom ticks length <0.0cm> withvalues 0 200 400 600 800 1000 / at 0 1 2 3 4 5 / / 0.00 0.00 0.25 1.20 0.50 3.30 0.75 3.40 1.00 2.70 1.25 2.20 1.50 1.80 1.75 1.50 2.00 1.10 2.25 0.90 2.50 0.70 2.75 0.58 3.00 0.50 3.25 0.45 3.50 0.30 3.75 0.25 4.00 0.20 4.25 0.17 4.50 0.14 4.75 0.12 5.00 0.10 /

<0.15cm,0.1cm> 1.7 0.0 1.7 2.3 /

0.5mm 0.00 0.00 0.10 0.40 0.10 1.05 0.20 2.30 0.30 3.00 0.41 3.45 0.60 3.40 0.90 2.70 1.10 2.20 2.00 1.10 2.75 0.50 3.75 0.25 5.50 0.10 /

Histogram of 28.334 gold samples (inch.dwt) in mine A of Witwatersrand.

units <1.0cm,1.0cm> x from 0.0 to 4.2, y from 0.0 to 4.5 <0.2cm> [0.2,0.4] from 0.0 4.1 to 0.0 4.5 left label A
b
s
.

f
r
e
q
u
e
n
c
y ticks short withvalues 50 100 150 200 250 300 350 400 / at 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 / / bottom ticks short withvalues 20 25 30 35 40 45 50 / at 0.05 0.7 1.35 2.0 2.65 3.30 3.95 / /

$/% /enspace Fe$ [0.1] at 4.8 0.0 0.13 0.0 0.26 0.05 0.39 0.10 0.52 0.02 0.65 0.20 0.78 0.20 0.91 0.25 1.04 0.30 1.17 0.55 1.30 0.80 1.43 1.30 1.56 1.90 1.69 2.60 1.82 3.00 1.95 3.25 2.08 3.60 2.21 3.80 2.34 3.80 2.47 4.00 2.60 3.95 2.73 3.60 2.86 2.95 2.99 2.10 3.12 1.50 3.25 1.20 3.38 0.80 3.51 0.55 3.64 0.30 3.77 0.20 3.90 0.10 4.03 0.02 /

Histogram of grade of magnetite of 4838 10'-samples.

units <1.0cm,1.0cm> x from 0.0 to 4.7, y from 0.0 to 5.0 left label R
e
l
a
t
i
v
e

f
r
e
q
u
e
n
c
y ticks short withvalues 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 / at 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 / / bottom ticks length <0cm> withvalues Error / at 0.5 0.0 / /

1.0 0.0 1.1 1.2 1.2 1.05 1.3 1.4 1.45 0.9 1.5 1.35 1.65 0.8 1.9 4.45 1.95 3.95 2.1 3.95 2.15 5.0 2.3 5.2 2.5 3.5 2.6 3.4 2.7 1.4 2.9 1.4 3.0 1.6 3.1 0.85 3.2 0.9 3.25 0.6 3.5 0.7 3.6 0.1 3.75 0.1 3.8 0.0 /

0.5 0.0 1.0 0.5 1.6 1.5 1.9 4.0 2.0 5.0 2.15 5.1 2.3 5.0 2.5 4.0 2.7 2.0 2.8 1.5 3.0 1.0 3.4 0.5 4.2 0.0 /

units <1.0cm,1.0cm> x from 0.0 to 4.7, y from 0.0 to 5.0 left label R
e
l
a
t
i
v
e

f
r
e
q
u
e
n
c
y ticks short withvalues 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 / at 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 / / bottom ticks length <0cm> withvalues Error / at 0.5 0.0 / /

0.4 0.0 0.45 0.05 0.6 0.0 0.75 0.1 0.85 0.0 0.9 0.0 0.95 0.6 1.1 0.25 1.22 0.22 1.3 1.3 1.4 1.05 1.5 2.2 1.65 2.1 1.8 3.15 1.95 3.2 2.0 4.2 2.13 4.6 2.25 3.6 2.35 3.8 2.48 3.6 2.6 2.7 2.7 2.8 2.9 1.7 3.0 0.6 3.1 0.6 3.2 1.1 3.3 0.6 3.45 0.5 3.55 0.05 3.88 0.0 4.0 0.3 4.1 0.0 4.2 0.1 4.3 0.1 /

0.6 0.0 1.0 0.5 1.2 1.0 1.3 1.3 1.5 1.4 1.8 3.1 1.98 4.1 2.1 4.25 2.22 4.1 2.85 1.7 3.5 0.3 3.8 0.15 4.2 0.0 /

Histograms of errors in two different prediction models of values for blocks of mining.

(iv)

(Koch and Link, 1970[13]):

units <.7cm,.7cm> x from 0 to 6, y from 0.0 to 6 <0.2cm> [0.15,0.3] from 2.3 5.5 to 2.8 5.0 <0.2cm> [0.15,0.3] from 4.2 4.5 to 3.5 4.2 left label R
e
l
a
t
i
v
e

f
r
e
q
u
e
n
c
i
e
s ticks long in withvalues 5 10 15 20 25 30 / at 1 2 3 4 5 6 / / bottom label Cu (log ppm) ticks long in withvalues $/bar{1}.75$ $/bar{1}.95$ .15 .35 .55 .75 / at 0.5 1.5 2.5 3.5 4.5 5.5 / / right ticks long in at 1 2 3 4 5 6 / / estimatedfrequencydistribution [lt] at 0.5 6.3 observedfrequencydistribution [lt] at 4 5.5 0.35 0.0 0.35 0.2 0.85 0.2 0.85 0.9 1.35 0.9 1.35 0.6 1.85 0.6 1.85 3.2 2.35 3.2 2.35 4.5 2.85 4.5 2.85 6.0 3.35 6.0 3.35 2.6 3.85 2.6 3.85 0.8 4.35 0.8 4.35 0.5 4.85 0.5 4.85 0.2 5.35 0.2 5.35 0 / 0.15 0.0 0.82 0.3 1.33 0.9 2.1 3.2 2.65 5 2.85 5.15 3.05 5 3.6 2.6 4.35 0.8 4.85 0.2 5.55 0.0 /

Histogram of 216 observations of free soluble copper in a background sediment in Zambia.

units <.7cm,.7cm> x from 0 to 6, y from 0.0 to 6 <0.2cm> [0.15,0.3] from 2.3 4.7 to 2.8 5.0 left label R
e
l
a
t
i
v
e

f
r
e
q
u
e
n
c
i
e
s ticks long in withvalues 5 10 15 20 25 / at 1 2 3 4 5 / / bottom label Cu (log ppm) ticks long in withvalues $/bar{1}$.75 $/bar{1}$.95 .15 .35 .55 .75 .95 / at 0.5 1.5 2.5 3.5 4.5 5.5 6.5 / / right ticks long in at 1 2 3 4 5 / / estimatedfrequencydistributionof the background [lt] at 0.5 5.5 2 standarddeviations at 4.5 4.2 3 standarddeviations at 6 3 possibleanomalies at 4.8 5.5 probableanomalies at 6.2 6 <0.1cm,0.1cm> 4.35 0 4.35 5.2 / 5.15 0 5.15 5.2 / 0.35 0 0.35 0.1 0.85 0.1 0.85 0.5 1.35 0.5 1.35 0.4 1.85 0.4 1.85 2.9 2.35 2.9 2.35 3.5 2.85 3.5 2.85 5.1 3.35 5.1 3.35 3.4 3.85 3.4 3.85 1.2 4.35 1.2 4.35 1.5 4.85 1.5 4.85 1.1 5.35 1.1 5.35 0.35 5.85 0.35 5.85 0.3 6.35 0.3 6.35 0.0 / 4.35 5.1 5.15 5.1 / 4.75 5.1 4.75 5.2 / 5.15 5.0 6.80 5.0 / 5.95 5.0 5.95 5.6 / 0.15 0.0 0.85 0.3 1.35 0.9 2.1 3.2 2.65 5 2.85 5.15 3.05 5 3.35 4 3.85 1.8 4.15 1.2 4.6 0.5 4.85 0.2 5.1 0.1 5.35 0.05 6.05 0.0 /

Histogram of 825 observations of free soluble copper, ``background'' and ``abberant region'' (anomalies).

Table: Estimates of Grades of Metal of the Don Thomás-Ganges Based on Cumulative Analyses of 18 Adjacent Bore Holes.

Point Estimates of Grades of Metal

Hole	Gold	Silver	Lead	Copper	Zinc
Nr.	[ppm]	[ppm]	[%]	[%]	[%]
1	0.15	17	1.8	0.09	1.4
2	0.20	66	2.9	0.08	7.3
3	0.53	172	4.2	0.09	6.8
4	0.40	172	4.4	0.13	6.4
5	0.32	316	6.2	0.24	6.1
6	0.27	277	5.4	0.22	5.3
7	0.41	272	6.2	0.25	5.4
8	0.37	242	5.7	0.24	5.3
9	0.33	294	6.1	0.26	5.7
10	0.30	265	5.5	0.24	5.1
11	0.27	249	5.6	0.23	5.8
12	0.34	249	5.4	0.26	6.1
13	0.31	250	5.8	0.26	7.6
14	0.29	253	7.7	0.33	9.6
15	0.27	239	7.3	0.33	9.3
16	0.28	241	7.0	0.32	8.9
17	0.27	236	6.7	0.35	9.0
18	0.27	278	6.6	0.36	8.6

90%-confidence Intervals

Gold		Silver		Lead		Copper		Zinc
[ppm]		[ppm]		[%]		[%]		[ppm]
below	above	below	above	below	above	below	above	below	above
0	0.48	0	377	0	10.2	0.02	0.14	0	44.4
0	1.51	0	492	0	8.4	0.05	0.13	0	16.9
0	1.04	0	355	2.0	6.9	0.04	0.22	0.5	12.2
0	0.80	0	649	2.0	10.4	0	0.50	2.0	10.2
0	0.65	8	546	1.7	9.1	0.02	0.42	1.7	8.9
0	0.84	52	491	2.8	9.5	0.07	0.42	2.4	8.3
0	0.74	49	436	2.7	8.7	0.10	0.39	2.8	7.8
0	0.66	101	487	3.4	8.8	0.13	0.39	3.4	7.9
0	0.59	87	443	2.9	8.1	0.12	0.36	2.9	7.4
0	0.54	87	411	3.2	7.9	0.13	0.34	3.5	8.1
0.06	0.61	103	396	3.3	7.5	0.15	0.37	3.9	8.3
0.06	0.56	116	384	3.7	7.9	0.16	0.36	4.3	10.9
0.05	0.52	130	377	3.9	11.6	0.18	0.48	4.9	14.3
0.05	0.49	123	356	3.7	11.0	0.19	0.47	4.9	13.7
0.08	0.49	132	349	3.6	10.5	0.18	0.45	4.7	13.1
0.07	0.46	134	338	3.4	10.0	0.21	0.49	5.1	12.9
0.09	0.46	158	398	3.5	9.7	0.23	0.49	4.9	12.3

(v)

102 Soil Samples on a Straight Line in a Distance of 50 m: Grades in ppm.

 

Table 2.7: Analyses of Soil Samples in ppm.

Sample Nr.	Cu	Co	Ni	Cr
1	28	30	285	281
2	25	35	336	259
3	27	47	436	252
4	25	41	448	237
5	21	37	408	170
6	25	46	472	208
7	11	27	385	84
8	16	32	463	89
9	11	27	398	84
10	17	35	415	217
11	27	44	432	292
12	26	40	406	297
13	25	30	307	230
14	26	27	254	202
15	21	22	180	151
16	20	21	163	142
17	21	23	171	138
18	18	19	140	145
19	18	18	130	128
20	18	21	119	142
21	17	17	108	117
22	16	17	97	117
23	16	17	97	89
24	16	16	93	74
25	17	15	103	86
26	13	17	98	85
27	18	17	102	95
28	15	12	100	77
29	16	15	96	82
30	16	15	87	80
31	18	17	129	99
32	18	19	133	98
33	19	18	131	99
34	19	18	127	95
35	18	15	117	101
36	17	15	110	77
37	18	15	105	68
38	14	14	91	53
39	16	14	87	55
40	15	14	75	49
41	15	14	82	94
42	14	13	81	45
43	15	14	73	45
44	14	10	71	36
45	17	17	88	37
46	17	12	90	46
47	16	14	80	44
48	15	13	80	38
49	13	11	74	40
50	15	12	63	33
51	15	13	76	35

Sample Nr.	Cu	Co	Ni	Cr
52	17	15	90	40
53	14	11	79	32
54	15	13	71	33
55	16	11	83	37
56	16	13	73	33
57	14	13	79	40
58	14	15	85	33
59	11	13	93	35
60	15	15	94	55
61	17	10	78	47
62	18	10	75	36
63	16	17	105	40
64	14	13	82	43
65	11	11	79	39
66	16	11	75	45
67	15	11	88	39
68	20	17	181	209
69	21	18	180	213
70	20	20	225	196
71	21	18	236	204
72	20	15	186	170
73	21	21	219	139
74	22	19	235	150
75	27	25	253	159
76	20	18	200	142
77	20	16	185	122
78	23	18	195	128
79	18	13	140	105
80	20	12	108	99
81	23	16	123	103
82	23	16	147	104
83	24	15	137	90
84	21	14	116	81
85	26	16	158	89
86	33	18	178	94
87	39	18	176	78
88	36	23	220	77
89	31	20	225	66
90	30	21	262	66
91	25	19	237	89
92	19	14	199	65
93	21	14	183	46
94	23	19	206	56
95	23	19	204	43
96	24	22	215	50
97	24	17	190	47
98	24	17	163	51
99	25	18	154	66
100	33	20	167	83
101	41	16	144	61
102	35	14	147	62