User:Cmglee/HTML

Maths

Common logarithm

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

${\begin{array}{rll}{\text{As found above,}}&\log _{10}0.012\approx {\bar {2}}.079181\\{\text{Since}}\;\;\log _{10}0.85&=\log _{10}(10^{-1}\times 8.5)=-1+\log _{10}8.5&\approx -1+0.929419={\bar {1}}.929419\;,\\\log _{10}(0.012\times 0.85)&=\log _{10}0.012+\log _{10}0.85&\approx {\bar {2}}.079181+{\bar {1}}.929419\\&=(-2+0.079181)+(-1+0.929419)&=-(2+1)+(0.079181+0.929419)\\&=-3+1.008600&=-2+0.008600\;^{*}\\&\approx \log _{10}(10^{-2})+\log _{10}(1.02)&=\log _{10}(0.01\times 1.02)\\&=\log _{10}(0.0102)\end{array}}$

* This step makes the mantissa between 0 and 1, so that its antilog (10^mantissa) can be looked up.

Mode (statistics)

When X has standard deviation ? = 0.25, the distribution of Y is weakly skewed. Using formulas for the log-normal distribution, we find:

${\begin{array}{rlll}{\text{mean}}&=e^{\mu +\sigma ^{2}/2}&=e^{0+0.25^{2}/2}&\approx 1.032\\{\text{mode}}&=e^{\mu -\sigma ^{2}}&=e^{0-0.25^{2}}&\approx 0.939\\{\text{median}}&=e^{\mu }&=e^{0}&=1\end{array}}$

Indeed, the median is about one third on the way from mean to mode.

When X has a larger standard deviation, ? = 1, the distribution of Y is strongly skewed. Now

${\begin{array}{rlll}{\text{mean}}&=e^{\mu +\sigma ^{2}/2}&=e^{0+1^{2}/2}&\approx 1.649\\{\text{mode}}&=e^{\mu -\sigma ^{2}}&=e^{0-1^{2}}&\approx 0.368\\{\text{median}}&=e^{\mu }&=e^{0}&=1\end{array}}$

Here, Pearson's rule of thumb fails.

Ratio of volumes of a cone, sphere and cylinder of the same radius and height

A cone, sphere and cylinder of radius r and height h.

The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere).

${\begin{array}{llll}{\text{Volume of the cone}}&={\tfrac {1}{3}}\pi r^{2}h&={\tfrac {1}{3}}\pi r^{2}(2r)&=({\tfrac {2}{3}}\pi r^{3})\times 1\\{\text{Volume of the sphere}}&={\tfrac {4}{3}}\pi r^{3}&&=({\tfrac {2}{3}}\pi r^{3})\times 2\\{\text{Volume of the cylinder}}&=\pi r^{2}h&=\pi r^{2}(2r)&=({\tfrac {2}{3}}\pi r^{3})\times 3\end{array}}$

The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.

Ratio of surface areas of a sphere and cylinder of the same radius and height

The above formulas can be used to show that the volumes of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere).

${\begin{array}{llll}{\text{Surface area of the sphere}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Surface area of the cylinder}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}$

The discovery of this ratio is credited to Archimedes.

Wikitext tables

Chroma subsampling

	4:1:1						4:2:0						4:2:2						4:4:4
Y'CrCb
	=						=						=						=
Y'
	+						+						+						+
	1	2	3	4	J = 4		1	2	3	4	J = 4		1	2	3	4	J = 4		1	2	3	4	J = 4
(Cr, Cb)	1				a = 1		1		2		a = 2		1		2		a = 2		1	2	3	4	a = 4
	1				b = 1						b = 0		1		2		b = 2		1	2	3	4	b = 4
	¼ horizontal resolution, full vertical resolution						½ horizontal resolution, ½ vertical resolution						½ horizontal resolution, full vertical resolution						full horizontal resolution, full vertical resolution

Template:Table of Phase Transitions

Phase transitions of matter ()

To From	Solid	Liquid	Gas	Plasma
Solid		Melting	Sublimation
Liquid	Freezing		Vaporization
Gas	Deposition	Condensation		Ionization
Plasma			Recombination

Template:DiagnosticTesting_Example

		Patients with bowel cancer (as confirmed on endoscopy)
	Total population (pop.) = 2030	Condition positive	Condition negative	Prevalence = (TP + FN) / pop. = (20 + 10) / 2030 ≈ 1.48%	Accuracy (ACC) = (TP + TN) / pop. = (20 + 1820) / 2030 ≈ 90.64%
Fecal occult blood screen test outcome	Test outcome positive	True positive (TP) = 20 (2030 × 1.48% × 67%)	False positive (FP) = 180 (2030 × (100% − 1.48%) × (100% − 91%))	Positive predictive value (PPV), precision = TP / (TP + FP) = 20 / (20 + 180) = 10%	False discovery rate (FDR) = FP / (TP + FP) = 180 / (20 + 180) = 90.0%
	Test outcome negative	False negative (FN) = 10 (2030 × 1.48% × (100% − 67%))	True negative (TN) = 1820 (2030 × (100% − 1.48%) × 91%)	False omission rate (FOR) = FN / (FN + TN) = 10 / (10 + 1820) ≈ 0.55%	Negative predictive value (NPV) = TN / (FN + TN) = 1820 / (10 + 1820) ≈ 99.45%
		True positive rate (TPR), recall, sensitivity = TP / (TP + FN) = 20 / (20 + 10) ≈ 66.7%	False positive rate (FPR), fall-out, probability of false alarm = FP / (FP + TN) = 180 / (180 + 1820) = 9.0%	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠ = (20 / 30) / (180 / 2000) ≈ 7.41	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠ ≈ 20.2	F1 score = 2 × ⁠precision × recall/precision + recall⁠ ≈ 0.174
		False negative rate (FNR), miss rate = FN / (TP + FN) = 10 / (20 + 10) ≈ 33.3%	Specificity, selectivity, true negative rate (TNR) = TN / (FP + TN) = 1820 / (180 + 1820) = 91%	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠ = (10 / 30) / (1820 / 2000) ≈ 0.366

Template:DiagnosticTesting Diagram

		Condition (as determined by "Gold standard")
		Condition Positive	Condition Negative
Test Outcome	Test Outcome Positive	True Positive	False Positive (Type I error)	Positive predictive value = Σ True Positive Σ Test Outcome Positive
	Test Outcome Negative	False Negative (Type II error)	True Negative	Negative predictive value = Σ True Negative Σ Test Outcome Negative
		Sensitivity = Σ True Positive Σ Condition Positive	Specificity = Σ True Negative Σ Condition Negative

2011_E._coli_O104:H4_outbreak

LEGEND	No food restrictions	Food sale/trade restrictions / ran tests
No cases
Suspected cases
Known cases
Deaths

Exposure to the outbreak start from April 21, 2011 with permanent or temporary stay in Germany, consumption of a food acquired in Germany or close contact with a HUS case.

Number of cases by country until 31 May 2011

Country	Deaths	Confirmed cases	Suspected cases
Germany	17	450	1 200
United Kingdom	0	3	-
Sweden	1	41	-
Denmark	0	14	26
Netherlands	0	2	-
France	0	0	3
Spain	0	1	-
Austria	0	-	-
Norway	0	-	-
Switzerland	0	2	-
Finland	0	-	1
Poland	0	1	-
United States	0	3	-

Uniform circular motion

Centripetal acceleration for some values of radius and magnitude of velocity

|v|   r		1 m/s 3.6 km/h 2.2 mph	2 m/s 7.2 km/h 4.5 mph	5 m/s 18 km/h 11 mph	10 m/s 36 km/h 22 mph	20 m/s 72 km/h 45 mph	50 m/s 180 km/h 110 mph	100 m/s 360 km/h 220 mph
		Slow walk		Bicycle		City car		Aerobatics
10 cm 3.9 in	Laboratory centrifuge	10 m/s² 1.0 g	40 m/s² 4.1 g	250 m/s² 25 g	1.0 km/s² 100 g	4.0 km/s² 410 g	25 km/s² 2500 g	100 km/s² 10000 g
20 cm 7.9 in		5.0 m/s² 0.51 g	20 m/s² 2.0 g	130 m/s² 13 g	500 m/s² 51 g	2.0 km/s² 200 g	13 km/s² 1300 g	50 km/s² 5100 g
50 cm 1.6 ft		2.0 m/s² 0.20 g	8.0 m/s² 0.82 g	50 m/s² 5.1 g	200 m/s² 20 g	800 m/s² 82 g	5.0 km/s² 510 g	20 km/s² 2000 g
1 m 3.3 ft	Playground carousel	1.0 m/s² 0.10 g	4.0 m/s² 0.41 g	25 m/s² 2.5 g	100 m/s² 10 g	400 m/s² 41 g	2.5 km/s² 250 g	10 km/s² 1000 g
2 m 6.6 ft		500 mm/s² 0.051 g	2.0 m/s² 0.20 g	13 m/s² 1.3 g	50 m/s² 5.1 g	200 m/s² 20 g	1.3 km/s² 130 g	5.0 km/s² 510 g
5 m 16 ft		200 mm/s² 0.020 g	800 mm/s² 0.082 g	5.0 m/s² 0.51 g	20 m/s² 2.0 g	80 m/s² 8.2 g	500 m/s² 51 g	2.0 km/s² 200 g
10 m 33 ft	Roller-coaster vertical loop	100 mm/s² 0.010 g	400 mm/s² 0.041 g	2.5 m/s² 0.25 g	10 m/s² 1.0 g	40 m/s² 4.1 g	250 m/s² 25 g	1.0 km/s² 100 g
20 m 66 ft		50 mm/s² 0.0051 g	200 mm/s² 0.020 g	1.3 m/s² 0.13 g	5.0 m/s² 0.51 g	20 m/s² 2 g	130 m/s² 13 g	500 m/s² 51 g
50 m 160 ft		20 mm/s² 0.0020 g	80 mm/s² 0.0082 g	500 mm/s² 0.051 g	2.0 m/s² 0.20 g	8.0 m/s² 0.82 g	50 m/s² 5.1 g	200 m/s² 20 g
100 m 330 ft	Freeway on-ramp	10 mm/s² 0.0010 g	40 mm/s² 0.0041 g	250 mm/s² 0.025 g	1.0 m/s² 0.10 g	4.0 m/s² 0.41 g	25 m/s² 2.5 g	100 m/s² 10 g
200 m 660 ft		5.0 mm/s² 0.00051 g	20 mm/s² 0.0020 g	130 m/s² 0.013 g	500 mm/s² 0.051 g	2.0 m/s² 0.20 g	13 m/s² 1.3 g	50 m/s² 5.1 g
500 m 1600 ft		2.0 mm/s² 0.00020 g	8.0 mm/s² 0.00082 g	50 mm/s² 0.0051 g	200 mm/s² 0.020 g	800 mm/s² 0.082 g	5.0 m/s² 0.51 g	20 m/s² 2.0 g
1 km 3300 ft	High-speed railway	1.0 mm/s² 0.00010 g	4.0 mm/s² 0.00041 g	25 mm/s² 0.0025 g	100 mm/s² 0.010 g	400 mm/s² 0.041 g	2.5 m/s² 0.25 g	10 m/s² 1.0 g

Chaffing and winnowing

	secure channel		insecure channel
Alice		Charles		Bob
constructs 4 packets, each containing one bit of her message and a valid MAC	Serial Bit MAC 1 1 234 2 0 890 3 0 456 4 1 678	adds 4 chaff packets with inverted bits and invalid MAC, shown in italics (chaffing)	Serial Bit MAC 1 0 321 1 1 234 2 0 890 2 1 987 3 0 456 3 1 543 4 0 765 4 1 678	discards packets with invalid MAC to recover the message (winnowing)
In this example, Alice wishes to send the message "1001" to Bob. For simplicity, assume that all even MAC are valid and odd ones are invalid.

Psychological pricing

Ending	Percentage
0	7.5
1	0.3
2	0.3
3	0.8
4	0.3
5	28.6
6	0.3
7	0.4
8	1.0
9	60.7

I'm My Own Grandpa

Narrator — Wife       |

Father — Stepdaughter       |

Narrator

Family tree showing how
the narrator of the song
is his own grandfather.

Solar eclipse

	Moon		Sun
	At perigee (nearest)	At apogee (farthest)	At perihelion (nearest)	At aphelion (farthest)
Mean radius, r	1,737.10 kilometres (1,079.38 miles)		696,000 kilometres (432,000 miles)
Distance, d	363,104 km (225,622 mi)	405,696 km (252,088 mi)	147,098,070 km (91,402,500 mi)	152,097,700 km (94,509,100 mi)
Angular diameter, 2×arctan(r / d)	32' 54" (0.5482°)	29' 26" (0.4907°)	32' 32" (0.5422°)	31' 28" (0.5244°)
Apparent size to scale
Rank in descending order	1st	4th	2nd	3rd

Angular diameter

Celestial body	Angular diameter	Relative size (10 pixels per arcsecond)
Sun	31.6′ – 32.7′	28.7 – 29.7 times the maximum value for Venus (orange bar below)
Moon	29.3′ – 34.1′	26.6 – 31.0 times the maximum value for Venus (orange bar below)
Venus	9.565″ – 66.012″
Jupiter	29.800″ – 50.115″
Saturn	14.991″ – 20.790″
Mars	3.492″ – 25.113″
Mercury	4.535″ – 13.019″
Uranus	3.340″ – 4.084″
Neptune	2.179″ – 2.373″
Ceres	0.330″ – 0.840″
Vesta	0.20" – 0.64"
Pluto	0.063″ – 0.115″
R Doradus	0.052″ – 0.062″
Betelgeuse	0.049″ – 0.060″
Eris	0.034" – 0.089″
Alphard	0.00909″
Alpha Centauri A	0.007″
Canopus	0.006″
Sirius	0.005936″
Altair	0.003″
Deneb	0.002″
Proxima Centauri	0.001″

Apparent magnitude

Visible to typical human eye	Apparent magnitude	Brightness relative to Vega	Number of stars brighter than apparent magnitude
Yes	-1	250%	1
	0	100%	4
	1	40%	15
	2	16%	48
	3	6.3%	171
	4	2.5%	513
	5	1.0%	1 602
	6	0.40%	4 800
No	7	0.16%	14 000
	8	0.063%	42 000
	9	0.025%	121 000
	10	0.010%	340 000

Magnitude (astronomy)

Apparent magnitude	Brightness relative to magnitude 0	Example		Apparent magnitude	Brightness relative to magnitude 0	Example		Apparent magnitude	Brightness relative to magnitude 0	Example
-27	6.3×1010	Sun		-7	630	SN 1006 supernova		13	6.3×10-6	3C 273 quasar
-26	2.5×1010			-6	250	International Space Station (max)		14	2.5×10-6	Pluto (max)
-25	1.0×1010			-5	100	Venus (max)		15	1.0×10-6
-24	4.0×109			-4	40			16	4.0×10-7	Charon (max)
-23	1.6×109			-3	16	Jupiter (max)		17	1.6×10-7
-22	6.3×108			-2	6.3	Mercury (max)		18	6.3×10-8
-21	2.5×108			-1	2.5	Sirius		19	2.5×10-8
-20	1.0×108			0	1.0	Vega		20	1.0×10-8
-19	4.0×107			1	0.40	Antares		21	4.0×10-9	Callirrhoe (small satellite of Jupiter)
-18	1.6×107			2	0.16	Polaris		22	1.6×10-9
-17	6.3×106			3	0.063	Andromeda galaxy		23	6.3×10-10
-16	2.5×106			4	0.025	Acubens		24	2.5×10-10
-15	1.0×106			5	0.010	Vesta asteroid (max)		25	1.0×10-10	Fenrir (small satellite of Saturn)
-14	4.0×105			6	4.0×10-3	typical limit of naked eye		26	4.0×10-11
-13	1.6×105	Full moon		7	1.6×10-3	Ceres (max)		27	1.6×10-11	visible light limit of 8m ground-based telescopes
-12	6.3×104			8	6.3×10-4	Neptune (max)		28	6.3×10-12
-11	2.5×104			9	2.5×10-4			29	2.5×10-12
-10	1.0×104			10	1.0×10-4	typical limit of 7x50 binoculars		30	1.0×10-12
-9	4.0×103	Iridium flare		11	4.0×10-5			31	4.0×10-13
-8	1.6×103			12	1.6×10-5			32	1.6×10-13	visible light limit of Hubble Space Telescope

Penang Ferry Service

Status	Configuration	Name	Year built	Deadweight tonnage	IMO ship identification number	Photograph
Retired	Passenger (upper deck) Vehicle (lower deck)	Pulau Labuan	1971	139	7038408
In service	Passenger (upper deck) Vehicle (lower deck)	Pulau Rawa	1975	262	7343736
In service	Passenger (upper deck) Vehicle (lower deck)	Pulau Talang Talang	1975	262	7343748
In service	Passenger (upper deck) Vehicle (lower deck)	Pulau Undan	1975	262	7343724
In service	Vehicle (both decks)	Pulau Rimau	1980	100	7911076
In service	Passenger (upper deck) Vehicle (lower deck)	Pulau Angsa	1981	100	8010491
In service	Vehicle (both decks)	Pulau Kapas	1981	280	8101082
In service	Vehicle (both decks)	Pulau Payar	2002	440	9254393
In service	Vehicle (both decks)	Pulau Pinang	2002	440	9275244

Day of Seven Billion

Age group	Estimated July 2010 population in millions
	Africa		Asia		Europe		Latin America and the Caribbean		Northern America		Oceania
0–4	155.3	155.3	360.4	360.4	39.67	39.67	53.83	53.83	23.54	23.54	3.079	3.079
5–9	136.2	136.2	354.5	354.5	37.01	37.01	55.52	55.52	22.66	22.66	2.875	2.875
10–14	120.2	120.2	364.8	364.8	37.32	37.32	55.12	55.12	21.72	21.72	2.844	2.844
15–19	108.1	108.1	374.6	374.6	42.54	42.54	54.11	54.11	23.93	23.93	2.829	2.829
20–24	97.21	97.21	379.4	379.4	51.28	51.28	52.08	52.08	24.00	24	2.838	2.838
25–29	83.89	83.89	343.8	343.8	53.33	53.33	49.47	49.47	24.42	24.42	2.721	2.721
30–34	69.28	69.28	310.5	310.5	52.96	52.96	45.52	45.52	22.17	22.17	2.497	2.497
35–39	55.60	55.6	315.4	315.4	52.88	52.88	41.34	41.34	22.71	22.71	2.594	2.594
40–44	45.15	45.15	297.5	297.5	53.46	53.46	37.64	37.64	23.03	23.03	2.382	2.382
45–49	37.82	37.82	248.4	248.4	55.15	55.15	34.33	34.33	25.74	25.74	2.362	2.362
50–54	31.68	31.68	212.2	212.2	53.08	53.08	28.68	28.68	24.91	24.91	2.128	2.128
55–59	25.68	25.68	188.5	188.5	48.30	48.3	23.72	23.72	21.72	21.72	1.862	1.862
60–64	20.11	20.11	135.1	135.1	41.80	41.8	18.07	18.07	18.62	18.62	1.676	1.676
65–69	14.78	14.78	100.0	100	31.57	31.57	13.88	13.88	13.72	13.72	1.227	1.227
70–74	10.43	10.43	78.13	78.13	32.85	32.85	10.66	10.66	10.25	10.25	0.939	0.939
75–79	6.367	6.367	53.71	53.71	24.09	24.09	7.564	7.564	8.223	8.223	0.702	0.702
80–84	3.100	3.1	29.75	29.75	17.89	17.89	4.870	4.87	6.755	6.755	0.564	0.564
85–89	1.036	1.036	12.70	12.7	9.446	9.446	2.460	2.46	4.201	4.201	0.318	0.318
90–94	0.229	0.229	3.882	3.882	2.689	2.689	0.921	0.921	1.714	1.714	0.120	0.12
95–99	0.030	0.03	0.775	0.775	0.795	0.795	0.253	0.253	0.425	0.425	0.031	0.031
100+	0.003	0.003	0.090	0.09	0.089	0.089	0.044	0.044	0.063	0.063	0.004	0.004