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A pie chart (or a circle graph) is a circularchart divided into sectors, illustrating proportion. In a pie chart, the arc length of each sector (and consequently its central angle and area), is proportional to the quantity it represents. When angles are measured with 1 turn as unit then a number of percent is identified with the same number of centiturns. Together, the sectors create a full disk. It is named for its resemblance to a pie which has been sliced. The earliest known pie chart is generally credited to William Playfair's Statistical Breviary of 1801.
The pie chart is perhaps the most ubiquitous statistical chart in the business world and the mass media. However, it has been criticized, and some recommend avoiding it, pointing out in particular that it is difficult to compare different sections of a given pie chart, or to compare data across different pie charts. Pie charts can be an effective way of displaying information in some cases, in particular if the intent is to compare the size of a slice with the whole pie, rather than comparing the slices among them. Pie charts work particularly well when the slices represent 25 to 50% of the data, but in general, other plots such as the bar chart or the dot plot, or non-graphical methods such as tables, may be more adapted for representing certain information.It also shows the frequency within certain groups of information.
The following example chart is based on preliminary results of the election for the European Parliament in 2004. The table lists the number of seats allocated to each party group, along with the derived percentage of the total that they each make up. The values in the last column, the derived central angle of each sector, is found by multiplying the percentage by 360Â°.
*Because of rounding, these totals do not add up to 100 and 360.
The size of each central angle is proportional to the size of the corresponding quantity, here the number of seats. Since the sum of the central angles has to be 360Â°, the central angle for a quantity that is a fraction Q of the total is 360Q degrees. In the example, the central angle for the largest group (European People's Party (EPP)) is 135.7Â° because 0.377 times 360, rounded to one decimal place(s), equals 135.7.
Use, effectiveness and visual perception
Pie charts are common in business and journalism, perhaps because they are perceived as being less "geeky" than other types of graph. However statisticians generally regard pie charts as a poor method of displaying information, and they are uncommon in scientific literature. One reason is that it is more difficult for comparisons to be made between the size of items in a chart when area is used instead of length and when different items are shown as different shapes. Stevens' power law states that visual area is perceived with a power of 0.7, compared to a power of 1.0 for length. This suggests that length is a better scale to use, since perceived differences would be linearly related to actual differences.
Further, in research performed at AT&T Bell Laboratories, it was shown that comparison by angle was less accurate than comparison by length. This can be illustrated with the diagram to the right, showing three pie charts, and, below each of them, the corresponding bar chart representing the same data. Most subjects have difficulty ordering the slices in the pie chart by size; when the bar chart is used the comparison is much easier.. Similarly, comparisons between data sets are easier using the bar chart. However, if the goal is to compare a given category (a slice of the pie) with the total (the whole pie) in a single chart and the multiple is close to 25 or 50 percent, then a pie chart can often be more effective than a bar graph.
Variants and similar charts
Polar area diagram
The polar area diagram is similar to a usual pie chart, except sectors are equal angles and differ rather in how far each sector extends from the center of the circle. The polar area diagram is used to plot cyclic phenomena (e.g., count of deaths by month). For example, if the count of deaths in each month for a year are to be plotted then there will be 12 sectors (one per month) all with the same angle of 30 degrees each. The radius of each sector would be proportional to the square root of the death count for the month, so the area of a sector represents the number of deaths in a month. If the death count in each month is subdivided by cause of death, it is possible to make multiple comparisons on one diagram, as is clearly seen in the form of polar area diagram famously developed by Florence Nightingale.
The first known use of polar area diagrams was by AndrÃ©-Michel Guerry, which he called courbes circulaires, in an 1829 paper showing seasonal and daily variation in wind direction over the year and births and deaths by hour of the day. LÃ©on Lalanne later used a polar diagram to show the frequency of wind directions around compass points in 1843. The wind rose is still used by meteorologists. Nightingale published her rose diagram in 1858. The name "coxcomb" is sometimes used erroneously. This was the name Nightingale used to refer to a book containing the diagrams rather than the diagrams themselves. It has been suggested that most of Nightingale's early reputation was built on her ability to give clear and concise presentations of data.
A useful variant of the polar area chart is the spie chart designed by Feitelson . This superimposes a normal pie chart with a modified polar area chart to permit the comparison of a set of data at two different states. For the first state, for example time 1, a normal pie chart is drawn. For the second state, the angles of the slices are the same as in the original pie chart, and the radii vary according to the change in the value of each variable. In addition to comparing a partition at two times (e.g. this year's budget distribution with last year's budget distribution), this is useful for visualizing hazards for population groups (e.g. the distribution of age and gener groups among road casualties compared with these groups's sizes in the general population). The R Graph Gallery provides an example.
Multi-level pie chart
Multi-level pie chart, also known as a radial tree c
The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 Ã— 9.
In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 Ã— 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 Ã— 25.
In 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)
The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like
1 Ã— 10 = 10 2 Ã— 10 = 20 3 Ã— 10 = 30 4 Ã— 10 = 40 5 Ã— 10 = 50 6 Ã— 10 = 60 7 Ã— 10 = 70 8 Ã— 10 = 80 9 Ã— 10 = 90
10 x 10 = 100 11 x 10 = 110 12 x 10 = 120 13 x 10 = 130 14 x 10 = 140 15 x 10 = 150 16 x 10 = 160 17 x 10 = 170 18 x 10 = 180 19 x 10 = 190 100 x 10 = 1000
This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.
Patterns in the tables
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:
â†’ â†’ 1 2 3 2 4 â†‘ 4 5 6 â†“ â†‘ â†“ 7 8 9 6 8 â†� â†� 0 0 Fig. 1 Fig. 2
For example, to memorize all the multiples of 7:
- Look at the 7 in the first picture and follow the arrow.
- The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
- The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
- After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
- Proceed in the same way until the last number, 3, which corresponds to 63.
- Next, use the 0 at the bottom. It corresponds to 70.
- Then, start again with the 7. This time it will correspond to 77.
- Continue like this.
Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.
In abstract algebra
Standards-based mathematics reform in the USA
In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.
A line chart or line graph is a type of graph, which displays information as a series of data points connected by straight line segments. It is a basic type of chart common in many fields. It is an extension of a scatter graph, and is created by connecting a series of points that represent individual measurements with line segments. A line chart is often used to visualize a trend in data over intervals of time â€“ a time seriesâ€“ thus the line is often drawn chronologically.
In the experimental sciences, data collected from experiments are often visualized by a graph that includes an overlaid mathematical function depicting the best-fit trend of the scattered data. This layer is referred to as a best-fit layer and the graph containing this layer is often referred to as a line graph.
For example, if one were to collect data on the speed of a body at certain points in time, one could visualize the data by a data table such as the following:
The table "visualization" is a great way of displaying exact values, but a very bad way of understanding the underlying patterns that those values represent. Because of these qualities, the table display is often erroneously conflated with the data itself; whereas it is just another visualization of the data.
Understanding the process described by the data in the table is aided by producing a graph or line chart of Speed versus Time. In this context, Versus (or the abbreviations vs and VS), separates the parameters appearing in an X-Y (two-dimensional) graph. The first argument indicates the dependent variable, usually appearing on the Y-axis, while the second argument indicates the independent variable, usually appearing on the X-axis. So, the graph of Speed versus Time would plot time along the x-axis and speed up the y-axis. Mathematically, if we denote time by the variable t, and speed by v, then the function plotted in the graph would be denoted v(t) indicating that v (the dependent variable) is a function of t.
It is simple to construct a "best-fit" layer consisting of a set of line segments connecting adjacent data points; however, such a "best-fit" is usually not an ideal representation of the trend of the underlying scatter data for the following reasons:
- It is highly improbable that the discontinuities in the slope of the best-fit would correspond exactly with the positions of the measurement values.
- It is highly unlikely that the experimental error in the data is negligible, yet the curve falls exactly through each of the data points.
A true best-fit layer should depict a continuous mathematical function whose parameters are determined by using a suitable error-minimization scheme, which appropriately weights the error in the data values.
In either case, the best-fit layer can reveal trends in the data. Further, measurements such as the gradient or the area under the curve can be made visually, leading to more conclusions or results from the data.
From Yahoo Answers
Answers:This one: http://neoparaiso.com/imprimir/multiplos-y-submultiplos.html It goes to 100 vertically and to 20 horizontally.
Answers:You probably mean that you would like to know all the times tables up 100. A good starting point is to learn the tables of squares up to 100 X 100 This will mean that if you knew what , say 18 x 18 is ( i.e. 324), then you can work out what 17 X19 is (its one less than 324, or 323) also 16 X 20 (trivial i know but it is 324 minus 4) or 320 and 15 X 21 ( which = 324-9) or 315 The pattern is, if you have a product of two numbers that are either side of a know square product ,then because the algebra of this is :- (x-1)(x+1) is identical to (x^2-1) i.e expanding the brackets, x^2 -x +x -1 which is x^2 -1. and (x-2)(x+2) is x^2-4, and (x-3)(x+3) is X^2 -9 and so on. I would advise getting some books on number manipulation. I have many puzzle books and mental arithmatic books. If you would like to know some titles, please e mail me.