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CUMULATIVE FREQUENCY GRAPH 137
Height in cm Frequency Cumulative Frequency (Number of Students less than Greatest Height in the Class interval)
140- 4 4 who are less than 145
145- 7 11
150- 17 28
155- 22 50
160- 24 74
165- 15 89
170- 9 98
175-180 2 100
The word cumulative sounds like the word accumulate, which means gathering together. In fact, we gather together the frequencies as we move down the table. In the first row of the cumulative frequency table there are 4 students with heights less than 145 cm. Therefore, the coordinates (145,4) are associated with this row of data. In the second row there are 11 students (made up of 4 + 7) with heights less than 150 cm. Therefore, the coordinates (150,11) are associated with the second row, and so on. By plotting, on a set of axes, the coordinates (145,4), (150,11), (155,28), (160,50), (165, 74), (170, 89), (175, 98), (180, 100), and, of course, (140, 0), we obtain a cumulative frequency graph of the heights of students at Luke’s school. The coordinates are joined with a series of straight-line segments. The cumulative frequency curve should start on the horizontal axis, with a cumulative frequency of zero, which is the point (140, 0). Provided the frequency is large enough, we can expect the cumulative frequency graph to be approximately S-shaped, and resemble the graph sketched in figure a. The cumulative frequency curve is sometimes called an ogive.
The broken horizontal axis in figure b indicates that there are some missing numbers. These numbers are omitted to save space, otherwise a large part of the Cartesian plane would have no graph drawn on it. The median is the middle of 100 students, which is the 50th person. The upper quartile (UQ) is the middle of the top half, which is the 75th person. The lower quartile (LQ) is the middle of the bottom half, which is the 25th person. The heights representing the median and the upper and the lower quartiles can be read from the graph, as accurately as possible:
Median = 160 cm The height of the 50th student
Upper quartile = 166 cm The height of the 75th student
Lower quartile = 154 cm The height of the 25th student
Interquartile range = 12 cm The upper quartile minus the lower quartile =
166 - 154
Different countries of the world have different currencies and when visiting another country it is necessary to exchange some of your money for some of theirs in order to be able to buy goods in that country. The rate of exchange of one currency for another varies from day to day, and your bank will have the latest rate.
Example 1. Ray and Jacky, who live in England, are taking an overseas trip to the United States to see some friends. They need to change their currency from pounds (<£) into dollars ($). They visit their bank and are told that one pound converts to 2.8563 dollars, that is, £\ = $2.8563. They decide to convert £ 1500 to dollars. How many dollars will they get?
Solution. To change pounds into dollars they must multiply the pounds by 2.8563, which is the conversion factor. This figure is often given to four or more decimal places. This kind of accuracy will be needed if a large amount of money is being exchanged. Write
<£1500 x 2.8563 = $4284.45 Multiply the number of pounds by the
They will get $4284.45 in exchange for <£1500.
CYCLIC QUADRILATERAL 139
Example 2. At the end of their time in the United States they have $157.00 that is unspent, and at the airport convert this back into pounds ready for use at home. How many pounds will they get if the conversion rate has not changed?
Solution. The conversion factor to change from dollars to pounds is 1/2- 8563, which is equal to 0.3501 to 4 dp. Write
$157.00 x 0.3501 = £54.9657 Multiply the number of dollars by the
They will get £54.91, to the nearest penny, in exchange for $157.
A cycle is one repetition of a periodic graph, which is a graph that repeats at regular intervals. An example of a periodic graph is a sine curve, whose equation is y = sinx. The cycle of this sine curve is highlighted in the figure and is from x = 0° to x = 360°. This portion of the curve is repeated every 360°, therefore we say the period of this cyclic curve is 360°. The period of a curve is defined using x values.
References: Amplitude, Circular Functions, Graphs.
References: Cycle, Cyclic Quadrilateral.
A quadrilateral is cyclic if its four vertices all lie on a circle. Since the four vertices all lie on the same circle, we say that the four points are concyclic points. In figure a,
140 CYCLIC QUADRILATERAL
ABCD is a cyclic quadrilateral, and the four points A, B, C, and D are coneyclic points. The following geometry theorems relate to the cyclic quadrilateral:
Theorem 1. The opposite angles of a cyclic quadrilateral are supplementary, which means they add up to 180°. In figure b,
AngleS + angle D = 180°