# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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The relationships between these units of area are as follows:

100 mm2 = 1 cm2 1,000,000 mm2 = 1 m2 10,000 cm2 = 1 m2

10,000 m2 = lha

1,000,000 m2 = 1km2 100 ha = 1km2

Large numbers may be written in index form. For example 10,000 = 104. Another unit of area, in the Imperial system of units, is the acre. One hectare is roughly

2.5 acres.

Methods for finding the areas of other shapes, such as the parallelogram, the trapezium, the triangle, etc., are given in the relevant entries.

References: Acre, Circle, Index, Metric Units, Survey, Radius, Rectangle.

40 ARITHMETIC MEAN

ARITHMETIC

This is the branch of mathematics that uses numbers in processes such as addition, subtraction, multiplication, division, and taking square roots.

ARITHMETIC MEAN

Arithmetic mean is often abbreviated to just the one word mean. It is one of three averages which are used in statistics, the other two being median and mode. The symbol for mean is x. Mean is often incorrectly referred to as “average.” The mean is a single quantity used to represent a set of quantities or a group. The method of finding the mean of a set of quantities is to add up all the quantities and divide this total sum by the number of quantities.

Example. Sarah has just started at the local child care center. She is in a small group of six children, whose ages are given in the table. Find the mean age of the children.

Name Sarah Luke Samuel Simon Francis

Age in years 2 3 4 1 4

Solution. Write

2+3 + 4+ 1 + 4

14

Sum the ages and then divide by the number of children

x = 2.8 Using calculator.

The mean age of the children is 2.8 years.

Sometimes the mean is calculated and then an extra quantity is added to the group as illustrated in the following example.

Example. The mean mass of a rowing eight at Plato College is 87.5 kilograms (abbreviated kg). The mass of the cox is 49.8 kg. What is the mean mass of the whole crew?

Solution. Write

The eight rowers weigh 8 x 87.5 = 700kg The whole crew weighs 700 + 49.8 = 749.8 kg Mass of original crew + cox

749.8

Mean mass of the crew =------------- Total mass + 9 crewmembers

9

The mean mass of the whole crew is 83.3 kg (to ldp).

ARITHMETIC MEAN 41

In the following example the mean is calculated when the data are contained in a frequency table.

Example. The frequency table shows the marks, out of 10, gained by 29 students who took a mathematics test. Find the mean.

Mark 01 23456789 10

Frequency 2133465221 0

Solution. The frequency row tells us, for example, that 6 students scored a mark of 5 out of 10. In order to find the sum of all the marks, we multiply the marks by the frequencies. Then, to obtain the mean, we divide this sum by the total number of students, which is 29. The symbol for mean is x:

^ _ (2 x 0) + (1 x 1) + (3 x 2) + (3 x 3) + (4 x 4) + (6 x 5) + (5 x 6) + (2 x 7) + (2 x 8) + (1 x9) + (0xl0) — 29

131

x = 4.5 (to 1 dp)

The mean mark in the test is 4.5.

Data that have a large range can be handled more easily if the data are grouped into class intervals. For example, if the students’ marks in a test have a large range, say, out of 100 instead of out of 10, the marks can grouped into class intervals of, say, 10 marks. When data are grouped into class intervals the individual score of each student is lost, but the mean can still be calculated. The middle of each class interval is used to represent the mark in the test obtained by the students, as illustrated in the example below.

Example. The grouped frequency table shows the percentage marks gained by 100 students who took a mathematics examination. Find the mean mark.

Mark % 0-9 10-19 20-29 30-39 40-49

Frequency 1 3 12 15 20

Mark % 50-59 60-69 70-79 80-89 90-100

Frequency 23 12 8 4 2

Solution. For more complicated statistical calculations, such as this example, it is convenient to use a vertical table. It is necessary to incorporate two extra columns, one for the middle mark and one for the product of the middle mark and the frequency. To obtain the sum of the marks of all the students, it is necessary to multiply the middle marks in each class interval by the corresponding frequencies, since the individual marks are not known.

42

ARROW GRAPH

Mark (x) Frequency (f ) Middle mark (m) f x m

0-9 1 4.5 4.5

10-19 3 14.5 43.5

20-29 12 24.5 294

30-39 15 34.5 517.5

40-49 20 44.5 890

50-59 23 54.5 1253.5

60-69 12 64.5 774

70-79 8 74.5 596

80-89 4 84.5 338

90-100 2 94.5 189

Totals 100 4900

The formula for the mean is

Sum of f X ti • 1 1 11 • , - Tfxm

x =------------------------ Which may be abbreviated to x = •>

Sum of / where £ means “the sum of”

4900 x =--------

100

The mean score in the examination is 49%.

References: Class Interval, Frequency Table, Median of a Set of Data, Mode.

ARRANGEMENTS

These are also called permutations. Reference: Combinations.

ARROW GRAPH

This is sometimes called an arrow diagram. Suppose there is a relation “is the capital of” between a set of cities and a set of countries. One pairing off could be “London is the capital of England,” and on the graph we would draw an arrow from London to England to show that this relation exists between them. Figure a is an arrow graph showing four possible pairings for the relation “is the capital of.” If the directions of the arrows were reversed, the result would be the inverse relation: “has as its capital”

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