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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
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
This is the branch of mathematics that uses numbers in processes such as addition, subtraction, multiplication, division, and taking square roots.
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
2+3 + 4+ 1 + 4
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?
The eight rowers weigh 8 x 87.5 = 700kg The whole crew weighs 700 + 49.8 = 749.8 kg Mass of original crew + cox
Mean mass of the crew =------------- Total mass + 9 crewmembers
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
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.
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 =--------
The mean score in the examination is 49%.
References: Class Interval, Frequency Table, Median of a Set of Data, Mode.
These are also called permutations. Reference: Combinations.
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”