Proportion

Proportion

If a "is proportional" to b (which is the same as 'a is in direct proportion with b') then as b increases, a increases. In fact, there is a constant number k with a = kb. We write a ∝ b if a is proportional to b.

The value of k will be the same for all values of a and b and so it can be found by substituting in values for a and b.

Example

If a ∝ b, and b = 10 when a = 5, find an equation connecting a and b.

a = kb (1)

Substitute the values of 5 and 10 into the equation to find k:

5 = 10k

so k = 1/2

substitute this into (1)

a = ½b

In this example we might then be asked to find the value of a when b = 2. Now that we have a formula connecting a and b (a = ½ b) we can subsitute b=2 to get a = 1.

Similarly, if m is proportional to n2, then m = kn2 for some constant number k.

If x and y are in direct proportion then the graph of y against x will be a straight line.

Inverse Proportion (HIGHER TIER)

If a and b are inversely proportionally to one another,

a ∝ 1/b

therefore a = k/b

In these examples, k is known as the constant of variation.

Example

If b is inversely proportional to the square of a, and when a = 3, b = 1, find the constant of variation.

b = k/a2 when a = 3,

b = 1

therefore 1 = k/32

therefore k = 9

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Standard Form

Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative.

The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Example

Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10¹³

It’s 10¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19

Example

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10^-6

It’s 10^-6 because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.

Manipulation in Standard Form

This is best explained with an example:

Example

The number p written in standard form is 8 × 10⁵

The number q written in standard form is 5 × 10^-2

Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:

8 × 5 × 10⁵ × 10^-2

= 40 × 10³ (Remember 10⁵ × 10^-2 = 10³)

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.

= 4 × 10⁴

Calculate p ÷ q.

Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits. (8 ÷ 5) × (10⁵ ÷ 10^-2) = 1.6 × 10⁷

Sampling

Sampling basically means selecting people/objects from a population in order to test the population for something. For example, we might want to find out how people are going to vote at the next election. Obviously we can't ask everyone in the country, so we ask a sample. 

When considering a particular population it is usually advisable to choose a sample in such a way that everyone is represented. This is not easy and requires careful thought about sample size and composition. Often questionnaires are devised to identify the required information. These need to be idiot proof, so questions need to cover all alternatives and give little scope for variation.

Example question

A bus company attempted to estimate the number of people who travel on local buses in a certain town. They telephoned 100 people in the town one evening and asked 'Have you travelled by bus in the last week?'

Nineteen people said 'Yes'. The bus company concluded that 19% of the town's population travel on local buses.

Give 3 criticisms of this method of estimation.

In answering this question, there are no 3 correct answers. As long as what you say is plausible and sensible, you should get the marks. For example, you might say:

• 100 people in a large town is not a large enough proportion of the population to give a good sample.
• People who travel on local buses once a fortnight may have said no to the question. They nevertheless travel on local buses.
• On the evening that the sample was carried out, anybody travelling by bus would be out.

Averages

Mean

There are three main types of average:

• mean -  The mean is what most people mean when they say 'average'. It is found by adding up all of the numbers you have to find the mean of, and dividing by the number of numbers. So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .
• mode - The mode is the number in a set of numbers which occurs the most. So the modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5, because there are more 5s than any other number.
• median - The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6

Histograms

Histograms are similar to bar charts apart from the consideration of areas. In a bar chart, all of the bars are the same width and the only thing that matters is the height of the bar. In a histogram, the area is the important thing.

Example

Draw a histogram for the following information.

Height (feet):	Frequency	Relative Frequency
0-2	0	0
2-4	1	1
4-5	4	8
5-6	8	16
6-8	2	2

(Ignore relative frequency for now). It is difficult to draw a bar chart for this information, because the class divisions for the height are not the same. The height is grouped 0-2, 2-4 etc, but not all of the groups are the same size. For example the 4-5 group is smaller than the 0-2 group.

When drawing a histogram, the y-axis is labelled 'relative frequency' or 'frequency density'. You must work out the relative frequency before you can draw a histogram. To do this, first you must choose a standard width of the groups. Some of the heights are grouped into 2s (0-2, 2-4, 6-8) and some into 1s (4-5, 5-6). Most are 2s, so we shall call the standard width 2. To make the areas match, we must double the values for frequency which have a class division of 1 (since 1 is half of 2). Therefore the figures in the 4-5 and the 5-6 columns must be doubled. If any of the class divisions were 4 (for example if there was a 8-12 group), these figures would be halved. This is because the area of this 'bar' will be twice the standard width of 2 unless we half the frequency.

If you are having problems working out the height of each of the bars, you can use the formula

Area of bar = frequency x standard width

Standard Deviation

The standard deviation measures the spread of the data about the mean value. It is useful in comparing sets of data which may have the same mean but a different range. For example, the mean of the following two is the same: 15, 15, 15, 14, 16 and 2, 7, 14, 22, 30. However, the second is clearly more spread out. If a set has a low standard deviation, the values are not spread out too much.

Just like when working out the mean, the method is different if the data is given to you in groups.

Non-Grouped Data

Non-grouped data is just a list of values. The standard deviation is given by the formula:

s means 'standard deviation'.
S means 'the sum of'.
 means 'the mean'

Example

Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14
First work out the mean: 10.222
Now, subtract the mean individually from each of the numbers given and square the result. This is equivalent to the (x - )² step. x refers to the values given in the question.

x	4	9	11	12	17	5	8	12	14
(x -  )2	38.7	1.49	0.60	3.16	45.9	27.3	4.94	3.16	14.3

Now add up these results (this is the 'sigma' in the formula): 139.55
Divide by n. n is the number of values, so in this case is 9. This gives us: 15.51
And finally, square root this: 3.94

The standard deviation can usually be calculated much more easily with a calculator and this may be acceptable in some exams. On my calculator, you go into the standard deviation mode (mode '.'). Then type in the first value, press 'data', type in the second value, press 'data'. Do this until you have typed in all the values, then press the standard deviation button (it will probably have a lower case sigma on it). Check your calculator's manual to see how to calculate it on yours.

NB: If you have a set of numbers (e.g. 1, 5, 2, 7, 3, 5 and 3), if each number is increased by the same amount (e.g. to 3, 7, 4, 9, 5, 7 and 5), the standard deviation will be the same and the mean will have increased by the amount each of the numbers were increased by (2 in this case). This is because the standard deviation measures the spread of the data. Increasing each of the numbers by 2 does not make the numbers any more spread out, it just shifts them all along.

Grouped Data

When dealing with grouped data, such as the following:

x	f
4	9
5	14
6	22
7	11
8	17

the formula for standard deviation becomes:

Try working out the standard deviation of the above data. You should get an answer of 1.32 .

You may be given the data in the form of groups, such as:

Number	Frequency
3.5 - 4.5	9
4.5 - 5.5	14
5.5 - 6.5	22
6.5 - 7.5	11
7.5 - 8.5	17

In such a circumstance, x is the midpoint of groups.

Sine and Cosine Rule

The Sine Rule

 The sine rule is an important rule relating the sides and angles of any triangle (it doesn't have to be right-angled!): If a, b and c are the lengths of the sides opposite the angles A, B and C in a triangle, then:  a   =   b  =  c  sinA    sinB  sinC If you wanted to find an angle, you can write this as:
 sinA =  sinB  = sinC
 a         b           c  

The Cosine Rule

 This also works in any triangle: c² = a² + b² - 2abcosC which can also be written as: a² = b² + c² - 2bccosA -

The area of a triangle

 The area of any triangle is ½ absinC (using the above notation). This formula is useful if you don't know the height of a triangle (since you need to know the height for ½ base × height).