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). 

Sin, Cos and Tan

A right-angled triangle is a triangle in which one of the angles is a right-angle. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.

In any right angled triangle, for any angle:

The sine of the angle = the length of the opposite side
                                   the length of the hypotenuse

The cosine of the angle = the length of the adjacent side
                                      the length of the hypotenuse

The tangent of the angle = the length of the opposite side
                                      the length of the adjacent side

So in shorthand notation:
sin = o/h   cos = a/h   tan = o/a
Often remembered by: soh cah toa

Example

Find the length of side x in the diagram below:

The angle is 60 degrees. We are given the hypotenuse and need to find the adjacent side. This formula which connects these three is:
cos(angle) = adjacent / hypotenuse
therefore, cos60 = x / 13
therefore, x = 13 × cos60 = 6.5
therefore the length of side x is 6.5cm.

This video will explain how the formulas work

The Graphs of Sin, Cos and Tan - (HIGHER TIER)The following graphs show the value of sinø, cosø and tanø against ø (ø represents an angle). From the sin graph we can see that sinø = 0 when ø = 0 degrees, 180 degrees and 360 degrees.

Note that the graph of tan has asymptotes (lines which the graph gets close to, but never crosses). These are the red lines (they aren't actually part of the graph).

Also notice that the graphs of sin, cos and tan are periodic. This means that they repeat themselves. Therefore sin(ø) = sin(360 + ø), for example.

Notice also the symmetry of the graphs. For example, cos is symmetrical in the y-axis, which means that cosø = cos(-ø). So, for example, cos(30) = cos(-30).
Also, sin x = sin (180 - x) because of the symmetry of sin in the line ø = 90.

Equation Of A Straight Line

Equations of straight lines are in the form y = mx + c (m and c are numbers). m is the gradient of the line and c is the y-intercept (where the graph crosses the y-axis). 

NB1: If you are given the equation of a straight-line and there is a number before the 'y', divide everything by this number to get y by itself, so that you can see what m and c are.
NB2: Parallel lines have equal gradients.

The above graph has equation y = (4/3)x - 2 (which is the same as 3y + 6 = 4x).
Gradient = change in y / change in x = 4 / 3
It cuts the y-axis at -2, and this is the constant in the equation.

Travel Graphs

Speed, Distance and Time

The following is a basic but important formula which applies when speed is constant (in other words the speed doesn't change):

Remember, when using any formula, the units must all be consistent. For example speed could be measured in m/s, distance in metres and time in seconds.

If speed does change, the average (mean) speed can be calculated:
Average speed = total distance travelled
                             total time taken

Units

In calculations, units must be consistent, so if the units in the question are not all the same (e.g. m/s, m and s or km/h, km and h), change the units before starting, as above.

The following is an example of how to change the units:

Example

Change 15km/h into m/s.
15km/h = 15/60 km/min               (1)
= 15/3600 km/s = 1/240 km/s      (2)
= 1000/240 m/s = 4.167 m/s        (3)

In line (1), we divide by 60 because there are 60 minutes in an hour. Often people have problems working out whether they need to divide or multiply by a certain number to change the units. If you think about it, in 1 minute, the object is going to travel less distance than in an hour. So we divide by 60, not multiply to get a smaller number.

Example

If a car travels at a speed of 10m/s for 3 minutes, how far will it travel?
Firstly, change the 3 minutes into 180 seconds, so that the units are consistent. Now rearrange the first equation to get distance = speed × time.
Therefore distance travelled = 10 × 180 = 1800m = 1.8km

Velocity and Acceleration

Velocity is the speed of a particle and its direction of motion (therefore velocity is a vector quantity, whereas speed is a scalar quantity).

When the velocity (speed) of a moving object is increasing we say that the object is accelerating. If the velocity decreases it is said to be decelerating. Acceleration is therefore the rate of change of velocity (change in velocity / time) and is measured in m/s².

Example

A car starts from rest and within 10 seconds is travelling at 10m/s. What is its acceleration?

Acceleration	=	change in velocity	=	10	=	1m/s²
		time		10

Distance-Time Graphs

These have the distance from a certain point on the vertical axis and the time on the horizontal axis. The velocity can be calculated by finding the gradient of the graph. If the graph is curved, this can be done by drawing a chord and finding its gradient (this will give average velocity) or by finding the gradient of a tangent to the graph (this will give the velocity at the instant where the tangent is drawn).

Velocity-Time Graphs/ Speed-Time Graphs

A velocity-time graph has the velocity or speed of an object on the vertical axis and time on the horizontal axis. The distance travelled can be calculated by finding the area under a velocity-time graph. If the graph is curved, there are a number of ways of estimating the area (see trapezium rule below). Acceleration is the gradient of a velocity-time graph and on curves can be calculated using chords or tangents, as above.

The distance travelled is the area under the graph.
The acceleration and deceleration can be found by finding the gradient of the lines.

On travel graphs, time always goes on the horizontal axis (because it is the independent variable).

Trapezium Rule

This is a useful method of estimating the area under a graph. You often need to find the area under a velocity-time graph since this is the distance travelled.

Area under a curved graph = ½ × d × (first + last + 2(sum of rest))

d is the distance between the values from where you will take your readings. In the above example, d = 1. Every 1 unit on the horizontal axis, we draw a line to the graph and across to the y axis.
'first' refers to the first value on the vertical axis, which is about 4 here.
'last' refers to the last value, which is about 5 (green line).]
'sum of rest' refers to the sum of the values on the vertical axis  where the yellow lines meet it.
Therefore area is roughly: ½ × 1 × (4 + 5 + 2(8 + 8.8 + 10.1 + 10.8 + 11.9 + 12 + 12.7 + 12.9 + 13 + 13.2 + 13.4))
= ½ × (9 + 2(126.8))
= ½ × 262.6
= 131.3 units²

Algebraic Fractions

Algebraic Fractions

Algebraic fractions are simply fractions with algebraic expressions on the top and/or bottom.

When adding or subtracting algebraic fractions, the first thing to do is to put them onto a common denominator (by cross multiplying).

e.g.      1      +      4    
         (x + 1)   (x + 6)

= 1(x + 6) + 4(x + 1)
       (x + 1)(x + 6)

= x + 6 + 4x + 4
   (x + 1)(x + 6)

=    5x + 10    
  (x + 1)(x + 6)

Solving equations

When solving equations containing algebraic fractions, first multiply both sides by a number/expression which removes the fractions.

Example

Solve    10    -2   =  1
         (x + 3)     x

multiply both sides by x(x + 3):
∴ 10x(x + 3) - 2x(x + 3) = x(x + 3)
     (x + 3)            x

∴ 10x - 2(x + 3) = x² + 3x      [after cancelling]
∴ 10x - 2x - 6 = x² + 3x
∴ x² - 5x + 6 = 0
∴ (x - 3)(x - 2) = 0
∴ either x = 3 or x =2

Inequalities

After studying this section, you will be able to:

• solve inequalities with one variable
  Solve Inequalities with two variable

NOTE:
When working with a double inequalities start by solving the two inequalities. An integer is any whole number, positive negative or zero.
When solving inequalities do not forget that multiplying or dividing by a negative number reverses the inequality sign:
−x > 3, becomes x < −3 (multiplying by −1). -Inequalities in two variables
NOTE:Remember an equation in the form y = 2x – 1 has gradient 2 and y-intercept (where it crosses the y-axis) of –1, 
in other words y = (gradient) x + (y -intercept)