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Further
Maths Notes
Common Mistakes
- Read the bold words in the exam!
- Always check data entry
- Remember to interpret data with the multipliers
specified (e.g. in thousands)
- Write equations in terms of variables
- Watch for a+bx and ax+b forms
- Remember to seasonalise data after predictions
- Ensure calculator is in degrees
- Always write true bearings as 90° T
- Always check the feasible region with a point
- Never delete an answer
- Least squares gradient has Sy on top
- When crashing, look for vertices common to multiple
paths
2007 Further Mathematics Units 3 and 4 Revision 2
○ Frequency tables ○ Dot plots ○ Line Plots ○ Stem-and-leaf plots
The main techniques used to organise data are:
Data is grouped before it is shown in any of the techniques above. There needs to be more than five and no more than fifteen groups. ○ Title ○ Label axes ○ Label important points ○ Use the key
When displaying data, remember:
Types of Data
Core - Univariate Data Page 1
The following summary statistics are usually used together: ○ Mean and standard deviation ○ Median and IQR ○ Mode and range Mode The mode is the most commonly occurring value. There can be two or more modes. The mode is not usually an accurate measure of centre, and is only used when there is a high number of scores. Median The median is the middle value. 50% of data lies either side of the median. If there are two middle values, it is their average. To calculate the median of grouped data, use a cumulative frequency table. The median does not take all values into account, and is therefore not affected by extreme values. The median is usually used for skewed distributions. It may not be equal to one of the values in the data set. Mean The mean is the average value. For ungrouped data: For grouped data: The mean is not necessarily equal to a value in the data set. The mean takes all data into account, and is therefore affected by extreme values. Range The range is the difference between the smallest value and the largest. It is influenced by outliers. For grouped data, the range is the difference between the highest and lowest groups with values. Interquartile range The IQR is the range between which 50% of values lie. It is not affected by outliers. It is best obtained using a calculator. Statistic Term number Q Median Q
Analysing data
Core - Univariate Data Page 3
The variance is the square of the standard deviation. Both values show the distribution of data around the mean.
- Both standard deviation and variance are always positive numbers.
- Both standard deviation and variance are affected by outliers.
- To estimate the standard deviation of a dataset, the following formula is used: This formula accounts for 95% of data being within two standard deviations of the mean.
The 68- 95 - 99.7 Rule
68% of values lie within one standard deviation of the mean. 95% of values lie within two standard deviations of the mean. 99.7% of values lie within three standard deviations of the mean.
Z-Scores
Z-scores are used to compare values between different sets of data. For example, the height of a girl can be compared with the height of a boy by comparing them with their gender and age. The formula for z-scores is: The following formulas can also be used to determine information: The z-scores then function like normal data distributions with a mean of zero and a standard deviation of one. The 68- 95 - 99.7 rule can then be applied to the standardised data.
Standard Deviation and Variance
Core - Univariate Data Page 4
This diagram shows values of Pearson's
Product-Moment Correlation Coefficient and
what strength relationship they represent.
Positive values always represent relationships with a positive gradient, and negative values always represent relationships with a negative gradient.
The Coefficient of Determination is equal to r². It describes the influence the independent variable
has on the dependent variable. It is usually expressed as a percentage. The standard analysis is:
The coefficient of determination calculated to be [ r² ] shows that [ r² %] of the variation in [dependent variable] can be explained by variation in [independent variable]. The other [(100- r² )]% of variation in [dependent variable] can be explained by other factors or influences. ○ It is designed for linear data only ○ It should be used with caution if outliers are present Note that:
Correlation
Core - Bivariate Data Page 6
Line of best fit
To calculate a line of best fit, first draw a line through the data so that approximately
half the data points are on each side. Then, find two points and use the following
equations to calculate the equation of the line:
When calculating the equation of the line of best fit, it is sometimes possible to estimate the y - intercept, however before doing so ensure that both axes start at zero. In this graph, it is easy to estimate the y-intercept to be 60, however the axes are broken so that would be incorrect.
Three Median Regression
The three-median regression line is formed as follows:
1. Plot the data on a scatterplot
2. Divide the points symmetrically into three groups
3. Find the median point of each group by finding the median of the x and y values
Use the slope of the outside points to draw a line that is one third of the way towards the
middle point.
To calculate the equation of the three-median regression line, use the following formulae:
Least Squares Regression
The least squares regression line is formed using the following formulae:
The Three-Median Regression line is usually used for data with outliers , and the Least Squares Regression line is usually used for data without outliers. Regressions Core - Bivariate Data Page 7
Time series data is simply data with a timeframe as the independent variable.
There are four ways in which it can be described:
1. Secular Trend
Data displays a trend (or secular trend) when a consistent increase or decrease can be
seen in the data over a significant period of time. A trend line can be fitted to such data.
2. Seasonality
Seasonal data changes or fluctuates at given intervals, with given intensities. For
example, sales of warm drinks might fluctuate in winter. This data can be deseasonalised.
3. Cyclic
Cyclic data shows fluctuations, but not at consistent intervals, amplitudes or seasons. This
includes data such as stock prices.
4. Random
Random data shows no pattern. All fluctuations occur by chance and cannot be
predicted.
Summarising Time Series Data Core - Bivariate Data Page 9
Time series data can be smoothed in two ways - moving average smoothing and median smoothing. Moving Average Smoothing Moving average smoothing is simple. Each smoothed data point is equal to the average of a specified number of points surrounding it. For example, if a five point moving average is chosen, then each consecutive group of five points is averaged to give the points in the smoothed data set. Median Smoothing Median smoothing is very similar to moving average smoothing, however the median of the points is taken instead of the average. This method is generally preferred to moving average smoothing when outliers are present.
The number of points to smooth to is generally equal to the number of data points in a
time cycle. For example, five points would be used for data from each working week,
and twelve points would be used for monthly data.
Deseasonalising Data A procedure exists for deseasonalising data. It involves the following formulae: Seasonal indices can also be used to comment on the relationship between seasons, for example: ○ A seasonal index of 1.3 means that the season is 30% above the average of the seasons ○ A seasonal index of 1.0 means that the season is equal to the average of the seasons ○ A seasonal index of 0.8 means that the season is 20% below the average of the seasons The sum of the seasonal indices is always equal to the number of seasons. Smoothing Time Series Data Core - Bivariate Data Page 10
Size of each interior angle: Size of each exterior angle: In a regular polygon, all sides and angles are equal.
The sum of the interior angles
of an n-sided polygon is:
The sum of the exterior angles
of any polygon is:
s= 360°
Right Angled Triangles The Pythagorean Theorem states that: Note that c always refers to the hypotenuse, however exams may refer to the sides differently. Regular Polygons Geometry and Trigonometry Page 12
Two figures are similar if they are the same shape and
the corresponding dimensions are in the same ratio.
Two triangles are similar if the angles are equal.
The sides of one similar shape are all proportional to the sides of the other, so ratios and fractions can be used to calculate the unknown lengths. Example:
Converting Units
Ratios between areas and volumes can be calculated from the length ratio. The following formulae are used: Length : Area : Volume : Ratios are used when converting units. Remember that when area and volume is converted, the ratios are squared and cubed respectively. Area: Length: Volume: Volume: Area: Length:
Ratios given in questions, unless otherwise
stated, are always length ratios.
Remember, the number of dimensions being scaled
matches the number of dimensions in the ratio.
Similarity and Ratios
Geometry and Trigonometry Page 13
The following table shows formulae used to calculate the volume and total surface area of simple shapes: Name Diagram Volume Total Surface Area Cuboid Sphere Cylinder Right square pyramid Cone s Note that for rectangular pyramids, the following formula can be used to determine the height: Volume and TSA Geometry and Trigonometry Page 15
Right Angled Triangles The Sine Rule (all triangles)
The sine rule is used with two sides
and two corresponding angles.
The Cosine Rule (all triangles)
The cosine rule is used with three
sides and one angle.
Trigonometry Geometry and Trigonometry Page 16
Degenerate graph Complete graph Connected graph Disconnected graph Simple graph Non-simple graphs (^) Subgraph Bipartite graph Planar Graphs and Euler's Formula A planar graph is one where edges do not cross. Some graphs can be redrawn to be planar, others not. Euler's formula is used to confirm whether graphs are planar or not. e Edges v Vertices f Regions or faces Note: the area outside the graph is counted as one region. Hint: if v is less than or equal to four, then the graph is planar. v 4 e 8 f 6
Original graph Redrawn graph^ Euler's formula Matrix Representation Networks can be represented using adjacency matrices. The numbers on the leading diagonal represent loops, and all undirected graphs are symmetrical about the leading diagonal. Graph type Number of edges with n vertices Complete Connected Loops A loop in an undirected network adds two to the degree of a vertex, and adds one to the leading diagonal of a matrix. For example: A B C D Node A has degree 3. Types of Networks Networks and Decision Mathematics Page 18
Paths A path is a sequence of steps between adjacent nodes. In this graph, B-D-A-C and C-A-B-D are two of the paths which could exist. B-C-D-A could not exist because B is not adjacent to C. Eulerian paths and circuits Eulerian paths and circuits pass each edge only once. Eulerian paths exist where there are two or less vertices with odd degree. Eulerian circuits exist when all vertices have even degree. On the graph above, D-A-C-D-B-A represents an Eulerian path, and no Eulerian circuits exist. Hamiltonian paths and circuits Hamiltonian paths and circuits are those which pass through each vertex only once. Examples in the above graph include B-A-D-C and D-C-A-B-D. Circuits A circuit is a sequence of steps between adjacent nodes which starts and finishes at the same vertex. In this graph, B-D-C-A-B and A-D-B-A-D-C-A are two of the circuits which could exist.
Trees
The nearest neighbour algorithm in which the longest edge, which doesn't disconnect the graph, is removed. This is repeated until no more edges can be removed.
Prim's algorithm which involves choosing random vertex as a starting graph and constantly building to it by adding the shortest edges which will connect it to another node.
A tree is a connected, simple graph with no circuits. A spanning tree is a subgraph of a connected graph which contains all the vertices of the original graph. The weight of a spanning tree is the combined weight of all its edges, and there are two ways in which the minimum- weight spanning tree can be found: The shortest path The shortest path algorithm (or Dijkstra's Algorithm) is used to find the shortest path between two nodes. It involves working from start to finish, labeling each vertex with the shortest path between it and the starting node. The Chinese Postman Problem The Chinese Postman Problem involves finding the shortest circuit (route) along a network (town) which covers all edges (roads) and starts and finishes at the same spot (the Post Office). It must be an Eulerian circuit if possible, otherwise the routes between the odd-degree vertices must be travelled twice. The Travelling Salesman Problem The Travelling Salesman Problem (TSP) involves finding the shortest circuit (route) which covers all vertices (houses) in a network (town) in the minimal distance. It must be a Hamiltonian circuit if possible, but there is no systematic way of solving this problem.
Paths, Circuits and Trees
Networks and Decision Mathematics Page 19
