Edexcel Mathematics S1 Important Points and Example Questions

A mathematical model is a simplification of a real world situation.

Some advantages: quick and easy to produce, simplify a more complex situation, help improve understanding of the real world as certain variables can readily be changed, enable predictions to be made about the future, provide control.

Some disadvantages: only give a partial description of the real situation, only work for a restricted range of values.

e.g. Give two reasons for using mathematical models.

Outline the stages that are needed to create a mathematical model.

Explain briefly the role of statistical tests in the process of mathematical modelling.

Variables associated with numbers are quantitative variables.

Variables not associated with numbers are qualitative variables.

A variable that can take any value in a given range is a continuous variable.

A variable that can take only specific values in a given range is a discrete variable.

In a grouped frequency table:

Groups are more commonly known as classes.

Class boundaries are outside the classes if the classes do not overlap, if they overlap, you leave them as they are.

Mid-point-the value in the middle of the classes.

Class width-distance within classes.

Cumilative frequency-running totals of the frequencies.

Mode-value that occurs most often. Not useful when data occurs only once.

Median-middle value when the data is put in order. n observations means divide n by two. If a whole number, find the midpoint of the corresponding term and the term above. If not a whole number, round up and pick the corresponing term. Used when there are extreme values.

Mean-sum of all observations divided by the total number of observations. Gives a true measure of the data but is affected by outliers.

Combined means-total sum of observations/ total number of observations.

To find the median use interpolation e.g.:

33.5        m        36.5                 m-33.5         35-27      Class boundaries on top.

---------------------------                ---------   =   ---------     Find m.

27          35          57                36.5-33.5       57-27      Cumilative frequencies.

Coding makes larger numbers easier to work with. To find the mean of the original data, find the mean of the coded data, equate this to the coding and solve.

range= highest value-lowest value.

IQR=upper quartile-lower quartile.

Lower quartile for discrete data: n/4, if whole number, find the midpoint of the corresponding term and the term above. If not a whole number, round up and pick the corresponding term.

Lower quartile for continuous data: n/4- use interpolation.

Upper quartile for discrete data: n/4 x 3, if whole number, find the midpoint of the corresponding term and the term above. If not a whole number, round up and pick the corresponding term.

Upper quartile for continuous data: n/4 x 3-use interpolation.

When using grouped data, you don't round up when using interpolation.

Percentiles-split data into 100 parts. To calculate the xth percentile, you find the value of the x/100th term. The n% to m% percentile range = Pm-Pn. You use interpolation to calculate percentiles. e.g. 90th percentile= 90/100 x n (where n is the number of terms. Then use interpolation to find the term.

Variance is the mean of the squares minus the square of the mean:

As variation is measured in units squared you usually take the square root of the variance to get the standard deviation. The symbol σ is another symbol used for standard deviation.

Adding and subtracting numbers doesn't change the standard deviation, but multiplying or dividing the data by a number does affect the standard deviation. To find the standard deviation of the original data, find the standard deviation of the coded data and either multiply this by what you divided the data by or divide this by what you multiplied the data by. 

A stem and leaf diagram is used to order and present data given to two or three significant figures. Each number is first split into its stem and leaf. It keeps the detail of the data but can be time consuming. It enables the shape of the data to be revealed and quartiles can easily be found. Two sets can be compared by looking at back to back stem and leaf diagrams.

An outlier is an extreme value that lies outside the overall pattern of the data. An outlier is any value which is greater than the upper quartile + 1.5 x IQR, or less than the lower quartile - 1.5 x IQR. To find outliers, here are the steps:

1. Find the lower quartile. 2. Find the lower quartile. 3. Find the interquartile range. 4. Use the outlier equation to find any outliers.

Box plots can be drawn to represent important features of the data. It shows the quartiles maximum and minimum values and also any outliers. Outliers are marked on with x's. If you don't know the actual minimum and maximum figures of the data, you use the outlier boundaries you have worked out.

When comparing data remember to talk about the location and spread of the data. Write your interpretation in context to the question.

Histograms give a good picture of how data is distributed. it enables you to see a rough location, the general shape of the data and how spread out the data are. A histogram is similar to a bar chart, but there are no gaps between bars and the area of the bar is proportional to the frequency.

To calculate the height of the bar  (the frequency density) use the formula: area of bar = k x frequency, and then frequency density = frequency / class width.

Class boundaries are on the x axis, and frequency density is on the y axis.

Positive skews in histograms: <---

Negative skews in histograms: --->

Using quartiles: median - lower quartile is less than upper quartile - median (+ive)

Using quartiles: median - lower quartile is more than upper quartile - median (-ive)

Using box plots: small distance between lower quartile and median = +ive.

Using box plots: small distance between upper quartile and median = -ive.

Measures of location: mode < median < mean = positive skew (think of arrows pointing in the same direction as if you were looking at a chart.)

Measures of location: mode > median > mean = negative skew (think of arrows pointing in the same direction as if you were looking at a chart.)

3(mean-median)/standard deviation.

This gives you a value and tells you how skewed the data are. The larger the number the greater the skew. The closer the number to 0 the more symmetrical the data. A negative number means a negative skew and a positive number means a positive skew.

Venn diagrams: different options:

A' (not A):

P (A') = 1- P(A)

A U B (A or B or both):

Addition rule: P(A U B) = P(A) + P(B) - P(A n B) (A or B or both = the probability of A plus the probability of B minus the probability of A and B.

A n B (A and B):

Rearrangement of the addition rule: P(A n B) = P(A) + P(B) - P(A U B) (A and B = the probability of A plus the probability of B minus the probability of A and B or both.

P (B|A) = P(B n A) / P(A)

Multiplication rule = P(B n A) = P(B|A) x P(A)

Conditional probabilities can be worked out using tree diagrams.

P (A n B) = P(A) X P(B)

P (A' n B') = P(A') X P(B')

Multiply along branches and add between them.

P(A) = P(A n B) + P(A n B')

Always think through what you need to work out. Do you already know some of the probabilities you need? Find sensible routes to the answer.

When two events have no outcomes in common, they are mutually exclusive. The addition rule applied to mutually exclusive events is: P (A U B) = P(A) + P(B)

When one event has no effect on the other, they are independent. The multiplication rule applied to independent events is: P(A n B) = P(A) x P(B)

Remember: independent with reference to Venn diagrams means overlapping, and mutually exclusive means not overlapping, but separate.

A and B are independent if P(A|B) = P(A) or P(B|A) = P(B) or P(A n B) = P(A) x P(B).

A and B are mutually exclusive if P(A n B) = 0. 

If both variables increase together, they are positively correlated.

If one variable increases as the other decreases, they are said to be negatively correlated.

if no straight line (linear) pattern can be seen there is said to be no correlation. 

When you are asked to 'interpret the correlation', you put what the correlation means in context of the question.

If quadrants are drawn on a scatter diagram, like this: 2|1 on the top and 4|3 on the bottom, for a positive correlation, most points lie in the 1st and 3rd quadrants. For a negative correlation, most points lie in the 2nd and 4th quadrants. For no correlation the points lie fairly equally in all 4 quadrants.

In correlation the variance is written as (the sum of x - the mean of x) squared, or as S**, and Syy but with ys in place of xs.

Co-variance is (the sum of x - the mean of x) x (the sum of y - the mean of y) divided by the number of terms. Or Sxy =  (the sum of x - the mean of x) x (the sum of y - the mean of y)

The formulae for calculating S**, Syy and Sxy are given in the formula booklet.

r is the product moment correlation coeffiecient. which is Sxy / the square root of S** x Syy

R is a measure of linear relationship:

r=1 is a perfect positive linear correlation

r=-1 is a perfect negative correlation.

r=0 is no linear correlation.

R is not affected by coding if you are multiplying, the same value is always obtained, but if adding or subtracting, it is affected.

An independent variable is one that is set independently of the other variable. It is plotted along the x axis.

A dependent variable is one whose values are determined by the values of the independent variable. It is plotted along the y axis.

For each point on a scattergraph you can express y in terms of x as y = (a + bx) +e, where e is the vertical distance from the line of best fit.

The equation of the regression line of y on x is y = a+bx where b = Sxy / S** and a = the mean of y - b multiplied by the mean of x.

Coding is sometimes used to simplify the calculations. To turn a coded regression line into an actual regression line you substitute the codes into the answer.

Interpolation is when you estimate the value of a dependent variable within the range of the data.

Extrapolation is when you estimate a value outside the range of the data. Values estimated by extrapolation can be unreliable.

A variable is represented by a symbol, and can take on any of a specified set of values.

Capital letters such as X are used for the random variable and a small letter such as x is used for a particular value of the random variable X.

The probability that X is equal to a particular value x is written as P(X=x) or p(x). 

To specify a discrete random variable completely, you need to know its set of possible values and the probability with which it takes each one.

You can draw up a table to show the probability of each outcome of an experiment. This is called a probability distribution.

You can also specify a discrete random variable as a function. For example, the random variable of throwing a fair die is P(X=x) = 1/6 for x = 1,2,3,4,5,6. This is known as a probability function.

For a discrete random variable the sum of all the probabilities must add up to one.

If a particular value of X is x, the probability that X is less than or equal to x, is written as F(x). F(x) is found by adding together all the probabilities for those outcomes that are equal to or less than x. This is written as: F(x) = P(X≤x)

Like a probability distribution, cumulative distribution function can be written as a table.

The expected value of x is defined as: X = E(X)= sum of x multiplied by P(X=X) = the sum of xp(x) (which is the same as P(X=x).

Var(X)= E(X²) - (E(X))²

E(aX+b)=aE(X)+b

Var(aX+b) = a²Var(X).

A variance is a measure of spread relative to the mean so adding a constant value to all the values of X will not affect this measure of spread hence the 'b' does not not change the variance/ The 'a²' is understandable if you remember that: Var(X)= E(X²) - (E(X))² so if each value of X is multiplied by the value of a the variance will be multiplied by a².

e.g.

(42x2)+(47x4)+(52x5)+(57x8)+(62x5)+(67x4)+(72x2)= E(X)

(42²x2)+(47²x4)+(52²x5)+(57²x8)+(62²x5)+(67²x4)+(72²x2)= E(X²)

The probability distribution for the score on a fair die is a discrete uniform distribution over the set of values because the values are discrete and it is called uniform because all of the probabilities are the same. Conditions for a discrete uniform distribution: a discrete random variable X us defined over a set of distinct values. Each value is equally likely.

In many cases X is defined over the set, in such cases the mean and variance are given by the following:

E(X) = (n+1)/2

Var(X) = ((n+1)(n-1))/12

For a normal distribution probabilities are given by areas under the curve in a similar way to which frequencies are found by areas under a histogram or frequency polygon.

Tables are provided to help us calculate probabilities for the standard normal distribution, Z.

The standard normal variable is usually denoted by Z and has a mean of 0 and a standard deviation of 1. The usual way of writing this is: P(Z>x) =1 - P(Z<x) since the total area under the curve =1. For continuous distribution like the normal, there is no difference between P(Z<z) and P(Z≤z).

You use the tables to work out values that are less than Z, e.g. P(Z<1.52), you shade in on the graph where 1.52 is less, and then look in the appendix for the value, which in this case is 0.9357.

You can also use tables to work out values that are greater than Z, e.g. P(Z>2.60) =1-0.9953 (value found in the Appendix.)

Tables do not give values of z < 0, so use symmetry and take the value you get in the appendix away from 1, as the required values will still be the same.

The table of percentage points of the normal distribution gives the value of z for various values of p=P(Z>z). Whenever possible this table should be used to find z given a value for p = P(Z>z).

If p(Z<a) is greater than 0.5, then a will be > 0.
If p(Z<a) is less than 0.5, then a is less than 0.
If p(Z>a) is less than 0.5, then a will be > 0.
If p(Z>a) is more than 0.5, then a will be < 0.

The random variable X having a mean of μ and a standard deviation of σ (or variance σ²) can be written as X~N(μ, σ²).

X~N(μ, σ²) can be transformed into the random variable Z~N(0,1²) by the formula: Z= (X-μ)/σ. 

 You round your z value to 2sf to use the nearest value in the tables.

If X~N(μ, σ²) and P(X>a) = α where α is a probability, you write this statement as P(Z>(a-μ/σ) = α.

Sometimes neither μ or σ is given, so you need to solve by simultaneous equations.