S1 Note Sample

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For use only in [the name of your school] January 2010

S1 Note

S1 Notes

(Edexcel)

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10

Definitions for S1Statistical Experiment

A test/investigation/process adopted for collecting data to provide evidence for or against a

hypothesis.

Explain briefly why mathematical models can help to improve our understanding of realworld problems

Simplifies a real world problem; enables us to gain a quicker / cheaper understanding of a real world

problem

Advantage and disadvantage of statistical model

Advantage : cheaper and quicker

Disadvantage : not fully accurate

Statistical models can be used to describe real world problems. Explain the process involved

in the formulation of a statistical model.

Observe real-world problem Devise a statistical model and collect data (Experimental) data collected Model used to make predictions Compare and observe against expected outcomes and test model; Statistical concepts are used to test how well the model describes the real-world problem Refine model if necessary.

A sample space

A list of all possible outcomes of an experiment

Event

Sub-set of possible outcomes of an experiment.

Normal Distribution

Bell shaped curve symmetrical about mean; mean = mode = median 95% of data lies within 2 standard deviations of mean 68.3% between one standard deviation of mean

2 conditions for skewnessPositive skew if and if( ) ( )3 2 2 1 0Q Q Q Q > Mean Median 0 > .

Negative skew if ( ) and if Me( )3 2 2 1 0Q Q Q Q < an Median 0 < .

Independent Events

( ) ( ) ( )P A B P A P B =

Mutually Exclusive Events

( )P A B = 0

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Explanatory and response variables

The response variable is the dependent variable. It depends on the explanatory variable (also called

the independent variable). So in a graph of length of life versus number of cigarettes smoked per

week, the dependent variable would be length of life. It depends (or may do) on the number of

cigarettes smoked per week.

Give two reasons to justify the use of statistical models

Used to simplify or represent a real world problem

Cheaper or quicker or easier (than the real situation) or more easily modified (any two lines)

To improve understanding of the real world problem B1

Used to predict outcomes from a real world problem (idea of predictions)

Describe the main features and uses of a box plot.



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Data

Discrete

Discrete data can only take certain values in any given range. Number of cars in a household is an

example of discrete data. The values do not have to be whole numbers (e.g. shoe size is discrete).

Continuous

Continuous data can take any value in a given range. So a persons height is continuous since it

could be any value within set limits.

Categorical

Categorical data is data which is not numerical, such as choice of breakfast cereal etc.

Data may be displayed as grouped data or ungrouped data.

We say that data is grouped when we present it in the following way:

Weight (w) Frequency

65- 3

70- 7

Or

Score (s) Frequency

5-9 2

10-14 5

NB: We can group discrete data or continuous data.

We must know how to interpret these groups,

So that

Weight (w)

65- 65 70w <

70- 70 75w <

Or

Score (s)

5-9 4.5 9.5s <

10-14 9.5 14.5s <



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Representation of DataHistograms, stem and leaf diagrams, box plots. Use to compare distributions. Back-to-back stem

and leaf diagrams may be required.

Stem and Leaf Diagrams

The stem and leaf diagram is a very useful way of grouping data whilst retaining the original data.

For example suppose we had the following scores from children in a Maths test:

85, 18, 38, 67, 43, 75, 78, 81, 92, 71, 52, 62, 49, 62, 82, 69, 55, 57, 95, 62,

We see that the smallest value is 18 and the largest is 95. The classes of stem and leaf diagrams

must be of equal width and so it would seem sensible to choose classes 10-19, 20-29, etc.

The stem in this case represents the tens and the leaf represents the units so we have the

following:

Scores in Maths Test

Stem (Tens) Leaf (Units)

1 8

2

3 8 7

4 3 9

5 2 5 7

6 7 2 2 9 2

7 5 8 1

8 5 1 2

9 2 5

We then arrange these in numerical order to give the following:

Scores in Maths Test

NB : the data must be in

order in a Stem and Leaf

Diagram.

Stem (Tens) Leaf (Units)

1 8

2

3 7 8

4 3 95 2 5 7

6 2 2 2 7 9

7 1 5 8

8 1 2 5

9 2 5

We should also include a key with the diagram, so we say

1 8 means 18

This diagram tells us the basic shape of the distribution. We can easily see the smallest and largest

values and we can see that the mode is 62. We can also use it to calculate , and .1Q 2Q 3Q

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NB: If we wanted to represent the interval 18-22 on a stem & leaf we could not make 1 the stem

since not all the numbers would begin with 1. What we could do is have a stem of 18 and then make

the leaf the number we add on to the stem. In this case our key would be:

18 0 means 18 and 18 4 means 22

Back to back stem diagrams

We can use these to compare two samples by using a back to back stem plot. In this we put stems

down the middle and then one set of data on the left and the on the on the right. So we might end up

with a diagram as follows:

Physics Maths

7 5 1 81 2

6 5 3 3 7 8

4 2 1 4 3 9

9 4 3 1 0 5 2 5 7

8 4 2 6 2 2 2 7 9

6 3 7 1 5 8

5 1 8 1 2 5

9 2 5

Our key here would be

In Physics 7 1 means 17

In Maths 1 8 means 18



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Histograms

Data that has been grouped can be represented using a histogram.

A histogram is made up of rectangles of varying widths and heights there are no gaps between the

blocks.

The key feature of a histogram is that the area of each block is proportional to the frequency

In order for the area to be equal (or proportional) to the frequency we plot frequency density on the

vertical axis, wherewidthclass

frequencydensityfrequency = . The class width is the width of the interval

(i.e. it runs from the lower boundary to the upper boundary)

Example Plot a histogram for the following:

Length (h) Frequency Class width Frequency

Density

650- 3 20 0.15670- 7 10 0.7

680- 20 10 2

690- 16 10 1.6

700-720 4 20 0.2

So the first block runs from 650 to 670 and has height 0.15 etc.

FD

Length

NB: If there are gaps between the stated upper limit of one class interval and the lower limit of

the next class interval then we need to fill those gaps as shown below. For example,

Length (m)

15-19 14.5 19.5x

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