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January 2010 S1 Note

S1 Notes (Edexcel)

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January 2010 S1 Note

Definitions for S1 Statistical 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 real world 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 ¾ Bell shaped curve ¾ symmetrical about mean; mean = mode = median ¾ 95% of data lies within 2 standard deviations of mean ¾ 68.3% between one standard deviation of mean 2 conditions for skewness Positive skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) > 0 and if Mean − Median > 0 . Negative skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) < 0 and if Mean − Median < 0 . Independent Events P ( A ∩ B) = P( A) × P( B) Mutually Exclusive Events P( A ∩ B) = 0

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January 2010 S1 Note

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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January 2010 S1 Note

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 person’s 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) 6570-

Frequency 3 7

Score (s) 5-9 10-14

Frequency 2 5

Or

NB: We can group discrete data or continuous data. We must know how to interpret these groups, So that Weight (w) 6570-

65 ≤ w < 70 70 ≤ w < 75

Or Score (s) 5-9 10-14

4.5 ≤ s < 9.5 9.5 ≤ s < 14.5

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January 2010 S1 Note

Representation of Data Histograms, 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) 1 2 3 4 5 6 7 8 9

Leaf (Units) 8 87 39 257 72292 581 512 25

We then arrange these in numerical order to give the following: Scores in Maths Test Stem (Tens) 1 2 3 4 5 6 7 8 9

Leaf (Units) 8

NB : the data must be in order in a Stem and Leaf Diagram.

78 39 257 22279 158 125 25

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 Q1 , Q2 and Q3 . Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

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January 2010 S1 Note

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 75 1 653 421 94310 842 63 51

1 2 3 4 5 6 7 8 9

Maths 8 78 39 257 22279 158 125 25

Our key here would be In Physics 7 1 means 17 In Maths 1 8 means 18

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January 2010 S1 Note

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 frequency vertical axis, where frequency density = . The class width is the width of the interval class width (i.e. it runs from the lower boundary to the upper boundary) Example Plot a histogram for the following: Length (h)

Frequency

Class width

650670680690700-720

3 7 20 16 4

20 10 10 10 20

Frequency Density 0.15 0.7 2 1.6 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, When question says “give a reason to justify the use of a histogram to represent these data”…. The answer is “Data is continuous”

Length (m) 15-19 20-24 25-29

14.5 ≤ x < 19.5 19.5 ≤ x < 24.5 24.5 ≤ x < 29.5

So the class width is 5. Not 19 − 15 = 4

NB: Be careful with age since “15-19” would mean 15 ≤ x < 20 since one is 19 until the moment before one’s 20th birthday. The shape of the histogram gives us information about the mean and the dispersion

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

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January 2010 S1 Note

Box Plot Diagram (or Box and whisker diagram)

This is a diagram used to illustrate the dispersion of data. There is a box which runs from the lower quartile, Q1 to the upper quartile Q3 with the median, Q2 marked on it. The whisker then goes from this box to the lowest value in one direction and to the highest value in the other. We end up with a diagram as follows: NB : It must have a horizontal axis with a scale on it. Lowest Value

Highest Value Q1

Q2

Q3

Skewness. Concepts outliers. Any rule to identify outliers will be specified in the question.

If the question refers to outliers then we should use a refined box plot where we fix the length of the whisker to, for example, 1.5 ( Q3 − Q1 ) at the end where an outlier lies. In this case we would mark with crosses those outliers which were outside of the whisker

Outlier

× Q3 + 1.5 ( Q3 − Q1 ) Q1

Q2

Q3

NB We can tell something about the skewness / symmetry of the distribution from the box plot. For example,

Lowest Value

Highest Value Q1

Q2

Q3

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January 2010 S1 Note We can see from the above that this is positively skewed ( Q3 − Q2 ) − ( Q2 − Q1 ) > 0 with a long tail of high values. Similarly,

Lowest Value

Highest Value Q1

Q2 Q3

The above is negatively skewed since ( Q3 − Q2 ) − ( Q2 − Q1 ) < 0 .

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January 2010 S1 Note

S1 Notes (Edexcel)

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For use only in [the name of your school]

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January 2010 S1 Note

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January 2010 S1 Note

Definitions for S1 Statistical 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 real world 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 ¾ Bell shaped curve ¾ symmetrical about mean; mean = mode = median ¾ 95% of data lies within 2 standard deviations of mean ¾ 68.3% between one standard deviation of mean 2 conditions for skewness Positive skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) > 0 and if Mean − Median > 0 . Negative skew if ( Q3 − Q2 ) − ( Q2 − Q1 ) < 0 and if Mean − Median < 0 . Independent Events P ( A ∩ B) = P( A) × P( B) Mutually Exclusive Events P( A ∩ B) = 0

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

10

For use only in [the name of your school]

January 2010 S1 Note

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.

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

11

For use only in [the name of your school]

January 2010 S1 Note

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 person’s 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) 6570-

Frequency 3 7

Score (s) 5-9 10-14

Frequency 2 5

Or

NB: We can group discrete data or continuous data. We must know how to interpret these groups, So that Weight (w) 6570-

65 ≤ w < 70 70 ≤ w < 75

Or Score (s) 5-9 10-14

4.5 ≤ s < 9.5 9.5 ≤ s < 14.5

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

12

For use only in [the name of your school]

January 2010 S1 Note

Representation of Data Histograms, 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) 1 2 3 4 5 6 7 8 9

Leaf (Units) 8 87 39 257 72292 581 512 25

We then arrange these in numerical order to give the following: Scores in Maths Test Stem (Tens) 1 2 3 4 5 6 7 8 9

Leaf (Units) 8

NB : the data must be in order in a Stem and Leaf Diagram.

78 39 257 22279 158 125 25

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 Q1 , Q2 and Q3 . Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

13

For use only in [the name of your school]

January 2010 S1 Note

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 75 1 653 421 94310 842 63 51

1 2 3 4 5 6 7 8 9

Maths 8 78 39 257 22279 158 125 25

Our key here would be In Physics 7 1 means 17 In Maths 1 8 means 18

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

14

For use only in [the name of your school]

January 2010 S1 Note

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 frequency vertical axis, where frequency density = . The class width is the width of the interval class width (i.e. it runs from the lower boundary to the upper boundary) Example Plot a histogram for the following: Length (h)

Frequency

Class width

650670680690700-720

3 7 20 16 4

20 10 10 10 20

Frequency Density 0.15 0.7 2 1.6 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, When question says “give a reason to justify the use of a histogram to represent these data”…. The answer is “Data is continuous”

Length (m) 15-19 20-24 25-29

14.5 ≤ x < 19.5 19.5 ≤ x < 24.5 24.5 ≤ x < 29.5

So the class width is 5. Not 19 − 15 = 4

NB: Be careful with age since “15-19” would mean 15 ≤ x < 20 since one is 19 until the moment before one’s 20th birthday. The shape of the histogram gives us information about the mean and the dispersion

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

15

For use only in [the name of your school]

January 2010 S1 Note

Box Plot Diagram (or Box and whisker diagram)

This is a diagram used to illustrate the dispersion of data. There is a box which runs from the lower quartile, Q1 to the upper quartile Q3 with the median, Q2 marked on it. The whisker then goes from this box to the lowest value in one direction and to the highest value in the other. We end up with a diagram as follows: NB : It must have a horizontal axis with a scale on it. Lowest Value

Highest Value Q1

Q2

Q3

Skewness. Concepts outliers. Any rule to identify outliers will be specified in the question.

If the question refers to outliers then we should use a refined box plot where we fix the length of the whisker to, for example, 1.5 ( Q3 − Q1 ) at the end where an outlier lies. In this case we would mark with crosses those outliers which were outside of the whisker

Outlier

× Q3 + 1.5 ( Q3 − Q1 ) Q1

Q2

Q3

NB We can tell something about the skewness / symmetry of the distribution from the box plot. For example,

Lowest Value

Highest Value Q1

Q2

Q3

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

16

For use only in [the name of your school]

January 2010 S1 Note We can see from the above that this is positively skewed ( Q3 − Q2 ) − ( Q2 − Q1 ) > 0 with a long tail of high values. Similarly,

Lowest Value

Highest Value Q1

Q2 Q3

The above is negatively skewed since ( Q3 − Q2 ) − ( Q2 − Q1 ) < 0 .

Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets

17

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