
Free Course

XSIQ

23 Hours

Assessment

Certification

50 Pts
Leaving Certificate  Probability and Statistics Higher Level

Description

Outcome

Certification

Probability and Statistics is one of two strands introduced in the first phase of the new Project Maths Course in the Irish curriculum. This topic covers up to half of the new Paper 2 in the Leaving Certificate Paper.
Statistics are used in real life to make sense of the information around us and how it affects us. Statistics looks at the data handling cycle and analysis of the data collected. This involves posing a question, collecting data on that question, presenting that data, analysing the data (using measures of spread and centre) and interpreting the results. In answering questions, it is essential that you can contextualise and justify your findings.
Probability is concerned with the likelihood of an event(s) happening. The information can be used to make informed decisions. The use of probability is commonly utilised in the world of finance, insurance and sport among others. Probability can also be used to infer the fairness of an event or series of events. It can be evaluated using a diagram or a rulebased approach. A combination of Probability and Statistics can be used to prove/disprove a given conjecture or statement (Hypothesis Testing (HL only)).
This Strand attempts to merge the mathematical aspects of Probability and Statistics with its reallife application. It is an interesting topic that is very accessible to all students. 
Students will learn about:1.1 Counting
  count the arrangements of n distinct objects (n!)
  count the number of ways of arranging r objects from n distinct objects
  count the number of ways of selecting r objects from n distinct objects
1.2 Concepts of probability
  discuss basic rules of probability (AND/OR, mutually exclusive) through the use of Venn diagrams
  calculate expected value and understand that this does not need to be one of the outcomes
  recognise the role of expected value in decision making and explore the issue of fair games
  extend their understanding of the basic rules of probability (AND/OR, mutually exclusive) through the use of formulae
  use the Addition Rule, Multiplication Rule (Independent events), Multiplication Rule (General case)
  solve problems involving conditional probability in a systematic way
1.3 Outcomes of random processes
  find the probability that two independent events both occur
  apply an understanding of Bernoulli trials
  solve problems involving up to 3 Bernoulli trials
  calculate the probability that the 1st success occurs on the nth Bernoulli trial where n is specified
  solve problems involving calculating the probability of k successes in n repeated Bernoulli trials (normal approximation not required)
  calculate the probability that the kth success occurs on the nth Bernoulli trial
  use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions
  solve problems involving reading probabilities from the normal distribution tables
1.4 Statistical reasoning with an aim to becoming a statistically aware consumer
  work with different types of bivariate data
1.5 Finding, collecting and organising data
  discuss different types of studies: sample surveys, observational studies and designed experiments
  design a plan and collect data on the basis of above knowledge
  recognise the importance of randomisation and the role of the control group in studies
  recognise biases, limitations and ethical issues of each type of study
  select a sample (stratified, cluster, quota – no formulae required, just definitions of these)
  design a plan and collect data on the basis of above knowledge
1.6 Representing data graphically and numerically1.6a Graphical
  describe the sample (both univariate and bivariate data) by selecting appropriate graphical or numerical methods
  explore the distribution of data, including concepts of symmetry and skewness
  compare data sets using appropriate displays, including backtoback stem and leaf plots
  determine the relationship between variables using scatterplots
  recognise that correlation is a value from 1 to +1 and that it measures the extent of the linear relationship between two variables
  match correlation coefficient values to appropriate scatter plots
  understand that correlation does not imply causality
  analyse plots of the data to explain differences in measures of centre and spread
  draw the line of best fit by eye
  make predictions based on the line of best fit
  calculate the correlation coefficient by calculator
1.6b Numerical
  recognise standard deviation and interquartile range as measures of variability
  use a calculator to calculate standard deviation
  find quartiles and the interquartile range
  use the interquartile range appropriately when analysing data
  recognise the existence of outliers
  recognise the effect of outliers
  use percentiles to assign relative standing
1.7 Analysing, interpreting and drawing inferences from data
  interpret a histogram in terms of distribution of data
  make decisions based on the empirical rule
  recognise the concept of a hypothesis test
  calculate the margin of error for a population proportion
  conduct a hypothesis test on a population proportion using the margin of error
1.8 Synthesis and problemsolving skills
  explore patterns and formulate conjectures
  explain findings
  justify conclusions
  communicate mathematics verbally and in written form
  apply their knowledge and skills to solve problems in familiar and unfamiliar contexts
  analyse information presented verbally and translate it into mathematical form
  devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions

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 A way to show your ability to learn and achieve high results
Modules List( 19 )

LEAVING CERTIFICATE  STATISTICS HIGHER LEVEL

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Module
1 Types of Data and Sampling
The aim of statistics is to help us make sense of large amounts of information and figure out what it means and how it affects us. Data must be gathered from samples and analysed and it is often the case that this data is either number based or word based. This gives rise to different types of data and samples. Terminology is very important in this topic.

Introduction to data
Start Topic

Overview of data
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Types of data
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Categorical data
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Numerical data
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Continuous and discrete data
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Sample types
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Module
2 Frequency Tables
Frequency tables help make the analysis of collected data much easier as they group the data into categories. They enable the mean, mode and median to be calculated more clearly.

Frequency and graphs  Overview
Start Topic

Frequency tables with nominal data
Start Topic

Frequency tables with discrete data
Start Topic

Frequency tables  Discrete data and summary statistics
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Mean from frequency tables  Discrete data
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Interpreting bar graphs
Start Topic


Module
3 Methods of Representing Data
Data can be presented in many pictorial forms. The graph used will vary depending on the data being presented.

Representing data
Start Topic

Line graphs
Start Topic

Line plots
Start Topic


Module
4 Pie Charts
Pie charts are used to display discrete numerical data or categorical data.

Pie charts
Start Topic

Pie charts  Worked example
Start Topic


Module
5 Histograms and Bar Charts
Histograms and Bar Charts are very similar but there are some important differences  no gaps between bars in histograms, bar charts show discrete data but histograms show continuous data and data is always grouped in histograms.

Histograms & bar graphs
Start Topic

Bar graphs
Start Topic


Module
6 StemandLeaf Plots
Stem & Leaf diagrams are similar to horizontal Bar Charts but are only suitable for small amounts of data. It is important that a Key is always included to explain the data presented.

Introduction to stemandleaf
Start Topic

Stemandleaf plots
Start Topic

Back to back stem plots
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Stemandleaf diagrams 1
Start Topic

Stemandleaf diagrams 2
Start Topic


Module
7 Skewness
It is possible to determine the distribution of data by looking at the shape of the histogram. There are 3 main shapes – symmetrical, positive/right and negative/left skewness.

Comparing mean, mode and median
Start Topic

Symmetry and skew of a distribution
Start Topic

Negative skewness – Left skewness
Start Topic

Positive skewness – Right skewness
Start Topic

Probability intervals
Start Topic

Comparing sample and population
Start Topic

Probability intervals  Examples
Start Topic


Module
8 Scatter Plots  Line of Best Fit
Scatter Plots are graphs that display and compare bivariate data (2 variables). Look for a relationship between the two variables and comment on the strength of the relationship. Using information on the graph, we can find the equation of a line that best describes this relationship.

Scatterplots
Start Topic

Strength of association
Start Topic

Introduction to the regression line
Start Topic

Finding the equation of a regression line
Start Topic

Intrepretation of slope and intercept
Start Topic

Practice question
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Scatter plots and linear models
Start Topic


Module
9 Correlation
Correlation is a measure of the strength of a relationship between bivariate data. Correlation can be classified as positive, negative or no correlation (0) and it is always a value between 1 and 1.

Correlation coefficient: r
Start Topic

Calculating r
Start Topic

Practice question
Start Topic

Correlation and causation
Start Topic


Module
10 Measures of Central Tendency
Analysing a large mass of data can be easily summarised using some key numbers – mean, mode and median. It is important that you can identify/calculate these values. Also, at times certain values may be more appropriate than others to use; therefore you must be able to justify your choice based on the information to hand.

Measures of central tendency: Mean, mode and median
Start Topic

Mode
Start Topic

Mean
Start Topic

Median
Start Topic

Mode, mean and median
Start Topic

Comparing mode, mean and median
Start Topic


Module
11 Measures of Central Spread
Measures of spread reflect the range over which the data is varied or spread out. The data ranges across four different quartiles which give the interquartile range. Standard deviation is a very important measure of spread that shows how far the data is spread from the mean. Outliers are extreme values that will affect analysis of any dataset.

Measures of central spread
Start Topic

Range of data
Start Topic

Interquartile range
Start Topic

Standard deviation as a measure of spread
Start Topic

Summarising data  Overview
Start Topic

Soccer activity
Start Topic


Module
12 Analyse Data
Having collected and presented data, conclusions must be drawn. Measure of Central Tendency and Spread must be used.

Mean from frequency tables  Discrete data
Start Topic

Frequency and graphs  Overview
Start Topic

Summarising data  Overview
Start Topic

Mode
Start Topic

Mean
Start Topic

Median
Start Topic

Mode, mean, median
Start Topic

Comparing mode, mean and median
Start Topic

Range of data
Start Topic

Interquartile range
Start Topic

Review  Summarising data
Start Topic

Standard deviation and normal distribution
Start Topic

Standard deviation and calculator
Start Topic


LEAVING CERTIFICATE  PROBABILITY HIGHER LEVEL

Module
13 Probability
Probability is concerned with the ‘chance’ that something may happen. Probability of events occurring may be calculated using rules or diagrams (tables/tree/Venn diagrams). Particular attention should also be paid to the terminology of this topic.

Fundamental Principle of Counting
Start Topic

Calculating the outcome
Start Topic

Permutations and combinations
Start Topic

Introduction to probability
Start Topic

Probabilities
Start Topic

Finding probabilities theoretically
Start Topic

Basic rules of probability
Start Topic

Venn diagrams
Start Topic

Conditional probability
Start Topic


Module
14 Expected Value
Expected value is used widely in insurance industries and casino games to determine the fairness of the result/payout. We can use this value to determine the potential loss/gain of an event for us. Expected value of 0 means a game is fair/equitable. The expected value does not have to be one of the possible outcomes of the event.

Probability and relative frequency
Start Topic

Shortrun coin tossing
Start Topic

Shortrun dice rolling
Start Topic

Predicting from past experience
Start Topic

Towards probability with coins
Start Topic

Towards probability with dice
Start Topic

Probability as longrun relative frequency
Start Topic

Mean and variance of a discrete random variable
Start Topic


Module
15 Binomial Distribution
Binomial distribution is a method of calculating probabilities that can only be applied in certain circumstances – Bernoulli trials. If the situation FITS it must be Bernoulli! Fixed number of trials; independent events; two possible outcomes; success/failure probability remains constant.

Binomial probability function and distribution
Start Topic

The number of successes in a given number of trials
Start Topic

Binomial distribution: Bernoulli trials
Start Topic

Bernoulli extended
Start Topic


Module
16 Normal Distribution
When data is arranged in order from lowest to highest, a pattern may emerge where the majority of the data may form a cluster around the middle. Graphing this information would produce a bellshaped curve known as the normal distribution curve. 3 values are important when dealing with the normal curve – 68%, 95% and 99.7%. Wide, flatter curves suggest greater spread/std. deviation, whereas narrow, steeper curves suggest smaller spread between values/std. deviations. Standard scores/zscores are the number of std. deviations a values lies above or below the mean.

Introduction to the normal distribution
Start Topic

The normal curve
Start Topic

Continuous random variables and the normal distribution
Start Topic

Calculation of probabilities for a normal distribution
Start Topic


Module
17 Hypothesis Testing
A hypothesis is a statement made about some characteristic of a population and hypothesis testing is a statistical method of proving/disproving it. The truth of the statement depends on if the statistic lies within a calculated confidenceinterval level.

Hypothesis testing
Start Topic

Hypothesis testing  Example
Start Topic

Hypothesis testing  Formal Procedure
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Worked example
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Module
18 Problem Solving
An example putting what we have learned so far into practice.

Random variables
Start Topic

Probability of events
Start Topic

Standard deviation as a measure of spread
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END OF COURSE INFORMATION

End of Course Information

Information about Certificate of Completion.

End of course information
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