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Leaving Certificate - Probability and Statistics Higher Level Free Course

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  • 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 rule-based 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 real-life 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 numerically
      1.6a 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 back-to-back 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 inter-quartile 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 problem-solving 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

  • All Alison courses are free to study. To successfully complete a course you must score 80% or higher in each course assessments. Upon successful completion of a course, you can choose to make your achievement formal by purchasing an official Alison Diploma, Certificate or PDF.

    Having an official Alison document is a great way to share your success. Plus it’s:

    • Ideal for including in CVs, job applications and portfolios
    • An indication of your ability to learn and achieve high results
    • An incentive to continue to empower yourself through learning
    • A tangible way of supporting the Alison mission to empower people everywhere through education.

Modules List( 19 )
  • For FREE online video tutorials for the Project M...
  • Module 1: Types of Data and Sampling
    • Introduction to data
    • Overview of data
    • Types of data
    • Categorical data
    • Numerical data
    • Continuous and discrete data
    • Sample types
  • Module 2: Frequency Tables
    • Frequency and graphs - Overview
    • Frequency tables with nominal data
    • Frequency tables with discrete data
    • Frequency tables - Discrete data and summary statistics
    • Mean from frequency tables - Discrete data
    • Interpreting bar graphs
  • Module 3: Methods of Representing Data
    • Representing data
    • Line graphs
    • Line plots
  • Module 4: Pie Charts
    • Pie charts
    • Pie charts - Worked example
  • Module 5: Histograms and Bar Charts
    • Histograms & bar graphs
    • Bar graphs
  • Module 6: Stem-and-Leaf Plots
    • Introduction to stem-and-leaf
    • Stem-and-leaf plots
    • Back to back stem plots
    • Stem-and-leaf diagrams 1
    • Stem-and-leaf diagrams 2
  • Module 7: Skewness
    • Comparing mean, mode and median
    • Symmetry and skew of a distribution
    • Negative skewness – Left skewness
    • Positive skewness – Right skewness
    • Probability intervals
    • Comparing sample and population
    • Probability intervals - Examples
  • Module 8: Scatter Plots - Line of Best Fit
    • Scatterplots
    • Strength of association
    • Introduction to the regression line
    • Finding the equation of a regression line
    • Intrepretation of slope and intercept
    • Practice question
    • Scatter plots and linear models
  • Module 9: Correlation
    • Correlation coefficient: r
    • Calculating r
    • Practice question
    • Correlation and causation
  • Module 10: Measures of Central Tendency
    • Measures of central tendency: Mean, mode and median
    • Mode
    • Mean
    • Median
    • Mode, mean and median
    • Comparing mode, mean and median
  • Module 11: Measures of Central Spread
    • Measures of central spread
    • Range of data
    • Inter-quartile range
    • Standard deviation as a measure of spread
    • Summarising data - Overview
    • Soccer activity
  • Module 12: Analyse Data
    • Mean from frequency tables - Discrete data
    • Frequency and graphs - Overview
    • Summarising data - Overview
    • Mode
    • Mean
    • Median
    • Mode, mean, median
    • Comparing mode, mean and median
    • Range of data
    • Inter-quartile range
    • Review - Summarising data
    • Standard deviation and normal distribution
    • Standard deviation and calculator
  • Module 13: Probability
    • Fundamental Principle of Counting
    • Calculating the outcome
    • Permutations and combinations
    • Introduction to probability
    • Probabilities
    • Finding probabilities theoretically
    • Basic rules of probability
    • Venn diagrams
    • Conditional probability
  • Module 14: Expected Value
    • Probability and relative frequency
    • Short-run coin tossing
    • Short-run dice rolling
    • Predicting from past experience
    • Towards probability with coins
    • Towards probability with dice
    • Probability as long-run relative frequency
    • Mean and variance of a discrete random variable
  • Module 15: Binomial Distribution
    • Binomial probability function and distribution
    • The number of successes in a given number of trials
    • Binomial distribution: Bernoulli trials
    • Bernoulli extended
  • Module 16: Normal Distribution
    • Introduction to the normal distribution
    • The normal curve
    • Continuous random variables and the normal distribution
    • Calculation of probabilities for a normal distribution
  • Module 17: Hypothesis Testing
    • Hypothesis testing
    • Hypothesis testing - Example
    • Hypothesis testing - Formal Procedure
    • Worked example
  • Module 18: Problem Solving
    • Random variables
    • Probability of events
    • Standard deviation as a measure of spread
  • End of Course Information
    • End of course information
Topics List ( 7 )
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.
Topics List ( 6 )
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.
Topics List ( 3 )
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.
Topics List ( 2 )
Module 4: Pie Charts
Pie charts are used to display discrete numerical data or categorical data.
Topics List ( 2 )
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.
Topics List ( 5 )
Module 6: Stem-and-Leaf 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.
Topics List ( 7 )
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.
Topics List ( 7 )
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.
Topics List ( 4 )
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.
Topics List ( 6 )
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.
Topics List ( 6 )
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 inter-quartile 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.
Topics List ( 9 )
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.
Topics List ( 8 )
Module 14: Expected Value
Expected value is used widely in insurance industries and casino games to determine the fairness of the result/pay-out. 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.
Topics List ( 4 )
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.
Topics List ( 4 )
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 bell-shaped 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/z-scores are the number of std. deviations a values lies above or below the mean.
Topics List ( 4 )
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 confidence-interval level.
Topics List ( 3 )
Module 18: Problem Solving
An example putting what we have learned so far into practice.
Topics List ( 1 )
End of Course Information
Information about Certificate of Completion.
Course Features
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    2-3 Hours

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  • Assessment


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    50 Pts

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