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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 4. Statistics & Probability / 4.6 Normal Distribution / 4.6.2 Calculations with Normal Distribution


4.6.2 Calculations with Normal Distribution


Calculating Normal Probabilities

Throughout this section we will use the random variable X tilde straight N left parenthesis mu comma space sigma squared right parenthesis. For X distributed normally, X can take any real number. Therefore any values mentioned in this section will be assumed to be real numbers.

How do I find probabilities using a normal distribution?

  • The area under a normal curve between the points x equals a and x equals b is equal to the probability straight P left parenthesis a less than X less than b right parenthesis
    • Remember for a normal distribution you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
      • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis a less or equal than X less or equal than b right parenthesis
  • You will be expected to use distribution functions on your GDC to find the probabilities when working with a normal distribution

How do I calculate P(X = x): the probability of a single value for a normal distribution? 

  • The probability of a single value is always zero for a normal distribution
    • You can picture this as the area of a single line is zero
  • straight P left parenthesis X equals x right parenthesis equals 0
  • Your GDC is likely to have a "Normal Probability Density" function
    • This is sometimes shortened to NPD, Normal PD or Normal Pdf
    • IGNORE THIS FUNCTION for this course!
    • This calculates the probability density function at a point NOT the probability

How do I calculate P(a < X < b): the probability of a range of values for a normal distribution? 

  • You need a GDC that can calculate cumulative normal probabilities
  • You want to use the "Normal Cumulative Distribution" function
    • This is sometimes shortened to NCD, Normal CD or Normal Cdf
  • You will need to enter:
    • The 'lower bound' - this is the value a
    • The 'upper bound' - this is the value b
    • The 'μ' value - this is the mean
    • The 'σ' value - this is the standard deviation
  • Check the order carefully as some calculators ask for standard deviation before mean
    • Remember it is the standard deviation
      • so if you have the variance then square root it
  • Always sketch a quick diagram to visualise which area you are looking for

How do I calculate P(X > a) or P(X < b) for a normal distribution? 

  • You will still use the "Normal Cumulative Distribution" function
  • straight P left parenthesis X greater than a right parenthesis can be estimated using an upper bound that is sufficiently bigger than the mean
    • Using a value that is more than 4 standard deviations bigger than the mean is quite accurate
    • Or an easier option is just to input lots of 9's for the upper bound (99999999... or 1099)
  • straight P left parenthesis X less than b right parenthesis can be estimated using a lower bound that is sufficiently smaller than the mean
    • Using a value that is more than 4 standard deviations smaller than the mean is quite accurate
    • Or an easier option is just to input lots of 9's for the lower bound with a negative sign (-99999999... or -1099)

Are there any useful identities?

  • straight P left parenthesis X less than mu right parenthesis equals straight P left parenthesis X greater than mu right parenthesis equals 0.5
  • As straight P left parenthesis X equals a right parenthesis equals 0 you can use:
    • straight P left parenthesis X less than a right parenthesis plus straight P left parenthesis X greater than a right parenthesis equals 1
    • straight P left parenthesis X greater than a right parenthesis equals 1 minus straight P left parenthesis X less than a right parenthesis
    • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis X less than b right parenthesis minus straight P left parenthesis X less than a right parenthesis
  • These are useful when:
    • The mean and/or standard deviation are unknown
    • You only have a diagram
    • You are working with the inverse distribution

Exam Tip

  • Check carefully whether you have entered the standard deviation or variance into your GDC

Worked Example

The random variable Y tilde straight N left parenthesis 20 comma 5 squared right parenthesis. Calculate:

i)
straight P left parenthesis Y equals 20 right parenthesis.

4-6-2-ib-ai-aa-sl-normal-prob-a-we-solution

ii)
straight P left parenthesis 18 less or equal than Y less than 27 right parenthesis.

4-6-2-ib-ai-aa-sl-normal-prob-b-we-solution

iii)
straight P left parenthesis Y greater than 29 right parenthesis

4-6-2-ib-ai-aa-sl-normal-prob-c-we-solution

Inverse Normal Distribution

Given the value of P(X < a) how do I find the value of a? 

  • Your GDC will have a function called "Inverse Normal Distribution"
    • Some calculators call this InvN
  • Given that straight P left parenthesis X less than a right parenthesis equals p you will need to enter:
    • The 'area' - this is the value p
      • Some calculators might ask for the 'tail' - this is the left tail as you know the area to the left of a
    • The 'μ' value - this is the mean
    • The 'σ' value - this is the standard deviation

Given the value of P(X > a)  how do I find the value of a? 

  • Given straight P left parenthesis X greater than a right parenthesis equals p
  • Use straight P left parenthesis X less than a right parenthesis equals 1 minus straight P left parenthesis X greater than a right parenthesis to rewrite this as
    • straight P left parenthesis X less than a right parenthesis equals 1 minus p
  • Then use the method for P(X < a) to find a
  • If your calculator does have the tail option (left, right or centre) then you can use the "Inverse Normal Distribution" function straightaway by:
    • Selecting 'right' for the tail
    • Entering the area as 'p'

Exam Tip

  • Always check your answer makes sense
    • If P(X < a) is less than 0.5 then a should be smaller than the mean
    • If P(X < a) is more than 0.5 then a should be bigger than the mean
    • A sketch will help you see this

Worked Example

The random variable  W tilde straight N left parenthesis 50 comma space 36 right parenthesis.

Find the value of w such that straight P left parenthesis W greater than w right parenthesis equals 0.175.

4-6-2-ib-ai-aa-sl-inverse-normal-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
                                                                    • 3.1.3 Arcs & Sectors
                                                                    • 3.2 Geometry of 3D Shapes
                                                                      • 3.2.1 3D Coordinate Geometry
                                                                        • 3.2.2 Volume & Surface Area
                                                                        • 3.3 Trigonometry
                                                                          • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                            • 3.3.2 Non Right-Angled Trigonometry
                                                                              • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                              • 3.4 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 4. Statistics & Probability
                                                                                                  • 4.1 Statistics Toolkit
                                                                                                    • 4.1.1 Sampling & Data Collection
                                                                                                      • 4.1.2 Statistical Measures
                                                                                                        • 4.1.3 Frequency Tables
                                                                                                          • 4.1.4 Linear Tranformations of Data
                                                                                                            • 4.1.5 Outliers
                                                                                                              • 4.1.6 Univariate Data
                                                                                                                • 4.1.7 Interpreting Data
                                                                                                                • 4.2 Correlation & Regression
                                                                                                                  • 4.2.1 Bivariate Data
                                                                                                                    • 4.2.2 Correlation & Regression
                                                                                                                    • 4.3 Probability
                                                                                                                      • 4.3.1 Probability & Types of Events
                                                                                                                        • 4.3.2 Conditional Probability
                                                                                                                          • 4.3.3 Sample Space Diagrams
                                                                                                                          • 4.4 Probability Distributions
                                                                                                                            • 4.4.1 Discrete Probability Distributions
                                                                                                                              • 4.4.2 Expected Values
                                                                                                                              • 4.5 Binomial Distribution
                                                                                                                                • 4.5.1 The Binomial Distribution
                                                                                                                                  • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                  • 4.6 Normal Distribution
                                                                                                                                    • 4.6.1 The Normal Distribution
                                                                                                                                      • 4.6.2 Calculations with Normal Distribution
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        Daniel Finlay

                                                                                                                                                                        Author: Daniel

                                                                                                                                                                        Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.


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