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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 4. Statistics & Probability / 4.6 Random Variables / 4.6.2 Unbiased Estimates


4.6.2 Unbiased Estimates


Unbiased Estimates

What is an unbiased estimator of a population parameter?

  • An estimator is a random variable that is used to estimate a population parameter
    • An estimate is the value produced by the estimator when a sample is used
  • An estimator is called unbiased if its expected value is equal to the population parameter
    • An estimate from an unbiased estimator is called an unbiased estimate
    • This means that the mean of the unbiased estimates will get closer to the population parameter as more samples are taken
  • The sample mean is an unbiased estimate for the population mean
  • The sample variance is not an unbiased estimate for the population variance
    • On average the sample variance will underestimate the population variance
    • As the sample size increases the sample variance gets closer to the unbiased estimate

What are the formulae for unbiased estimates of the mean and variance of a population?

  • A sample of n data values (x1, x2, ... etc) can be used to find unbiased estimates for the mean and variance of the population
  • An unbiased estimate for the mean μ of a population can be calculated using
    •  x with bar on top equals fraction numerator sum x over denominator n end fraction
  • An unbiased estimate for the variance σ² of a population can be calculated using
    • s subscript n minus 1 end subscript superscript 2 equals fraction numerator n over denominator n minus 1 end fraction s subscript n superscript 2
    • This is given in the formula booklet
    • s subscript n superscript 2 is the variance of the sample data
      • s subscript n superscript 2 equals fraction numerator sum left parenthesis x minus x with bar on top right parenthesis squared over denominator n end fraction equals fraction numerator sum x squared over denominator n end fraction minus open parentheses x with bar on top close parentheses squared
  • Different calculators can use different notations for s subscript n minus 1 end subscript superscript 2
    • sigma subscript n minus 1 end subscript superscript 2, s subscript blank superscript 2, s with hat on top subscript blank superscript 2 are notations you might see
    • You may also see the square roots of these

Is sn-1 an unbiased estimate for the standard deviation?

  • Unfortunately sn-1 is not an unbiased estimate for the standard deviation of the population
  • It is better to work with the unbiased variance rather than standard deviation
  • There is not a formula for an unbiased estimate for the standard deviation that works for all populations
    • Therefore you will not be asked to find one in your exam

How do I show the sample mean is an unbiased estimate for the population mean?

  • You do not need to learn this proof
    • It is simply here to help with your understanding
  • Suppose the population of X has mean μ and variance σ²  
  • Take a sample of n observations
    • X1, X2, ..., Xn
    • E(Xi) = μ
  • Using the formula for a linear combination of n independent variables:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight E invisible function application open parentheses X with bar on top close parentheses end cell equals cell straight E invisible function application open parentheses fraction numerator X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n over denominator n end fraction close parentheses end cell row blank equals cell fraction numerator straight E invisible function application open parentheses X subscript 1 close parentheses plus straight E invisible function application open parentheses X subscript 2 close parentheses plus blank horizontal ellipsis plus straight E open parentheses X subscript n close parentheses over denominator n end fraction end cell row blank equals cell fraction numerator mu plus mu plus blank horizontal ellipsis plus mu blank over denominator n end fraction end cell row blank equals cell fraction numerator n mu over denominator n end fraction end cell row blank equals mu end table
  • As table attributes columnalign right center left columnspacing 0px end attributes row cell straight E invisible function application open parentheses X with bar on top close parentheses end cell equals mu end table this shows the formula will produce an unbiased estimate for the population mean

Why is there a divisor of n-1 in the unbiased estimate for the variance?

  • You do not need to learn this proof
    • It is simply here to help with your understanding
  • Suppose the population of X has mean μ and variance σ²  
  • Take a sample of n observations
    • X1, X2, ..., Xn
    • E(Xi) = μ
    • Var(Xi) = σ2
  • Using the formula for a linear combination of n independent variables:

table attributes columnalign right center left columnspacing 0px end attributes row cell Var invisible function application open parentheses X with bar on top close parentheses end cell equals cell Var invisible function application open parentheses fraction numerator X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n over denominator n end fraction close parentheses end cell row blank equals cell fraction numerator Var invisible function application open parentheses X subscript 1 close parentheses plus Var invisible function application open parentheses X subscript 2 close parentheses plus blank horizontal ellipsis plus Var open parentheses X subscript n close parentheses over denominator n squared end fraction end cell row blank equals cell fraction numerator sigma squared plus sigma squared plus blank horizontal ellipsis plus sigma squared blank over denominator n squared end fraction end cell row blank equals cell fraction numerator n sigma squared over denominator n squared end fraction end cell row blank equals cell sigma squared over n end cell end table
  • It can be shown that straight E left parenthesis X with bar on top squared right parenthesis equals mu squared plus sigma squared over n
    • This comes from rearranging Var invisible function application open parentheses X with bar on top close parentheses equals straight E invisible function application open parentheses X with bar on top squared close parentheses minus open square brackets straight E invisible function application open parentheses X with bar on top close parentheses close square brackets squared
  • It can be shown that straight E left parenthesis X squared right parenthesis equals straight E open parentheses X subscript i squared close parentheses equals mu squared plus sigma squared
    • This comes from rearranging Var invisible function application open parentheses X close parentheses equals straight E invisible function application open parentheses X squared close parentheses minus open square brackets straight E invisible function application open parentheses X close parentheses close square brackets squared
  • Using the formula for a linear combination of n independent variables:
table attributes columnalign right center left columnspacing 0px end attributes row cell straight E invisible function application open parentheses S subscript n superscript 2 close parentheses end cell equals cell straight E invisible function application open parentheses fraction numerator sum X subscript i superscript 2 over denominator n end fraction minus X with minus on top squared close parentheses end cell row blank equals cell fraction numerator sum straight E invisible function application left parenthesis X subscript i superscript 2 right parenthesis blank over denominator n end fraction minus straight E invisible function application open parentheses X with bar on top squared close parentheses end cell row blank equals cell fraction numerator sum open parentheses mu squared plus sigma squared close parentheses over denominator n end fraction minus open parentheses mu squared plus sigma squared over n close parentheses end cell row blank equals cell fraction numerator n open parentheses mu squared plus sigma squared close parentheses over denominator n end fraction minus open parentheses mu squared plus sigma squared over n close parentheses end cell row blank equals cell mu squared plus sigma squared minus open parentheses mu squared plus sigma squared over n close parentheses end cell row blank equals cell sigma squared minus sigma squared over n end cell row blank equals cell fraction numerator n sigma squared minus sigma squared over denominator n end fraction end cell row blank equals cell fraction numerator n minus 1 over denominator n end fraction sigma ² end cell end table

  • As straight E open parentheses S subscript n superscript 2 close parentheses not equal to straight sigma squared this shows that the sample variance is not unbiased
    • You need to multiply by fraction numerator n over denominator n minus 1 end fraction
    • straight E open parentheses S subscript n minus 1 end subscript superscript 2 close parentheses equals straight sigma squared

Exam Tip

  • Check the wording of the exam question carefully to determine which of the following you are given:
    • The population variance: sigma squared
    • The sample variance: s subscript n superscript 2
    • An unbiased estimate for the population variance: s subscript n minus 1 end subscript superscript 2

Worked Example

The times, X minutes, spent on daily revision of a random sample of 50 IB students from the UK are summarised as follows.

n equals 50 sum x equals 6174 s subscript n superscript 2 equals 1384.3

Calculate unbiased estimates of the population mean and variance of the times spent on daily revision by IB students in the UK.

4-6-2-ib-ai-hl-unbiased-estimates-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 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
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse 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
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 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 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      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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