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IBMathsDPApplications & Interpretation (AI)HLRevision Notes4. Statistics & ProbabilityStatistics ToolkitMeasures of Dispersion

Measures of Dispersion (DP IB Applications & Interpretation (AI)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Roger B

Updated on

DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SLDP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

Quartiles & range

What are quartiles?

  • Quartiles are measures of location

  • Quartiles divide a population or data set into four equal sections

    • The lower quartile, Q1 splits the lowest 25% from the highest 75%

    • The median, Q2 splits the lowest 50% from the highest 50%

    • The upper quartile, Q3 splits the lowest 75% from the highest 25%

  • There are different methods for finding quartiles

    • Values obtained by hand and using technology may differ

  • You will be expected to use your GDC to calculate the quartiles

What are the range and interquartile range?

  • The range and interquartile range are both measures of dispersion

    • They describe how spread out the data is

  • The range is the largest value of the data minus the smallest value of the data

  • The interquartile range is the range of the central 50% of data

    • It is the upper quartile minus the lower quartile

      IQR equals Q subscript 3 minus Q subscript 1

      • This is given in the exam formula booklet

  • The units for the range and interquartile range are the same as the units for the data

Worked Example

Find the range and interquartile range for the data set given below.

 43                        29                        70                        51                        64                       43

4-1-2-ib-ai-aa-sl-quartiles-range-we-solution

Standard deviation & variance

What are the standard deviation and variance?

  • The standard deviation and variance are both measures of dispersion

    • They describe how spread out the data is in relation to the mean

  • The variance is the mean of the squares of the differences between the values and the mean

    • Variance is denoted sigma squared

  • The standard deviation is the square-root of the variance

    • Standard deviation is denoted sigma space (Greek letter 'sigma')

  • The units for the standard deviation are the same as the units for the data

  • The units for the variance are the square of the units for the data

How are the standard deviation and variance calculated for ungrouped data?

  • In the exam you will be expected to use the statistics function on your GDC to calculate the standard deviation and the variance

  • Calculating the standard deviation and the variance by hand may deepen your understanding

  • The formula for variance is sigma squared equals fraction numerator sum from i equals 1 to k of space f subscript i left parenthesis x subscript i minus mu right parenthesis squared over denominator n end fraction

    • space f subscript i is the number of times (frequency) that each value x subscript i occurs

    • This can be rewritten as

sigma squared equals fraction numerator sum from i equals 1 to k of space f subscript i x subscript i squared over denominator n end fraction minus mu squared

  • The formula for standard deviation is sigma equals square root of fraction numerator sum from i equals 1 to k of space f subscript i left parenthesis x subscript i minus mu right parenthesis squared over denominator n end fraction end root

    • This can be rewritten as

sigma equals square root of fraction numerator sum from i equals 1 to k of space f subscript i x subscript i squared over denominator n end fraction minus mu squared end root

  • You do not need to learn these formulae as you will use your GDC to calculate these

Worked Example

Find the variance and standard deviation for the data set given below.

 43                        29                        70                        51                        64                       43

4-1-2-ib-ai-aa-sl-standev-variance-we-solution

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    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.

    Roger B

    Reviewer: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.