Statistical Measures (DP IB Maths: AI HL)

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Mean, Mode, Median

What are the mean, mode and median?

  • Mean, median and mode are measures of central tendency
    • They describe where the centre of the data is
  • They are all types of averages
  • In statistics it is important to be specific about which average you are referring to
  • The units for the mean, mode and median are the same as the units for the data

How are the mean, mode, and median calculated for ungrouped data?

  • The mode is the value that occurs most often in a data set
    • It is possible for there to be more than one mode
    • It is possible for there to be no mode
      • In this case do not say the mode is zero
  • The median is the middle value when the data is in order of size
    • If there are two values in the middle then the median is the midpoint of the two values
  • The mean is the sum of all the values divided by the number of values

begin mathsize 22px style x with bar on top equals 1 over n sum from i equals 1 to n of x subscript i end style

    • Where sum from i equals 1 to n of x subscript i equals x subscript 1 plus x subscript 2 plus... plus x subscript n is the sum of the n pieces of data
    • The mean can be represented by the symbol μ
  • Your GDC can calculate these statistical measures if you input the data using the statistics mode

Worked example

Find the mode, median and mode for the data set given below.

 43                        29                        70                        51                        64                       43

4-1-2-ib-ai-aa-sl-mean-median-mode-we-solution

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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

begin mathsize 22px style IQR equals Q subscript 3 minus Q subscript 1 end style

      • This is given in the 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

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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 σ2
  • The standard deviation is the square-root of the variance
    • Standard deviation is denoted σ
  • 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
    • 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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Dan

Author: Dan

Expertise: Maths

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.