Frequency Tables (DP IB Maths: AI HL)

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Dan

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Dan

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Ungrouped Data

How are frequency tables used for ungrouped data?

  • Frequency tables can be used for ungrouped data when you have lots of the same values within a data set
    • They can be used to collect and present data easily
  • If the value 4 has a frequency of 3 this means that there are three 4’s in the data set

How are measures of central tendency calculated from frequency tables with ungrouped data?

  • The mode is the value that has the highest frequency
  • The median is the middle value
    • Use cumulative frequencies (running totals) to find the median
  • The mean can be calculated by
    • Multiplying each value xi by its frequency fi
    • Summing to get Σfixi
    • Dividing by the total frequency n = Σfi
    • This is given in the formula booklet

begin mathsize 22px style x with bar on top equals fraction numerator begin display style sum from i equals 1 to k of end style f subscript i x subscript i over denominator n end fraction end style

    • Your GDC can calculate these statistical measures if you input the values and their frequencies using the statistics mode

How are measures of dispersion calculated from frequency tables with ungrouped data?

  • The range is the largest value of the data minus the smallest value of the data
  • The interquartile range is calculated by

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

    • The quartiles can be found by using your GDC and inputting the values and their frequencies
  • The standard deviation and variance can be calculated by hand using the formulae
    • Variance

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

    • Standard deviation

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 be expected to use your GDC to find the standard deviation and variance
      • You may want to see these formulae to deepen your understanding

Examiner Tip

  • Always check whether your answers make sense when using your GDC
    • The value for a measure of central tendency should be within the range of data

Worked example

The frequency table below gives information number of pets owned by 30 students in a class.

Number of pets 0 1 2 3
Frequency 11 5 8 6

 

Find

a)
the mode.

4-1-3-ib-ai-aa-sl-ungrouped-data-a-we-solution

b)
the median.

4-1-3-ib-ai-aa-sl-ungrouped-data-b-we-solution

c)
the mean.

4-1-3-ib-ai-aa-sl-ungrouped-data-c-we-solution

d)
the standard deviation.

4-1-3-ib-ai-aa-sl-ungrouped-data-d-we-solution

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Grouped Data

How are frequency tables used for grouped data?

  • Frequency tables can be used for grouped data when you have lots of the same values within the same interval
    • Class intervals will be written using inequalities and without gaps
      • 10 less or equal than x less than 20 and 20 less or equal than x less than 30
    • If the class interval 10 less or equal than x less than 20 has a frequency of 3 this means there are three values in that interval
      • You do not know the exact data values when you are given grouped data

How are measures of central tendency calculated from frequency tables with grouped data?

  • The modal class is the class that has the highest frequency
    • This is for equal class intervals only
  • The median is the middle value
    • The exact value can not be calculated but it can be estimated by using a cumulative frequency graph
  • The exact mean can not be calculated as you do not have the raw data
  • The mean can be estimated by
    • Identifying the mid-interval value (midpoint) xi for each class
    • Multiplying each value by the class frequency fi
    • Summing to get Σfixi
    • Dividing by the total frequency = Σfi
    • This is given in the formula booklet

begin mathsize 22px style x with bar on top equals fraction numerator begin display style sum from i equals 1 to k of end style f subscript i x subscript i over denominator n end fraction end style

    • Your GDC can estimate the mean if you input the mid-interval values and the class frequencies using the statistics mode

How are measures of dispersion calculated from frequency tables with grouped data?

  • The exact range can not be calculated as the largest and smallest values are unknown
  • The interquartile range can be estimated by

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

    • Estimates of the quartiles can be found by using a cumulative frequency graph
  • The standard deviation and variance can be estimated using the mid-interval values xi in the formulae
    • Variance

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

    • Standard deviation

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 be expected to use your GDC to estimate the standard deviation and variance using the mid-interval values
      • You may want to use these formulae to deepen your understanding

Examiner Tip

  • As you can only estimate statistical measures from a grouped frequency table it is good practice to indicate that the values are not exact
    • You can do this by rounding values rather than leaving as surds and fractions
    • x with bar on top equals 0.333 (3sf) rather than x with bar on top equals 1 third

Worked example

The table below shows the heights in cm of a group of 25 students.

Height, h Frequency
150 less or equal than h less than 155 3
155 less or equal than h less than 160 5
160 less or equal than h less than 165 9
165 less or equal than h less than 170 7
170 less or equal than h less than 175 1
a)
Write down the modal class.

4-1-3-ib-ai-aa-sl-grouped-data-a-we-solution

b)
Write down the mid-interval value of the modal class.

4-1-3-ib-ai-aa-sl-grouped-data-b-we-solution

c)
Calculate an estimate for the mean height.

4-1-3-ib-ai-aa-sl-grouped-data-c-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.