Univariate Data (DP IB Maths: AI HL)

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Box Plots

Univariate data is data that is in one variable.

What is a box plot (box and whisker diagram)?

  • A box plot is a graph that clearly shows key statistics from a data set
    • It shows the median, quartiles, minimum and maximum values and outliers
    • It does not show any other individual data items
  • The middle 50% of the data will be represented by the box section of the graph and the lower and upper 25% of the data will be represented by each of the whiskers
  • Any outliers are represented with a cross on the outside of the whiskers
    • If there is an outlier then the whisker will end at the value before the outlier
  • Only one axis is used when graphing a box plot
  • It is still important to make sure the axis has a clear, even scale and is labelled with units

2-2-2-box-plot-diagram-1

What are box plots useful for?

  • Box plots can clearly show the shape of the distribution
    • If a box plot is symmetrical about the median then the data could be normally distributed
  • Box plots are often used for comparing two sets of data
    • Two box plots will be drawn next to each other using the same axis
    • They are useful for comparing data because it is easy to see the main shape of the distribution of the data from a box plot
      • You can easily compare the medians and interquartile ranges

Examiner Tip

  • In an exam you can use your GDC to draw a box plot if you have the raw data
    • You calculator's box plot can also include outliers so this is a good way to check

Worked example

The distances, in metres, travelled by 15 snails in a one-minute period are recorded and shown below: 

0.5,   0.7,   1.0,   1.1,   1.2,   1.2,   1.2,   1.3,   1.4,   1.4,   1.4,   1.4,   1.5,   1.5,   1.5           

a)
i)
Find the values of Q subscript 1 comma space Q subscript 2 and Q subscript 3.
ii)
Find the interquartile range.
iii)
Identify any outliers.

4-1-6-ib-ai-aa-sl-box-plots-a-we-solution

b)
Draw a box plot for the data.

4-1-6-ib-ai-aa-sl-box-plots-b-we-solution

                         

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Cumulative Frequency Graphs

What is cumulative frequency?

  • The cumulative frequency of x is the running total of the frequencies for the values that are less than or equal to x
  • For grouped data you use the upper boundary of a class interval to find the cumulative frequency of that class

What is a cumulative frequency graph?

  • A cumulative frequency graph is used with data that has been organised into a grouped frequency table
  • Some coordinates are plotted
    • The x-coordinates are the upper boundaries of the class intervals
    • The y-coordinates are the cumulative frequencies of that class interval
  • The coordinates are then joined together by hand using a smooth increasing curve

What are cumulative frequency graphs useful for?

  • They can be used to estimate statistical measures
    • Draw a horizontal line from the y-axis to the curve
      • For the median: draw the line at 50% of the total frequency
      • For the lower quartile: draw the line at 25% of the total frequency
      • For the upper quartile: draw the line at 75% of the total frequency
      • For the pth percentile: draw the line at p% of the total frequency
    • Draw a vertical line down from the curve to the x-axis
    • This x-value is the relevant statistical measure
  • They can used to estimate the number of values that are bigger/small than a given value
    • Draw a vertical line from the given value on the x-axis to the curve
    • Draw a horizontal line from the curve to the y-axis
    • This value is an estimate for how many values are less than or equal to the given value
      • To estimate the number that is greater than the value subtract this number from the total frequency
    • They can be used to estimate the interquartile range IQR equals Q subscript 3 minus Q subscript 1
    • They can be used to construct a box plot for grouped data

Worked example

The cumulative frequency graph below shows the lengths in cm, l, of 30 puppies in a training group.

cumulative-frequency-graph-2-2-2

a)
Given that the interval 40 less or equal than l less than 45 was used when collecting data, find the frequency of this class.

4-1-6-ib-ai-aa-sl-cum-freq-a-we-solution

b)
Use the graph to find an estimate for the interquartile range of the lengths.

4-1-6-ib-ai-aa-sl-cum-freq-b-we-solution

c)
Estimate the percentage of puppies with length more than 51 cm.

4-1-6-ib-ai-aa-sl-cum-freq-c-we-solution

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Histograms

What is a (frequency) histogram?

  • A frequency histogram clearly shows the frequency of class intervals
    • The classes will have equal class intervals
    • The frequency will be on the y-axis
    • The bar for a class interval will begin at the lower boundary and end at the upper boundary
  • A frequency histogram is similar to a bar chart
    • A bar chart is used for qualitative or discrete data and has gaps between the bars
    • A frequency histogram is used for continuous data and has no gaps between bars

What are (frequency) histograms useful for?

  • They show the modal class clearly
  • They show the shape of the distribution
    • It is important the class intervals are of equal width
  • They can show whether the variable can be modelled by a normal distribution
    • If the shape is symmetrical and bell-shaped

Worked example

The table below and its corresponding histogram show the mass, in kg, of some new born bottlenose dolphins.

Mass, m kg Frequency
4 less or equal than m less than 8 4
8 less or equal than m less than 12 15
12 less or equal than m less than 16 19
16 less or equal than m less than 20 10
20 less or equal than m less than 24 6
a)
Draw a frequency histogram to represent the data.

4-1-6-ib-ai-aa-sl-histogram-a-we-solution

b)
Write down the modal class.

4-1-6-ib-ai-aa-sl-histogram-b-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.