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Univariate Data (DP IB Maths: AI HL)
Revision Note
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
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
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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
- Draw a horizontal line from the y-axis to the curve
- 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
- They can be used to construct a box plot for grouped data
Worked example
The cumulative frequency graph below shows the lengths in cm, , of 30 puppies in a training group.
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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, kg | Frequency |
4 | |
15 | |
19 | |
10 | |
6 |
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