4.1.6 Univariate Data

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

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 p^th 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 
  • They can be used to construct a box plot for grouped data

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