Quartiles and Comparing Distributions

A distribution’s quartiles are three points that split the data set into four equal parts. The first quartile (Q1) is the median of the lower half, the second quartile (Q2) is the overall median, and the third quartile (Q3) is the median of the upper half.
The Interquartile Range (IQR) is the difference between Q3 and Q1. This statistical measure identifies the middle 50% of values when ordered from smallest to largest. The IQR helps to assess variability and potential outliers.
Comparing distributions of two or more sets can be done through measures of central tendency (mean, median) and measures of spread (range, IQR, standard deviation).
Box-and-whisker plots, or box plots, enable comparison of distributions across different data sets. These graphical displays show five-number summary: minimum, Q1, median, Q3, and maximum.
The shape of a distribution can be discerned from its box plot. A symmetrical distribution has its median midway between Q1 and Q3. If the median is closer to Q1, the distribution is skewed to the right, while if it is nearer Q3, the distribution is skewed to the left.
The concept of outliers is essential when comparing distributions. Outliers are extreme values that lie beyond the ‘whiskers’ in the box plot. They might signal errors, or they might be indicative of variability in data.
Cumulative frequency graphs can also offer insights when comparing distributions. By plotting the cumulative frequency against the upper class boundaries, you can visualise and compare distributions.
Always consider real-life situations, as problems in these papers often contextualise statistical principles. Be clear on how to interpret and compare the information given in various contextual problems.
Practice interpreting quartiles, median, range, and other statistical measures from various types of data presentations such as tables, graphs, or charts, as you may encounter multiple formats in your assessments.