Statistics and Probability - Data presentation (histograms, cumulative frequency, box plots)

Histograms with Unequal Class Widths: Unlike bar charts, the area of each bar in a histogram represents the frequency. For continuous data with unequal class widths, we plot Frequency Density on the vertical y-axis. The bars are drawn touching each other to show the continuous nature of the data, and the height of each bar is determined by the frequency density.

Cumulative Frequency (CF): A cumulative frequency table shows a running total of frequencies. To construct a CF curve (ogive), points are plotted using the upper class boundary for the x-coordinate and the cumulative frequency for the y-coordinate. Visually, this creates a smooth 'S-shaped' curve that starts at the lower boundary of the first class interval with a frequency of 0.

Quartiles and the Median: The median ($Q_2$) is the middle value, found at the $50\%$ mark of the total frequency. The Lower Quartile ($Q_1$) is the $25\%$ mark, and the Upper Quartile ($Q_3$) is the $75\%$ mark. On a cumulative frequency graph, these are found by drawing a horizontal line from the calculated y-value (e.g., $0.25n$) to the curve, and then a vertical line down to the x-axis.

The Interquartile Range (IQR): The $IQR$ is a measure of statistical dispersion, representing the range of the middle $50\%$ of the data. It is calculated as the difference between the upper and lower quartiles. Unlike the range, it is not affected by extreme outliers, making it a more robust measure of spread.

Box-and-Whisker Plots: This visual tool summarizes the 'five-number summary': Minimum, $Q_1$, Median, $Q_3$, and Maximum. It consists of a central box spanning from $Q_1$ to $Q_3$, a vertical line through the box at the Median, and horizontal 'whiskers' extending to the minimum and maximum values. It clearly shows the symmetry or skewness of the distribution.

Identifying Outliers: Outliers are extreme values that fall significantly outside the rest of the data. In IB Mathematics, an outlier is formally defined as any value smaller than $Q_1 - 1.5 \times IQR$ or larger than $Q_3 + 1.5 \times IQR$. On a box plot, these are often plotted as individual points or crosses beyond the whiskers.

Interpreting Distribution Shape: A distribution is 'Positively Skewed' if the 'tail' or whiskers extend further to the right (higher values), and the median is closer to the left of the box. It is 'Negatively Skewed' if the tail extends further to the left (lower values) and the median is closer to the right of the box.