Statistics and Probability - Data Presentation (Histograms, Box Plots, Cumulative Frequency)

Histograms: These are used for continuous data where the area of each bar represents the frequency. If class widths are unequal, we must plot Frequency Density on the y-axis instead of frequency. Visually, a histogram with unequal class widths will have bars of different widths, and the height is adjusted so that $Area = Frequency$. The calculation used is $Frequency Density = \frac{Frequency}{Class Width}$.

Cumulative Frequency Curves (Ogives): This graph represents the running total of frequencies. We plot the cumulative frequency on the y-axis against the upper class boundary of each interval on the x-axis. Visually, this results in a smooth S-shaped curve. It is used to estimate the median ($50\%$ of the data), quartiles ($25\%$ and $75\%$), and percentiles by reading across from the y-axis to the curve and then down to the x-axis.

Box-and-Whisker Plots: A visual summary of the five-number summary: Minimum, Lower Quartile ($Q_{1}$), Median ($Q_{2}$), Upper Quartile ($Q_{3}$), and Maximum. The 'box' spans from $Q_{1}$ to $Q_{3}$, representing the Interquartile Range ($IQR$), with a vertical line inside marking the median. 'Whiskers' extend to the minimum and maximum values that are not outliers.

Outliers and the $1.5 \times IQR$ Rule: Outliers are extreme values that fall significantly outside the main body of data. Mathematically, a value is an outlier if it is less than $Q_{1} - 1.5 \times IQR$ or greater than $Q_{3} + 1.5 \times IQR$. Visually, outliers are represented on a box plot as individual points (dots or crosses) beyond the whiskers.

Data Skewness: Skewness refers to the symmetry of the data distribution. In a box plot, if the median line is closer to $Q_{1}$, the data is 'positively skewed' (tail to the right). If the median is closer to $Q_{3}$, it is 'negatively skewed' (tail to the left). Visually, on a histogram, positive skew shows a 'hump' on the left and a long tail stretching to the right.

Frequency Polygons: A frequency polygon is a line graph used to represent the shape of a frequency distribution. It is created by joining the midpoints of the tops of the bars in a histogram with straight lines. Visually, it provides a simplified view of the data's peak and spread, allowing for easy comparison between multiple datasets on the same axes.