Statistics and Probability - Descriptive Statistics (Measures of Central Tendency and Dispersion)

Types of Data: Statistical data is classified as either Discrete (countable values like the number of students) or Continuous (measurable values like height or time). Visually, discrete data is often represented by bar charts with gaps between bars, whereas continuous data is shown using histograms where bars are adjacent to represent a continuous scale of intervals.

Measures of Central Tendency: These include the Mean (arithmetic average $\bar{x}$), Median (the middle value of an ordered set), and Mode (the most frequent value). In a perfectly symmetrical normal distribution curve, these three measures coincide at the center peak. In skewed distributions, the mean is pulled toward the tail: if the tail stretches to the right (positive skew), the mean is typically greater than the median.

Measures of Dispersion - Range and IQR: Dispersion describes how spread out the data is. The Range is the difference between the maximum and minimum values. The Interquartile Range ($IQR$) represents the middle $50\%$ of the data. Visually, this is seen in a Box-and-Whisker plot where the central box spans from the first quartile ($Q_1$) to the third quartile ($Q_3$), and the length of this box represents the $IQR$.

Standard Deviation and Variance: The Standard Deviation ($\sigma$) measures the average distance of each data point from the mean. A small $\sigma$ indicates data points are clustered closely around the mean, resulting in a narrow, tall bell curve. A large $\sigma$ indicates high variability, resulting in a wider, flatter curve. Variance is simply the square of the standard deviation ($\sigma^2$).

Cumulative Frequency and Ogives: Cumulative frequency is the running total of frequencies. When plotted against the upper class boundaries, it forms an 'S-shaped' curve known as an ogive. This visual tool is used to estimate the median (at the $50\%$ mark on the y-axis), quartiles (at $25\%$ and $75\%$), and percentiles by drawing a horizontal line to the curve and then a vertical line down to the x-axis.

Outliers and Box Plots: An outlier is an extreme value that lies significantly outside the general pattern of the data. Mathematically, an outlier is often defined as any value less than $Q_1 - 1.5 \times IQR$ or greater than $Q_3 + 1.5 \times IQR$. On a box plot, these are visually identified as individual points or crosses ($x$) beyond the ends of the whiskers.

Grouped Data Analysis: For data presented in intervals, the mean is estimated using the mid-interval values ($x_i$). Visually, the frequency density in a histogram (where $Frequency Density = \frac{Frequency}{Class Width}$) ensures that the area of each bar is proportional to the frequency, which is particularly important when class widths are unequal.