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

Discrete vs. Continuous Data: Discrete data consists of isolated values (like the number of students), while continuous data can take any value within a range (like height). On a graph, discrete data is often represented by bar charts with gaps, whereas continuous data is shown using histograms where bars are adjacent to represent the numerical continuum.

Measures of Central Tendency: These describe the center of a data set. The mean ($\bar{x}$) is the arithmetic average, the median ($M$) is the middle value when data is ordered, and the mode is the most frequent value. Visually, in a perfectly symmetrical 'Bell Curve' or Normal Distribution, the mean, median, and mode all overlap at the highest central peak.

Measures of Dispersion: These describe the spread of the data. The Range is the difference between the maximum and minimum values. The Interquartile Range ($IQR$) measures the spread of the middle 50% of the data. Standard Deviation ($\sigma$) measures how much values typically deviate from the mean. A low $\sigma$ results in a steep, narrow curve, while a high $\sigma$ creates a flat, wide curve.

The Box-and-Whisker Plot: A visual summary of the 'Five-Number Summary': Minimum, lower quartile ($Q_{1}$), median ($Q_{2}$), upper quartile ($Q_{3}$), and maximum. It consists of a central box spanning from $Q_{1}$ to $Q_{3}$ with a line at the median, and 'whiskers' extending to the minimum and maximum values (excluding outliers).

Cumulative Frequency Curves: Also known as an 'Ogive', this is a line graph where the y-axis represents the running total of frequencies. The curve typically follows an 'S-shape'. By drawing a horizontal line from $50\%$ of the total frequency to the curve and then down to the x-axis, one can visually estimate the median.

Outliers and Skewness: An outlier is an extreme value that differs significantly from others, often identified if it is more than $1.5 \times IQR$ away from the quartiles. Visually, data is 'Positively Skewed' (right-skewed) if it has a long tail on the right side, meaning the mean is typically greater than the median.

Grouped Data and Mid-interval Values: When data is given in intervals (e.g., $10 \le x < 20$), we use the mid-interval value ($x_{m}$) to estimate the mean. This assumes data is uniformly distributed within each class. On a histogram, the area of each bar represents the frequency of that specific class interval.