Statistics and Probability - Representation of data: frequency tables, histograms, and cumulative frequency graphs

Frequency Tables for Grouped Data: These tables organize continuous data into intervals or 'classes' (e.g., $10 \le x < 20$). Visually, the table consists of a column for the class intervals and a column for the frequency ($f$), which represents how many data points fall within each range.

Histograms: Unlike bar charts, histograms are used for continuous data and have no gaps between bars. The horizontal $x$-axis represents the continuous scale (e.g., time, mass), and the area of each bar is proportional to the frequency. For bars of equal width, the height of the bar represents the frequency.

Frequency Density: When class widths are unequal in a histogram, the height of the bar is the frequency density rather than the frequency. Visually, this ensures that the area ($width \times height$) correctly represents the frequency, preventing wider classes from looking artificially 'larger' than narrow ones.

Cumulative Frequency: This is a 'running total' of frequencies. To calculate it, you add each class frequency to the sum of all previous frequencies. In a table, this results in a non-decreasing sequence of values ending at the total number of data points ($n$).

Cumulative Frequency Graphs (Ogives): This graph is created by plotting the cumulative frequency on the $y$-axis against the upper class boundary of each interval on the $x$-axis. The points are connected with a smooth S-shaped curve or straight lines, starting from the lower boundary of the first class at zero frequency.

Median and Quartiles from Graphs: To find the median ($Q_2$) visually, locate the $\frac{n}{2}$ position on the $y$-axis, move horizontally to the curve, and then vertically down to the $x$-axis. Similarly, the Lower Quartile ($Q_1$) is at $\frac{n}{4}$ and the Upper Quartile ($Q_3$) is at $\frac{3n}{4}$.

Interquartile Range (IQR) and Box Plots: The IQR is the difference between $Q_3$ and $Q_1$, representing the spread of the middle $50\%$ of the data. This data can be visually summarized in a Box Plot, where a central box spans from $Q_1$ to $Q_3$ with a line at the median, and 'whiskers' extend to the minimum and maximum values.

Estimated Mean from Grouped Data: Because exact values are unknown in grouped tables, the mean is estimated using the midpoint ($x$) of each class. Visually, we assume all data points in a bar are concentrated at the center of that bar's interval.