U11 Statistical Charts (I) - Essential Formulas

This section covers the fundamental concepts and formulas for representing and interpreting univariate data using various statistical charts, including stem-and-leaf diagrams, histograms, and cumulative frequency polygons.

1 Stem-and-Leaf Diagrams

Structure and Interpretation

A stem-and-leaf diagram displays the shape of a data distribution while preserving the original data values. Each data point is split into a stem (leading digit(s)) and a leaf (trailing digit). For example, the value $58$ has stem $5$ and leaf $8$. The key is essential: $5 | 8$ represents $58$.

$$ \text{Data value} = (\text{Stem} \times 10) + \text{Leaf} $$
Stem | Leaf 3 | 2 5 7 4 | 0 1 3 8 5 | 2 4 6 8 9 6 | 1 3 5 7 | 0 4 Key: 5 | 2 = 52

2 Histograms and Frequency Density

Frequency Density for Unequal Class Intervals

For a histogram, the area of each bar represents the frequency of that class. When class intervals are unequal, the height of the bar is the frequency density.

$$ \text{Frequency Density} = \frac{\text{Frequency of Class}}{\text{Class Width}} $$

The total frequency is equal to the total area of all bars in the histogram.

3 Cumulative Frequency Polygons (Ogive)

Plotting and Quartiles

A cumulative frequency polygon is drawn by plotting the upper class boundary against the cumulative frequency. Points are joined by straight lines. It is used to estimate medians, quartiles, and percentiles.

$$ Q_1 \text{ at } \frac{n}{4}, \quad \text{Median } (Q_2) \text{ at } \frac{n}{2}, \quad Q_3 \text{ at } \frac{3n}{4} $$

where $n$ is the total cumulative frequency. The interquartile range (IQR) is $IQR = Q_3 - Q_1$.

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now