8.1 Dispersion
Concept of Dispersion
Dispersion measures how spread out a set of numerical values is relative to its central value (such as the mean or median).
- Small Dispersion: Data values are clustered tightly around the central value.
- Large Dispersion: Data values are widely spread out from the central value.
Stem-and-Leaf Plot and Dot Plot
- Stem-and-Leaf Plot: Displays data distribution retaining individual data values divided into a stem (leading digit) and leaf (trailing digit).
- Dot Plot: Shows the frequency of individual data points plotted along a continuous number line.
8.2 Measures of Dispersion
Range and Interquartile Range
- Range: Difference between the largest value and the smallest value.
$$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$$
- Quartiles ($Q_1, Q_2, Q_3$):
- First Quartile ($Q_1$): The median of the lower half of the dataset ($25\%$ point).
- Second Quartile ($Q_2$ / Median): The middle value ($50\%$ point).
- Third Quartile ($Q_3$): The median of the upper half of the dataset ($75\%$ point).
- Interquartile Range (IQR): Difference between the third quartile and the first quartile.
$$\text{Interquartile Range} = Q_3 - Q_1$$
Variance and Standard Deviation
These measures evaluate how far each data point in a set lies from the mean ($\bar{x}$).
- Mean ($\bar{x}$):
$$\bar{x} = \frac{\sum x}{N}$$
- Variance ($\sigma^2$): The mean of the squared differences from the mean.
$$\sigma^2 = \frac{\sum (x - \bar{x})^2}{N} = \frac{\sum x^2}{N} - \bar{x}^2$$
- Standard Deviation ($\sigma$): The square root of variance, measuring spread in original data units.
$$\sigma = \sqrt{\frac{\sum x^2}{N} - \bar{x}^2}$$
Box Plot and Outliers
- A Box Plot visually displays a 5-number summary: $\text{Minimum, } Q_1, \text{Median } (Q_2), Q_3, \text{ and Maximum}$.
- Outliers: Extreme data values that lie significantly outside the main pattern.
$$\text{Lower Outlier Boundary} = Q_1 - 1.5 \times (Q_3 - Q_1)$$
$$\text{Upper Outlier Boundary} = Q_3 + 1.5 \times (Q_3 - Q_1)$$
Comparing Sets of Data
- A smaller standard deviation or interquartile range indicates that the data is more consistent or uniform.
- When extreme values (outliers) are present, the interquartile range is preferred over the range or standard deviation as a measure of dispersion because it is unaffected by outliers.