1
Range & Interquartile Range
Measures of spread between extremities and quartiles:
$$\text{Range} = \text{Max} - \text{Min}$$
$$\text{IQR} = Q_3 - Q_1$$
IQR is resistant to outliers, unlike range.
2
Variance & Standard Deviation
Average squared deviation from the mean $\bar{x}$:
$$\sigma^2 = \frac{\sum x^2}{N} - (\bar{x})^2$$
$$\sigma = \sqrt{\sigma^2}$$
3
Effects of Data Transformation
When every data value undergoes an arithmetic operation:
Add constant $k$ ($x + k$): Mean becomes $\bar{x} + k$. Dispersion (Range, IQR, $\sigma$) remains UNCHANGED!
Multiply by $k$ ($kx$): Mean becomes $k\bar{x}$, Range $\times k$, IQR $\times k$, $\sigma \times k$, Variance $\times k^2$!
SPM Box-Plot Interpretation: Identifying Outliers & Skewness
Outlier Limits: A value $x$ is an extreme outlier if:
$$x < Q_1 - 1.5(\text{IQR}) \quad \text{or} \quad x > Q_3 + 1.5(\text{IQR})$$
Skewness:
• If median line is closer to $Q_1$, the distribution is positively skewed (skewed right).
• If median line is centered, the distribution is symmetric.
• If median line is closer to $Q_3$, the distribution is negatively skewed (skewed left).
• If median line is closer to $Q_1$, the distribution is positively skewed (skewed right).
• If median line is centered, the distribution is symmetric.
• If median line is closer to $Q_3$, the distribution is negatively skewed (skewed left).