U30 Uses and Abuses of Statistics: Essential Formulas

Truncated Y-Axis

Starting the vertical axis at a value $y_0 > 0$ exaggerates small differences. The visual ratio of two bars with heights $h_1$ and $h_2$ becomes $\frac{h_1 - y_0}{h_2 - y_0}$, which is not equal to the true ratio $\frac{h_1}{h_2}$.

Choosing a Favorable Measure of Central Tendency

For a data set $x_1, x_2, \dots, x_n$, the mean $\bar{x}$, median, and mode can differ significantly, especially in skewed distributions. The mean is sensitive to extreme values (outliers).

If a data set is $\{1, 2, 2, 3, 100\}$, the mean is $21.6$, while the median is $2$. Using the mean here would be misleading.

Pearson's Correlation Coefficient

A high correlation $r$, close to $+1$ or $-1$, does not imply causation. The formula for the correlation coefficient is:

A third, lurking variable may cause the relationship. Remember: "$r$ measures linear association, not cause."

Non-Representative Samples

If a sample of size $n$ is drawn from a population of size $N$, but the sampling frame excludes a subgroup, estimates like the sample proportion $\hat{p}$ will be biased. The true population parameter $p$ satisfies $p = \frac{X}{N}$, but the biased estimate $\hat{p}_{b}$ is $\frac{X_s}{n}$, where $X_s$ is from a non-representative sample.

Leading questions can shift responses. For example, "Do you support the necessary policy A?" vs. "Do you support policy A?" yield different response proportions.