Statistics Reference

Statistics and Data Displays

Descriptive statistics summarize a data set; displays and standardized scores reveal its shape, spread, and unusual values.

Solved Examples

Jump directly to a worked example by summary measure or data-display task.

Measure	Meaning	Use It When
Mean	Arithmetic average.	Data are roughly balanced without extreme outliers.
Median	Middle ordered value.	Data are skewed or include outliers.
Interquartile range	$IQR=Q_3-Q_1$.	You need the spread of the middle half.
Z-score	Number of standard deviations from the mean.	You need to compare a value to a distribution.
Correlation	Direction and strength of a linear association.	You are reading a scatter plot, not proving cause.

Mean

$$\bar{x}=\frac{\text{sum of values}}{\text{number of values}}$$

Range and IQR

$$\text{range}=\max-\min,\qquad IQR=Q_3-Q_1$$

Z-score

$$z=\frac{x-\mu}{\sigma}$$

Correlation describes association, not causation. A third variable can influence both quantities in a scatter plot.

Find the mean of $2,4,4,6,9$.

Solution Steps

$$2+4+4+6+9=25$$$$n=5$$$$\boxed{\bar{x}=\frac{25}{5}=5}$$

For $1,2,3,4,5,6,7$, find the median, $Q_1$, and $Q_3$.

Solution Steps

$$\text{median}=4$$$$\{1,2,3\}\Rightarrow Q_1=2$$$$\{5,6,7\}\Rightarrow\boxed{Q_3=6}$$

A score of $80$ comes from a distribution with mean $70$ and standard deviation $5$. Find its z-score.

Solution Steps

$$80-70=10$$$$z=\frac{10}{5}=2$$$$\boxed{\text{The score is 2 standard deviations above the mean.}}$$

As weekly study time increases, quiz scores generally increase in a scatter plot. Describe the association.

Solution Steps

$$\text{points trend upward from left to right}$$$$\boxed{\text{positive correlation}}$$$$\text{association does not by itself prove causation}$$