Descriptive Statistics

Measures of center

For a dataset, "center" can mean different things.

Mean (average): sum the values, divide by the count.

$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

Median: the middle value when the data is sorted. If $n$ is even, average the two middle values.

Mode: the value that appears most often. (A dataset can have multiple modes or none.)

Example. Data: $\{2, 4, 4, 6, 9\}$.

• Mean: $(2+4+4+6+9)/5 = 25/5 = 5$
• Median: middle value is $4$
• Mode: $4$ (appears twice)

Mean vs. median — when does it matter?

The mean is sensitive to outliers. The median isn't.

Salaries at a small company: $\{40k, 42k, 45k, 48k, 500k\}$.

• Mean: $135k$ (misleading — only the CEO earns near that)
• Median: $45k$ (a much more honest sense of the typical employee)

Rule of thumb: use the median when data is skewed or has outliers.

Measures of spread

Range = max − min. Simple but ignores everything in the middle.

Variance ($\sigma^2$ or $s^2$): average squared distance from the mean.

$\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$

(For a sample variance you divide by $n-1$ instead of $n$ — Bessel's correction.)

Standard deviation ($\sigma$ or $s$): the square root of the variance.

$\sigma = \sqrt{\sigma^2}$

Standard deviation lives in the same units as the data, which is why it's the spread measure people quote.

A worked example

Data: $\{2, 4, 4, 4, 5, 5, 7, 9\}$. Mean = $5$.

$x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
2	-3	9
4	-1	1
4	-1	1
4	-1	1
5	0	0
5	0	0
7	2	4
9	4	16

Sum of squared deviations = 32. Variance (population) = $32/8 = 4$. Standard deviation = $\sqrt{4} = 2$.

Key takeaways

• Mean, median, mode all describe the center — choose based on the data.
• Mean is pulled around by outliers; median isn't.
• Standard deviation is the most-used measure of spread.