Describing Center and Spread

Manual: General · Subject: Statistics

Compute and compare summary statistics that describe the typical value and variability of data.

Numerical Summaries

Mean, median, and mode

The mean uses every value and is sensitive to extreme observations. The median is the middle value and is resistant to outliers. The mode identifies the most frequent value and is useful for categorical or repeated data.

Key formulas

Statistic	Formula	Interpretation
Sample mean	$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$	Average of sample values
Sample variance	$s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2$	Average squared deviation from the mean
Sample standard deviation	$s=\sqrt{s^2}$	Typical distance from the mean
Range	$\max(x)-\min(x)$	Total spread from smallest to largest

ℹ️

Why use $n-1$ ?

The denominator $n-1$ in sample variance corrects the tendency of sample deviations to underestimate population variability.

Measures of spread

Range

Easy to compute
Uses only two values
Very sensitive to outliers

IQR

Spread of middle 50%
Resistant to outliers
Best paired with the median

Choosing a summary

1

Step 1: Inspect the distribution for skewness and outliers.
2

Step 2: Use mean and standard deviation for roughly symmetric data.
3

Step 3: Use median and IQR for skewed data or data with outliers.
4

Step 4: Compare groups using the same summaries for consistency.

Which summary is most resistant to outliers?

Mean Median Standard deviation Range

If two data sets have the same mean but one has a larger standard deviation, what does that mean?

Neuroanatomy Explorer

Memory Deck

Match the Pairs

Concept Constellation

Describing Center and Spread

Numerical Summaries

Mean, median, and mode

Measures of spread

Choosing a summary