One-Variable Data

Variable type

Categorical data place observations into groups. Quantitative data record numerical amounts. Quantitative variables can be discrete counts or continuous measurements.

The variable type determines meaningful graphs and statistics. Averaging category codes is not meaningful.

Measures of centre

The mean is the arithmetic balance point, the median the middle ordered value, and the mode the most frequent value.

Mean uses every numerical value and is sensitive to outliers. Median is resistant and often better for skewed distributions. Mode is useful for common categories or values.

Measures of spread

Range is maximum minus minimum. Interquartile range is

Standard deviation measures spread around the mean. Pair median with IQR for skewed/outlier-heavy data and mean with standard deviation for roughly symmetric data without strong outliers.

Worked example

Always order data before finding median and quartiles.

Common mistakes

Calculating median before ordering data. Order first.

Using a histogram for categorical data. Use bars with separated categories.

Reporting mean alone for a skewed distribution. Include shape and resistant spread.

Calling every distant value an error. Investigate potential outliers.

Generalizing from a biased sample because the graph looks clear. Design matters.

Quick self-check

• Is the variable categorical or quantitative, discrete or continuous?
• Does the display match the variable type?
• What are shape, centre, spread, and unusual features?
• Are mean/SD or median/IQR the better pair?
• Were quartiles and possible outliers found consistently?
• Do comparison and generalization claims respect sample size and collection method?