Measures of Central Tendency

Mode

  • Definition: For nominal variables, the mode is the most frequent category (the category with the largest frequency N).

  • Purpose: Provides a quick summary of the type of case that is typical in the sample/population.

  • Formula/idea: The mode is the category with the maximum frequency.

  • Example (regional deaths): modal region = Middle East & North Africa with absolute frequency 10,819.

Median

  • Definition: The middle score in a distribution; for ordinal data it is the middle category; for interval/ratio data it is the value that splits the distribution in half.

  • How to find (ordinal):

    • Step 1: Arrange values from low to high.

    • Step 2: Determine the observation that splits the distribution.

  • Key formula: Position of the median (for ordered data) = extMedianposition=N+12ext{Median position} = \frac{N+1}{2}.

  • Examples:

    • Aussie Respondents, 2018 (ordinal): (1783+1)/2 = 892; 892 falls in category 5 (Neutral) → Median = Neutral.

    • Aussie Respondents, 2018 (second dataset): (1798+1)/2 = 899.5; 899 (and 900) fall in the category “probably should have the right” → Median = “probably should have the right.”

    • Distance travelled by offenders (interval/ratio): (21+1)/2 = 11th score; 11th score = 5.09 km → Median = 5.09 km.

  • Note: For ordinal data, the median identifies the central category; for interval/ratio data, it’s a numeric value.

Mean

  • Definition: Uses all data points; accounts for frequency and position of scores.

  • Mean =xˉ=n∑i=1nxi​

  • Examples:

    • Incarceration history (example data): Sum = 109, n = 32 → xˉ=10932=3.40625 .\bar{x} = \frac{109}{32} = 3.40625\,.

    • Dollars stolen (example data): Sum = 2745, n = 13 → xˉ=274513=211.15 .\bar{x} = \frac{2745}{13} = 211.15\,.

  • Caution: Extreme values can skew the mean.

Choosing the Right Measure

  • If the variable is categorical/nominal: use mode.

  • If numeric with extreme scores: use median.

  • If numeric with no extreme scores: use mean.

Scale items and measurement nuance

  • Very controversial: ordinal items with many categories can be treated as interval/ratio if there are at least 7 response categories.

  • In that case, the mean is acceptable to report.

  • Example (7-point or larger scale): Euthanasia justifiable? 1 = Never, 10 = Always; Median = 8; Mean = 6.69 (ordinal but treated as interval/ratio).

Excel tip

  • You can compute mode, median, and mean in Excel using the Descriptive Statistics Toolpak.

  • Example output (for reference): Mode = 2; Median = 2; Mean = 3.40625 (for a given dataset).

Quick recap

  • Mode = most frequent category (nominal data).

  • Median = middle value/category (ordinal or interval/ratio data; use the position (N+1)/2).

  • Mean = average of all data points; sensitive to extreme values.

  • Choose: mode (nominal), median (numeric with outliers), mean (numeric without outliers).

Summary

  • Know how to determine the mode, median, and mean.

  • Know when each measure is suitable/appropriate.