Measures of Central Tendency
Core rules
- Arithmetic mean = Σx/n; for grouped data use midpoints and frequencies, or assumed mean and step-deviation methods.
- Median is the middle value; for grouped data, median = l + ((n/2 − cf)/f)h.
- Mode is the most frequent value; grouped mode = l + ((f1−f0)/(2f1−f0−f2))h.
- Geometric mean GM = (x1...xn)^(1/n), used for rates; harmonic mean HM = n/Σ(1/x), used for averages of ratios.
- Combined mean = (n1x̄1 + n2x̄2)/(n1+n2).
- Empirical relation for moderately skewed data: mode = 3 median − 2 mean.
Worked example
A class has 30 boys with mean mark 60 and 20 girls with mean mark 55. Combined mean = (30×60 + 20×55)/(30+20) = (1800+1100)/50 = 2900/50 = 58. The combined mean lies between 55 and 60 and is closer to 60 because boys are more numerous.
Traps
- Using the wrong frequencies in the median formula; cumulative frequency must be less-than type.
- Forgetting to adjust class boundaries for inclusive classes.
- Treating mode as unique; a distribution can be bimodal.
- Using the mean for open-end classes without stating an assumption.
- Confusing combined mean with a simple average of the two means: (60+55)/2 = 57.5 is wrong here.