Statistics: Measures of Central Tendency and Dispersion
Statistics: Measures of Central Tendency and Dispersion tests summarising data and interpreting spread. Expect mean by direct, assumed-mean and step-deviation methods, median and mode of grouped data, range, and variance or standard deviation.
Core rules
- Mean = Σfx/Σf for grouped data; step deviation uses d=(x−A)/h and mean=A+h(Σfd/Σf).
- Median class is where cumulative frequency reaches n/2; median=l+[n/2−cf]/f×h.
- Mode=l+[(f₁−f₀)/(2f₁−f₀−f₂)]×h.
- Range=max−min; variance=Σf(x−x̄)²/Σf; standard deviation is its positive square root.
- Class marks are midpoints; use them for x in grouped calculations.
Worked example
For classes 0–10, 10–20, 20–30 with frequencies 5, 8, 7, n=20. Cumulative frequencies are 5, 13, 20; n/2=10 falls at 10–20, so median class is 10–20. With l=10, cf=5, f=8, h=10: median=10+[(10−5)/8]×10=16.25.
Traps
Do not use class limits as x-values in grouped mean; use class marks. The median class is not always the modal class. In step deviation, keep the sign of d. For standard deviation, do not forget the square root at the end, and do not divide by n−1 unless the question specifies sample standard deviation.