Measures of central tendency — A/L Combined Mathematics

What must I remember about Measures of central tendency?

• Measures of Central Tendency: Statistical summary metrics identifying the central or typical value around which data cluster: Mean, Median, and Mode.
• Arithmetic Mean x̄: Ungrouped data: x̄ = (∑ x_i) / n; Grouped frequency distribution: x̄ = (∑ f_i * x_i) / (∑ f_i) = (∑ f_i * x_i) / N.
• Assumed Mean method: Let d_i = x_i - A (where A is assumed mean); then x̄ = A + (∑ f_i * d_i) / N; with step-deviation u_i = (x_i - A)/c: x̄ = A + c * (∑ f_i * u_i) / N.
• Median (M_e): The middle observation dividing ordered data into two equal halves (50% below, 50% above); for grouped data: M_e = L + [(N/2 - F) / f_m] * c, where L is lower boundary of median class, F is cumulative frequency preceding median class, f_m is frequency of median class, c is class width.
• Mode (M_o): The most frequently occurring value; for grouped data modal class: M_o = L + [Δ1 / (Δ1 + Δ2)] * c, where Δ1 = f_m - f₁ and Δ2 = f_m - f₂ (f_m = modal class freq, f₁ = preceding, f₂ = succeeding).

Test yourself (1 minute)

1. Sensitivity to Outliers

   Mean is heavily distorted by extreme values / outliers; Median is ROBUST against extreme outliers and preferred for skewed income/economic data.

2. Empirical relationship in skewed distributions

   Mean - Mode ≈ 3 * (Mean - Median).

3. Algebraic property of Mean

   The algebraic sum of deviations of all observations from their arithmetic mean is identically ZERO: ∑ (x_i - x̄) = 0.

Practise Measures of central tendency questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 8

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Year 2 · Term 3