Measures of Position in Statistics

Measures of Position (Partition Values)

• Partition values divide data into equal parts: Median, Quartiles, Deciles, Percentiles.
• Median splits data into two equal halves.
• Quartiles divide data into four equal parts, identified by Q1 and Q3.
• Deciles split data into ten equal segments.
• Percentiles segment data into one hundred parts.
• Formulas for partition values involve the rank, lower boundary, cumulative frequency, and respective fractions.

Measures of Dispersion

• Dispersion indicates the extent of variation from a central value.
• Key measures of dispersion include Range, Quartile Deviation, Mean Deviation, and Standard Deviation.

Range

• Range is the difference between the maximum (H) and minimum (L) data values: R = H - L.
• Coefficient of Range gauges variability and is unit-independent: Coefficient of R = (H - L) / (H + L).
• Range is less reliable as it only accounts for extremes.

Quartile Deviation

• Quartile Deviation formula: Q = (Q3 - Q1) / 2.
• Coefficient of Quartile Deviation connects quartile values: Coefficient of Q = (Q3 - Q1) / (Q3 + Q1).
• Focuses on 50% of data, making it less reliable for overall variability.

Mean Deviation

• Mean Deviation from a central value (A): MD(A) = (1/n) Σ( fi |xi - A |) / N.
• Can derive from mean, median, or mode; least when calculated from median.
• Coefficient of Mean Deviation enhances usability: Coefficient of MD(A) = MD(A) / A.

Standard Deviation (σ)

• Formula: σ = √[(1/n) Σ(fi (xi - x̄)²)] / N.
• Coefficient of Standard Deviation is normalized: Coefficient of σ = σ / x̄.
• Effective in handling variability, overcoming sign issues present in Mean Deviation.

Variance

• Variance (σ²) measures the square of the standard deviation: σ² = (1/N) Σ(fi (xi - x̄)²).
• Coefficient of Variation = (σ / x̄) × 100 % indicates relative variability between different data series.

Combining Standard Deviations

• For pooled standard deviation of two series:
• Formula: σ = √[ (n1σ₁² + n2σ₂²) / (n1 + n2)] where n1 and n2 denote sizes, x1 and x2 are means, σ1 and σ2 are individual standard deviations.

Shortcut Method for Standard Deviation

• Particularly beneficial for large values.
• Define di as (xi - A) / h for simplification, leading to the equation for required computations.