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1. Quartile Deviation (Q.D.)

Q.D. = (Q3 - Q1) / 2
Coefficient of Q.D. = (Q3 - Q1) / (Q3 + Q1)

Class Position Calculation: Q1 Class = (N/4)-th item | Q3 Class = (3N/4)-th item

Q = L + [ (Position - cf) / f ] × i

Required Table Setup:

Class Interval (C.I.) Frequency (f) Cumulative Frequency (cf)
Continuous Range Given value Successive row totals (f₁ + f₂ + ...)
Total N = Σf ---

2. Mean Deviation (M.D.)

M.D. (from Mean) = Σf|m - X̄| / N
M.D. (from Median) = Σf|m - Me| / N

Required Table Setup (Applies to both Mean / Median paths):

Class Interval Frequency (f) Midpoint (m) fm cf (For Median path only) Absolute Deviation |m - Center| Product f|m - Center|
Continuous Range Given values (Lower+Upper)/2 f × m Cumulative row tally Positive value absolute difference f × |m - Center|
Total N = Σf --- Σfm --- --- Σf|m - Center|

3. Standard Deviation (σ) Master Framework

There are four distinct tactical calculation methods used to evaluate Standard Deviation in continuous collections:

A. The Direct Method

σ = √[ (Σfm² / N) - (Σfm / N)² ]
Class Interval Frequency (f) Midpoint (m) fm m² fm²
Data Range Given row Mid Value f × m m × m f × m²
Total N = Σf --- Σfm --- Σfm²

B. The Actual Mean Method

σ = √[ Σfx² / N ]     (where x = m - X̄)
Class Interval Frequency (f) Midpoint (m) fm Deviation x = (m - X̄) x² fx²
Data Range Given row Mid Value f × m Value minus true mean Squared divergence f × x²
Total N = Σf --- Σfm --- --- Σfx²

C. The Assumed Mean Method

σ = √[ (Σfd² / N) - (Σfd / N)² ]     (where d = m - A)
Class Interval Frequency (f) Midpoint (m) d = m - A fd d² fd²
Data Range Given row Mid Value Midpoint minus chosen A f × d d × d f × d²
Total N = Σf --- --- Σfd --- Σfd²

D. The Step Deviation Method

σ = √[ (Σfd'² / N) - (Σfd / N)² ] × i     (where d' = (m - A) / i)
Class Interval Frequency (f) Midpoint (m) d' = (m - A) / i fd' (d')² f(d')²
Data Range Given row Mid Value Scaled Step Deviation f × d' d' × d' f × (d')²
Total N = Σf --- --- Σfd' --- Σfd'²

4. Box and Whisker Plot

A Box and Whisker Plot is a visual graph used to show data distribution based on a Five-Number Summary.

The Five-Number Summary:

  1. Minimum Value (Min): The lowest value in the dataset.
  2. Lower Quartile (Q1): Splits the lowest 25% of data.
  3. Median (Q2): The middle value splits the data at 50%.
  4. Upper Quartile (Q3): Splits the highest 25% of data.
  5. Maximum Value (Max): The highest value in the dataset.

Visual Structure:

Min Q1 Median Q3 Max |-----------[============|============]-----------| <--- Interquartile Range ---> (IQR)

Key Metrics & Outlier Boundaries

  • Interquartile Range (IQR): Measures the width of the box.
    IQR = Q3 - Q1
  • Outliers Lower Boundary: Any score below this is an outlier.
    Lower Fence = Q1 - (1.5 × IQR)
  • Outliers Upper Boundary: Any score above this is an outlier.
    Upper Fence = Q3 + (1.5 × IQR)