Summary of "Measure of Dispersion | Dispersion & its types | Range & coefficient of range | Standard deviation"

Measure of Dispersion (Range, Coefficient of Range, Standard Deviation)

Overview

This lecture covers measures of dispersion: what dispersion is, why it matters, how to compute range and coefficient of range, and how to compute standard deviation (and related measures) for individual, discrete (frequency), and continuous (class-interval) data. Several worked examples are shown and practical tips are given for exam questions and pharmaceutical-statistics contexts.

Main ideas and concepts

Dispersion (variability): the amount of change or spread in a dataset. More variation → higher dispersion; less variation → lower dispersion.
Importance: helps with data understanding, analysis, interpretation, visualization, communication and decision-making.
Types (qualitative): high dispersion (wide spread) and low dispersion (narrow spread).
Common causes: variability in data collection, measurement error, natural population variability, extreme values/outliers.

Formulas, methods and step-by-step procedures

1. Range

Definition: range = largest value − smallest value.
Individual values: identify max and min, subtract.
Class intervals with incorrectly specified limits (e.g., 31–40, 41–50 instead of continuous boundaries):
- Convert class limits to class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit (example: 31 → 30.5, 40 → 40.5).
- Then range = largest boundary − smallest boundary.
- Note: the 0.5 offset is the usual convention for integer class widths; adjust the offset according to the precision of your data if different.
Examples:
- Individual: max 25, min 9 → range = 16.
- Interval: max 70, min 10 → range = 60.
- Incorrect interval corrected: 90.5 − 30.5 → range = 60.

2. Coefficient of range

Formula: coefficient of range = (largest − smallest) / (largest + smallest).
Examples:
- (25 − 9) / (25 + 9) = 16 / 34 ≈ 0.47
- (70 − 10) / (70 + 10) = 60 / 80 = 0.75

3. Standard deviation (population-type formula used in lecture)

Basic formula (individual series, exact mean):
- σ = sqrt( Σ(x − x̄)^2 / n )
- x̄ is the mean, n is the number of observations.
- This is the population standard deviation (division by n). The sample version (division by n − 1) is not covered in the lecture.
Procedure (individual series):
1. Compute mean x̄ = Σx / n.
2. Compute deviations d_i = x_i − x̄.
3. Square them: d_i^2.
4. Sum squares: Σd_i^2.
5. Divide by n and take square root: σ = sqrt(Σd_i^2 / n).
Worked example:
- x = {11, 14, 15, 17, 18}
- x̄ = (11 + 14 + 15 + 17 + 18) / 5 = 15
- Σ(x − x̄)^2 = 30 → σ = sqrt(30 / 5) = sqrt(6) ≈ 2.45
Coefficient of variation (CV):
- CV (%) = (σ / x̄) × 100
- Example: CV = (2.45 / 15) × 100 ≈ 16.33%

4. Assumed-mean method (when mean x̄ is a decimal)

Use an assumed mean a to simplify arithmetic when the mean is non-integer:
- For each observation compute d = x − a.
- Compute Σd and Σd^2.
- Formula used in the lecture:
  - σ = sqrt( (Σd^2 / n) − (Σd / n)^2 )
- This is algebraically equivalent to σ = sqrt(Σ(x − x̄)^2 / n) but often easier to compute.
Worked example:
- Ten x values with x̄ = 49.4 → choose a = 48
- Σd = 14, Σd^2 = 1146, n = 10
- Σd^2 / n = 114.6; (Σd / n) = 1.4; (Σd / n)^2 = 1.96
- Variance = 114.6 − 1.96 = 112.64 → SD = sqrt(112.64) ≈ 10.61
- Variance = σ^2, so variance here = 112.64

5. Discrete and continuous series (frequencies included)

Discrete series (with frequencies f):
- Replace Σ(x − x̄)^2 with Σ f (x − x̄)^2 and divide by n = Σf.
Continuous series (class intervals):
- Use class midpoints as x values; include frequencies f.
- If mean is fractional, use the assumed-mean method with d = midpoint − a and include f.
- n = Σf (total frequency).
Key point: formulas are the same except each term is weighted by frequency f and n is the total frequency.

Practical tips and exam advice

Always copy the data carefully; transcription errors produce wrong results.
Decide the question type by what is provided:
- Only x values → individual series; n = number of x values.
- x with f values → discrete series; n = Σf.
- x in intervals with f → continuous series; use midpoints and n = Σf.
Use the assumed-mean method when the mean is not a convenient integer to simplify computation.
Adjust class limits to class boundaries (subtract 0.5 and add 0.5 for integer data) before computing range or midpoints if intervals are not continuous.
The lecture uses population formulas (division by n); check whether your exam or application requires sample formulas (division by n − 1).

Terminology recap

Dispersion = spread/variability in data.
Range = max − min.
Coefficient of range = (max − min) / (max + min).
Standard deviation, σ = sqrt(Σ(x − x̄)^2 / n).
Variance = σ^2.
Coefficient of variation = (σ / x̄) × 100.

Speakers / sources featured

Primary lecturer (unnamed instructor).
Examples / referenced persons and sources:
- AB de Villiers (example batsman A)
- Ravindra Jadeja (example batsman B)
- Depth of Biology (application / lecture series referenced)
- Course/program contexts mentioned: BBA, BCA, Class 11, B.Tech, B.Pharmacy