Organization and Presentation of Data

1. DEFINITIONS AND NOTATIONS

• Population

• A population is the complete collection of objects, items, individuals, or elements from which information is gathered.
• It can be finite or infinite.
• “Population” is not restricted to humans; it refers to any set of entities subject to statistical study (machines, plants, accidents, etc.).

• Object (Individual, Element)

• The smallest unit in the population on which information is actually collected.

• Character (Variable, Attribute)

• The common aspect of interest that is measured on every object.
• Nature of a character:
  • Qualitative (Categorical) – non-numerical values (e.g., car color, gender).
  • Quantitative (Numerical) – numerical values:
    • Discrete – countable, usually integers (e.g., number of children in a family, daily car accidents).
    • Continuous – can take any real value within an interval (e.g., height, amount of rainfall).

• Illustrative Examples

1. Daily car-accident study (Tripoli)

   • Population $(M)$: All car accidents reported in Tripoli.
   • Object: A single accident.
   • Characters:
     • Number of injured (discrete quantitative).
     • Severity level (qualitative).

2. Hospital survey

   • Population: All patients seen in hospital $H$.
   • Object: One patient.
   • Characters:
     • Age, Length (continuous quantitative).
     • Gender, Profession (qualitative).

• Sample

• Observing all individuals may be impractical. Instead we select a subset of size $n$ where $n \ll \text{Card}(M)$.
• A sample inherits the same characters as the population.

▸ Ordered Sample

• Observations are arranged in ascending order by the character of interest.
  Example (Age): Nada 35.5 < Jamal 40 < Samir 68.

▸ Exhaustive Sample (Without Replacement)

• After each observation the individual is removed and cannot reappear in the sample (e.g., today’s car accidents).

2. FREQUENCY DISTRIBUTION TABLES

• Frequency Definitions

• Absolute frequency (effective): ${n<em>i}$ – number of objects with modality $x</em>i$.
• Relative frequency: ${f<em>i = \dfrac{n</em>i}{n}}$.
• Percentage: ${P<em>i = f</em>i \times 100}$ with $\sum P_i = 100$.
• Cumulative absolute frequency: ${N<em>i = \sum</em>{j=1}^{i} n_j}$.
• Cumulative relative frequency: ${F<em>i = \sum</em>{j=1}^{i} f<em>j = \dfrac{N</em>i}{n}}$ (ranges $0 \le F_i \le 1$).

• Frequency Tables

▸ Discrete Quantitative Character

Example (children per family, sample size 10):

▸ Continuous Quantitative Character

• Sort data ascending.
• Create $k$ disjoint classes $[L<em>i, L</em>{i+1}[\,, i=1..k$.
  • Width (amplitude): ${a<em>i = L</em>{i+1}-L_i}$.
  • Class center: ${c<em>i = \dfrac{L</em>i + L_{i+1}}{2}}$.
• Record ${n<em>i, N</em>i, f<em>i, F</em>i}$ as above.

Example (12 body temperatures, $a=1$):

3. GRAPHICAL REPRESENTATION

3.1 Qualitative Characters

• Bar Chart: One rectangle per modality $x<em>i$, uniform width; height $\propto n</em>i$ or $f_i$.
• Pie (Circular) Chart: Slice angle $\alpha<em>i = f</em>i \times 360^\circ = \dfrac{n_i}{n} \times 360^\circ$.
  Example (10 accidents): Not serious 6, Serious 3, Very serious 1 → $\alpha = 216^\circ,108^\circ,36^\circ$.

3.2 Discrete Quantitative Characters

• Bar Chart (vertical segments on $x_i$).
• Frequency Polygon: Connect tops of bars with straight lines.
• Cumulative Frequency Polygon: Plot $N<em>i$ (or $F</em>i$) vs $x_i$, join by segments; step function rising from 0 to $n$ (or 1).

3.3 Continuous Quantitative Characters

• Histogram: Rectangle width $a<em>i$, height $n</em>i^<em>$ or $f<em>i^</em>$ where
  $n</em>i^* = \dfrac{n<em>i}{a</em>i}\; a$ and $f<em>i^* = \dfrac{f</em>i}{a_i}\; a$ (corrected for unequal widths, with reference amplitude $a$).
  If all classes share the same width, corrected and raw frequencies coincide.
• Frequency Polygon: Connect class-center tops of histogram.
• Cumulative Frequency Curve (Ogive): Continuous increasing curve of $N(x)$ or $F(x)$ from 0 up to $n$ (or 1). Independent of class widths.

4. PRACTICE / EXERCISE OVERVIEW

• Newborn Weights (50 observations)

• Classes $[2;2.4[, [2.4;2.8[, [2.8;3.2[, [3.2;3.6[, [3.6;4[\,$ with effectives 6,10,20,10,4.
• Tasks: Define population/character, compute #(\text{weight}\ge3.1\,\text{kg}).

• University Math Grades (25 scores)

• Build frequency table with $a = 5$, then draw histogram, frequency polygon, cumulative curve.

• Medical Degrees in France (1970-84)

| Profession | Diplomas | % Women |
| Doctors | 106 759 | 38% |
| Pharmacists | 43 924 | 60% |
| Dentists | 25 965 | 36% |
| Midwives | 8 215 | 100% |

• Nature: Mixed (quantitative count & percentage).
• Plot bar/pie chart; compute female counts = \text{Total} \times \text{% Women} and graph.

• Wheat Ear Heights

• Frequency table given (9 height ranges).
• Determine population, character (continuous quantitative).
• Compute $f<em>i, F</em>i$; draw histogram, cumulative curve; answer queries:
  i) $P(h\le34\text{ cm})$, ii) $P(34\le h\le66\text{ cm})$, iii) P(h>48\text{ cm}).

• Temperature-Class Distribution

• Classes with unequal widths; build histogram using corrected effectives, calculate cumulative frequencies, graph ogive.

5. KEY TAKE-AWAYS & CONNECTIONS

• Always define Population → Object → Character before data handling.
• Sampling reduces cost but demands clarity (ordered/exhaustive).
• Frequencies (absolute/relative/percentage/cumulative) are the backbone of descriptive statistics.
• Graph choice depends on character type:
  • Qualitative → Bar or Pie.
  • Discrete quantitative → Bar, polygon.
  • Continuous quantitative → Histogram, frequency polygon, ogive.
• Corrected heights in histograms guarantee area proportional to frequency when class widths differ.
• Cumulative functions $N(x),F(x)$ are monotone increasing, right-continuous; useful for medians, quartiles, percentiles.
• Each graphical method furnishes a visual insight into distribution shape, central tendency, spread, and outliers.