Grade 12 Statistics – Comprehensive Notes on Measures of Central Tendency & Foundational Concepts

Virtual Classroom Etiquette

• Be on time; punctuality mirrors professional practice.

• Mute yourself when not speaking to minimize background noise.

• Turn on video to increase engagement and accountability.

• Join from a quiet, stationary location; avoid switching rooms.

• No food or drinks visible on-camera—maintains decorum.

• Come prepared (materials, software, stable connection).

• Show respect in chat and voice interactions.

• Use the “raise hand” feature before speaking; fosters orderly discussion.

Classroom Rules (Physical or Virtual)

• Respect • yourself • teacher • classmates • learning environment
  → Hashtag philosophy: #giverespect

• Arrive on time, stay on task, and be ready daily #beready

• Take responsibility for your learning #yougotthis

• Keep personal electronics put away unless instructed
  → #teamnophonesinclass

• Maintain a positive attitude #bepositive

Statistics: Core Definition

• Statistics = science of collecting, organizing, presenting, analysing, and interpreting data in order to support effective decision-making.

Branches & Functions of Statistics

• Descriptive Statistics
  • Summarises data using tables, graphs, and numerics (mean, median, etc.).

• Inferential Statistics
  • Draws conclusions about a population from a sample (estimation, hypothesis testing).

Types of Data

• Qualitative (Categorical)
  • Non-numeric labels: gender, hair colour, nationality, profession.

• Quantitative (Numerical)
  • Measurable quantities: height, weight, age, salary.

Population vs. Sample

• Population: complete set of individuals/objects/events of interest.

• Sample: subset of the population used for analysis to save cost/time.

Measures of Central Tendency

(“centre” of a data set)

• Mean (Arithmetic Average)
  • $\bar{x}=\dfrac{\sum x}{n}$
  • All observations contribute; sensitive to outliers; quantitative data only.

• Median
  • Middle value after ordering; resistant to outliers; works with ordinal or higher scales.

• Mode
  • Most frequent value; applies to qualitative & quantitative; can be bimodal/multimodal or have no mode.

When to use:

• Data with extreme outliers → choose median or mode.

• Nominal data (e.g., favourite brand) → mode only.

• Symmetric, outlier-free quantitative data → mean preferred.

Pros & Cons Summary

Mean

• Pros: widely used; utilises every value; single unique number.

• Cons: highly affected by extreme values.
  Median

• Pros: robust to extremes; single value.

• Cons: ignores magnitude of all but central observations.
  Mode

• Pros: only choice for nominal data; simple.

• Cons: may be non-unique or non-existent; not reliable for inference.

Computing Central Tendencies (Ungrouped Data)

Mean

1. Sum all raw scores.

2. Divide by count $n$.
   Example: $\bar{x}=\dfrac{12+13+14+15+16}{5}=14$

Median

1. Order data.

2. Locate position $\dfrac{n+1}{2}$.
   • If integer → exact value.
   • If x.5 → average two surrounding values.
   Example (10 scores): position $\dfrac{10+1}{2}=5.5$ → average of 5th & 6th values.

Mode

1. Count frequency of each value.

2. Highest frequency = mode(s).
   • If all equal → no mode.

Example Interpretation (Social-Media Hours)

Data: 5,6,5,4,4,3,1,1,8,7

• Mode = 5 & 4 (bimodal) → most students spend 4-5 h/day.

• Mean ≈ 4.4 h (pulled upward by 8 & 7).

• Median = 4.5 h.
  Choosing median gives typical time without overweighting heavy users.
  Limitation: ignores dual peak (students at 1 h and 4-5 h).

Transition to Grouped Data

• Raw scores consolidated into Class Intervals for large datasets.

Terminology Recap

• Class Interval: e.g., 1–5, 6–10.

• Class Limits: lowest & highest integers in interval.

• Class Size $i$: constant width (upper–lower = 5, etc.).

• Frequency $f$: count per class.

• Class Boundary: true limits (whole-number data ⇒ −0.5 / +0.5).

• Class Mark $m$ (midpoint): $m=\dfrac{LL+UL}{2}$.

Mean (Grouped Data – Midpoint Method)

$\bar{X}=\dfrac{\sum f m}{\sum f}$
Steps:

1. Compute midpoints $m$ for each class.

2. Multiply by frequency $f$, sum products.

3. Divide by total $N$.
   Interpretation: compare individual scores to $\bar{X}$ (e.g., below mean ⇒ needs improvement).

Median (Grouped Data)

Formula:
\tilde{X}=LB+\Bigl(\dfrac{\frac{N}{2}-<cf}{f_m}\Bigr)i ere:

• $LB$ = lower boundary of median class.

• $N$ = total observations.

• <cf = cumulative frequency before median class.

• $f_m$ = frequency of median class.

• $i$ = class size.
  Procedure:

1. Build “less-than” cumulative frequency column.

2. Find first class with cumulative ≥ $\frac{N}{2}$ → median class.

3. Substitute into formula.

Mode (Grouped Data)

Formula:
$\text{Mode}=LB+\Bigl(\dfrac{d<em>1}{d</em>1+d_2}\Bigr)i$

• Modal class: class with highest frequency.

• $d<em>1=f</em>{MC}-f_{above}$

• $d<em>2=f</em>{MC}-f_{below}$

• $LB$, $i$ as defined earlier.

Worked Example (40 Science Scores)

Classes (10-14 … 50-54) with frequencies given.

1. Mean computed as $\bar{X}=33.63$ → benchmark.

2. Median & Mode require cumulative frequencies and differences according to formulas.

3. Interpretation: Students < 33.63 underperformed; > 33.63 excelled.

Conversion: Raw to Grouped Data (Profile-Picture Reactions)

Steps Demonstrated

1. Range = Highest − Lowest = $50-12=38$.

2. Decide number of classes (e.g., k = 6).

3. Class size $i=\lceil \dfrac{\text{Range}}{k}\rceil=\lceil 38/6 \rceil=7$.

4. Construct Frequency Distribution Table (FDT) with 6 classes of width 7.

5. Add lower boundaries, cumulative frequencies (<cf) to enable median computation.

Real-World Data Examples & Relevance

• PHILIPPINE Income Groups (2018): poverty thresholds, household counts, population sizes.
  • Useful for choosing median income due to heavy right-skew (few rich, many poor).

• Global Online Retail Sales (2018-2020): illustrates growing e-commerce share; mean vs. median growth differs across countries.

Measures of Variability (Mentioned for Continuity)

(Though detailed formulas not in transcript, remember they quantify spread.)
Common: Range, Quartile Deviation, Variance $\sigma^2$, Standard Deviation $\sigma$.

Ethical & Practical Considerations

• Choice of central tendency affects policy: e.g., poverty line based on median income avoids distortion by millionaires.

• In education, reporting only mean may hide clusters of struggling/high-achieving students (necessitating mode & distribution plots).

• Data privacy: when collecting classroom statistics, ensure anonymization.

Quick Reference Formulas

• Ungrouped Mean: $\bar{x}=\dfrac{\sum x}{n}$

• Ungrouped Median Position: $\dfrac{n+1}{2}$

• Grouped Mean: $\bar{X}=\dfrac{\sum fm}{N}$

• Grouped Median: LB+\Bigl(\dfrac{N/2-<cf}{f_m}\Bigr)i

• Grouped Mode: $LB+\Bigl(\dfrac{d<em>1}{d</em>1+d_2}\Bigr)i$

Practice Activities (for Self-Study)

1. Decide the best central tendency measure for each dataset (income groups, online sales, etc.).

2. Compute mean/median/mode for:
   • $15,19,37,20,18,17,22,23,17,17$
   • Business-Math grades list (interpret results).

3. Grouped Data Problem: Sport-club weights (40-99 kg) – calculate all three measures and discuss which is most informative.

Key Takeaways

• Central tendency condenses information into a single representative value; selection depends on scale, distribution shape, and presence of outliers.

• Grouped formulas adjust for lost individual detail via class midpoints and boundaries.

• Robust statistical literacy equips Grade 12 learners for evidence-based reasoning in academics, business, and social issues.