Statistics — Measures of Central Tendency

• Computing mean, median, and mode for both ungrouped and grouped data.
• Weighted mean and combined mean for merged datasets.
• Choosing the appropriate measure for different types of data.
• Recognising skewness from the relationship between mean, median, and mode.

Level 1 — Mean, median, mode (ungrouped)

Mean (x̄): Σxᵢ/n — sum of all values divided by count. Affected by outliers.

Median: Middle value when data sorted. For n values: (n+1)/2-th term (odd n); average of n/2-th and (n/2+1)-th (even n).

Mode: Most frequently occurring value. Data can be unimodal, bimodal, or multimodal. Stable to outliers.

Empirical relation (approximately): Mode ≈ 3 Median − 2 Mean (holds for moderately skewed distributions).

Level 2 — Grouped data formulas

Where: L = lower class boundary of median/modal class, h = class width, f = frequency, cf = cumulative frequency before class, f₀, f₁, f₂ = frequencies before, of, and after modal class.

Weighted mean: x̄_w = Σwᵢxᵢ / Σwᵢ (different items have different weights/importance).

Combined mean: If two groups have means x̄₁, x̄₂ and sizes n₁, n₂: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

JEE tip: Assumed mean method reduces calculation effort. Choose A close to data centre; h = class width for step deviation.

NCERT spotlight — Ogive and median graphically

Cumulative frequency curve (ogive): Plot cumulative frequency vs upper class boundary. Draw line at cumulative frequency n/2 → read off median on x-axis. "Less than" ogive and "more than" ogive intersect at median.

Frequency polygon: Connect midpoints of class tops — area equals area of histogram. Used for comparing two distributions visually.

Skewness: Mean > Median > Mode (positive/right skew — tail on right). Mean < Median < Mode (negative/left skew). Symmetric: Mean = Median = Mode.

Find mean, median, and mode for the grouped data: Class: 10-20, 20-30, 30-40, 40-50, 50-60; Frequency: 4, 8, 11, 7, 5.

Step 1 — n = 4+8+11+7+5 = 35; midpoints xᵢ: 15, 25, 35, 45, 55.
Step 2 — Mean (direct): Σfᵢxᵢ = 4×15 + 8×25 + 11×35 + 7×45 + 5×55
         = 60 + 200 + 385 + 315 + 275 = 1235.
         x̄ = 1235/35 = 35.29.
Step 3 — Cumulative frequencies: 4, 12, 23, 30, 35.
         Median class: n/2 = 17.5 → falls in 30-40 (cf=12, f=11).
         Median = 30 + ((17.5 − 12)/11) × 10 = 30 + 5.5 = 35.5 ≈ 35.5.
Step 4 — Modal class: highest frequency = 11 (30-40).
         f₀=8, f₁=11, f₂=7; h=10, L=30.
         Mode = 30 + ((11−8)/(22−8−7)) × 10 = 30 + (3/7)×10 = 30 + 4.3 = 34.3.
Step 5 — Check empirical relation: 3×35.5 − 2×35.29 = 106.5 − 70.6 = 35.9 ≈ 34.3 (rough check ✓).

Applications — average in everyday life

Weighted mean: CGPA uses credit-weighted marks. Combined mean: overall class average from two sections. Mode: most popular shoe size for inventory. Median: income distribution (avoids distortion by billionaires).

• Five numbers: 12, 15, 18, 22, 28. Find mean and median.
• Two classes of 30 and 40 students have means 70 and 75. Find combined mean.
• For grouped data with modal class 40-50, f₀=3, f₁=8, f₂=5, h=10: find mode.

Open the Practice tab for graded questions on Measures of Central Tendency.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.