Statistics: Notes और MCQ Practice

आसान भाषा में:

Statistics का उद्देश्य है data को collect, organise, summarize और interpret करना ताकि सही decisions ले सकें. Mean, Median, Mode, Range, Variance, Standard Deviation जैसे measures हमें data की central tendency और spread बताते हैं.

Important Formulas / Rules

• Mean = (Σ x) ÷ n
• Median = middle value (sorted data)

(if n even → average of two middle values)

• Mode = most frequent value
• Range = Max − Min
• Variance (σ²) = Σ (x − Mean)² ÷ n
• Standard Deviation (σ) = √Variance
• Percentage change = (New − Old) ÷ Old × 100

Quick Tricks

1. Mean shortcut – अगर data में कई बार same number repeat हो रहा है, तो (sum of distinct numbers × frequency) ÷ n से जल्दी निकालें.
2. Median without full sorting – अगर data already grouped, तो cumulative frequency > n/2 वाले class का lower bound + [(n/2 − CF_before) ÷ f] × class width.

Common Mistakes

• Confusing population vs sample variance – population uses ÷ n, sample uses ÷ (n‑1).
• Ignoring units – variance की unit square होती है, इसलिए answer को interpret करते समय √Variance (Standard Deviation) लेना याद रखें.

Statistics

1. Concept

Statistics deals with collection, organization, summarization and interpretation of data. In SSC exams the focus is on:

• Measures of central tendency: Mean, Median, Mode
• Measures of dispersion: Range, Variance, Standard Deviation
• Weighted mean for data with different frequencies
• Basic Probability (simple events, complementary, independent)
• Permutations & Combinations (nPr, nCr) for counting arrangements.

2. Important formulas / rules / facts

• Mean = (Sum of all observations) ÷ (Number of observations)
• Median = middle value after arranging data in ascending order (if even number, average of two middle values)
• Mode = value occurring most frequently
• Range = Largest value – Smallest value
• Variance (σ²) = Σ (x – Mean)² ÷ N (for population)
• Standard Deviation (σ) = √Variance
• Weighted Mean = (Σ (value × weight)) ÷ (Σ weight)
• Probability of an event = (Number of favourable outcomes) ÷ (Total number of equally likely outcomes)
• Complementary rule: P(not A) = 1 – P(A)
• nPr = n! ÷ (n‑r)!
• nCr = n! ÷ [r! × (n‑r)!]
• Factorial shortcut: n! = n × (n‑1) × … × 1

3. Solved examples (exam style)

Example 1 – Simple mean

Find the mean of the numbers: 12, 15, 20, 25, 28.

Solution

1. Sum = 12+15+20+25+28 = 100
2. Number of observations = 5
3. Mean = 100 ÷ 5 = 20

Example 2 – Weighted mean

A student scored 78 in English (weight 2) and 85 in Mathematics (weight 3). Find the overall mean score.

Solution

1. Σ(value × weight) = 78×2 + 85×3 = 156 + 255 = 411
2. Σ weight = 2 + 3 = 5
3. Weighted Mean = 411 ÷ 5 = 82.2

Example 3 – Probability

A bag contains 4 red, 5 blue and 6 green balls. One ball is drawn at random. What is the probability that it is not blue?

Solution

1. Total balls = 4+5+6 = 15
2. Favourable (not blue) = 4 red + 6 green = 10
3. P(not blue) = 10 ÷ 15 = 2 ÷ 3 ≈ 0.6667

(or use complement: P(blue)=5/15=1/3, so P(not blue)=1‑1/3=2/3)

Example 4 – Combination

In how many ways can a committee of 3 persons be formed from 7 candidates?

Solution

Use nCr = 7C3 = 7! ÷ (3!×4!) = (7×6×5) ÷ (3×2×1) = 35 ways.

Example 5 – Median of grouped data

The following frequencies are given for marks obtained by 20 students:

Marks — 0‑20 — 21‑40 — 41‑60 — 61‑80 — 81‑100

Frequency — 2 — 4 — 8 — 4 — 2

Find the median.

Solution

1. Cumulative frequencies: 2, 6, 14, 18, 20.
2. N/2 = 20/2 = 10 → median lies in the 3rd class (41‑60) because cumulative frequency just before it is 6.
3. L = lower limit of median class = 41
4. f = frequency of median class = 8
5. CF = cumulative frequency before median class = 6
6. w = class width = 20

Median = L + [(N/2 – CF) ÷ f] × w

= 41 + [(10‑6) ÷ 8] × 20

= 41 + (4/8)×20 = 41 + 0.5×20 = 41 + 10 = 51

So median = 51 marks.

4. Shortcuts and tricks

• Mean shortcut: When numbers are in an arithmetic series, mean = (first + last) ÷ 2.
• Weighted mean quick: If weights sum to 100, treat the numerator as a percentage directly.
• Probability complement: It is often faster to compute the complement (e.g., “not A”) than to count favourable cases.
• nCr without full factorial: Cancel common terms. Example 12C5 = (12×11×10×9×8) ÷ (5×4×3×2×1).
• Median of even set: Instead of adding two middle numbers then dividing, use average of the two numbers directly.
• Variance shortcut: σ² = (Σx² ÷ N) – (Mean)². Compute Σx² once and reuse.

5. Common mistakes to avoid

• Forgetting to arrange data in order before finding median.
• Mixing up nCr (order not important) with nPr (order important).
• Using percentage directly without converting to decimal in probability problems.
• Ignoring the class width while calculating median for grouped data.
• Adding frequencies twice when computing weighted mean.
• Assuming independence when events are actually dependent; always check the wording.