Stats1_Formula_Dissection_Guide

Stats 1: Formula Dissection & Memory Guide 🧠

IMPORTANT

Purpose: Break down EVERY Stats formula piece by piece. Understand what each part means, when to use it, and how to remember it forever.

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📊 Formula Index by Week

Week	Key Formulas	Count
3: Central Tendency	Mean, Weighted Mean, Combined Mean	5
4: Dispersion	Range, Variance, SD, Correlation	8
5: Counting	Permutations, Circular, Factorial	6
6: Combinations	nCr, Binomial Coefficient	4
7: Probability	Basic Rules, Addition, Multiplication	5
8: Conditional	Bayes’ Theorem, Independence	4

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Week 3: Central Tendency

Formula 1: Mean (Simple)

📐 The Formula

$\bar{x}=\frac{\sum x_i}{n}=\frac{x_1+x_2+...+x_n}{n}$

🔍 Anatomy

• $\bar{x}$ (x-bar): The mean/average
• $\sum x_i$: Sum of ALL values
• $n$: Total count of values

🧠 Memory Trick

“Add them all up, divide by how many”

• Total ÷ Count

📝 Practice Questions

1. Find mean of: 2, 4, 6, 8, 10

   • Answer: $(2+4+6+8+10)/5=30/5=6$

2. Mean of 5 numbers is 12. Find sum of numbers.

   • Answer: Sum = Mean × n = $12\times 5=60$

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Formula 2: Weighted Mean

📐 The Formula

$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

🔍 Anatomy

• $f_i$: Frequency (how many times value appears)
• $x_i$: Value
• $f_i{x}_i$: Value × its frequency
• Logic: If 5 appears 10 times, it contributes $5\times 10=50$ to sum

🧠 Memory Trick

“Multiply each value by how many times it appears, then average”

📝 Practice Questions

1. Values: [2, 6, 11], Frequencies: [10, 5, 3]. Find mean.

   • Answer: $\frac{10(2)+5(6)+3(11)}{10+5+3}=\frac{20+30+33}{18}=\frac{83}{18}=4.61$

2. Marks distribution: 60 (5 students), 70 (8 students), 80 (7 students). Find mean.

   • Answer: $\frac{5(60)+8(70)+7(80)}{20}=\frac{300+560+560}{20}=72$

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Formula 3: Combined Mean

📐 The Formula

${\bar{x}}_{combined}=\frac{n_1{\bar{x}}_1+n_2{\bar{x}}_2}{n_1+n_2}$

🔍 Anatomy

• $n_1,n_2$: Sizes of two groups
• ${\bar{x}}_1,{\bar{x}}_2$: Means of two groups
• Logic: Convert means back to totals, add totals, divide by combined count

🧠 Memory Trick

“Weighted average of the two means”

• Bigger group’s mean counts more!

📝 Practice Questions

1. Group A: 10 students, mean 75. Group B: 15 students, mean 80. Combined mean?

   • Answer: $\frac{10(75)+15(80)}{25}=\frac{750+1200}{25}=78$

2. Two classes: 20 students (avg 65), 30 students (avg 70). Combined?

   • Answer: $\frac{20(65)+30(70)}{50}=\frac{1300+2100}{50}=68$

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Formula 4: Correcting the Mean

📐 The Formula

$\text{New Mean}=\frac{\text{Old Total}-\text{Wrong Value}+\text{Correct Value}}{n}$

🔍 Anatomy

• Old Total: Original mean × n
• Correction: Subtract wrong, add correct
• New Mean: Corrected total ÷ n

📝 Practice Questions

1. Mean of 5 numbers is 8. One value 7 was wrongly read as 11. Correct mean?
   • Answer:
     • Old total = $8\times 5=40$
     • Correct total = $40-11+7=36$
     • New mean = $36/5=7.2$

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Week 4: Dispersion & Correlation

Formula 5: Range

📐 The Formula

$\text{Range}=\text{Max}-\text{Min}$

🧠 Memory Trick

“Biggest minus smallest”

📝 Practice Questions

1. Data: 12, 18, 25, 30, 22. Find range.
   • Answer: $30-12=18$

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Formula 6: Variance

📐 The Formula

${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}$

🔍 Anatomy

• $(x_i-\bar{x})$: Deviation (distance from mean)
• $(x_i-\bar{x}{)}^2$: Squared deviation (always positive)
• Average of squared deviations: How spread out on average

🧠 Memory Trick

“Average of squared distances from mean”

⚠️ Common Mistakes

• ❌ Forgetting to square the deviations
• ❌ Using $n-1$ for sample (we use $n$ for population)

📝 Practice Questions

1. Data: 2, 4, 6. Find variance.
   • Mean = 4
   • Deviations: -2, 0, 2
   • Squared: 4, 0, 4
   • Variance = $(4+0+4)/3=8/3=2.67$

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Formula 7: Standard Deviation

📐 The Formula

$\sigma =\sqrt{{\sigma }^2}=\sqrt{\frac{\sum (x_i-\bar{x}{)}^2}{n}}$

🔍 Anatomy

• SD = √Variance
• Units: Same as original data

🧠 Memory Trick

“Square root of variance”

• Variance is squared, SD brings it back to original units

📝 Practice Questions

1. Variance = 16. Find SD.
   • Answer: $\sqrt{16}=4$

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Formula 8: Effect of Transformations on SD

📐 The Rules

• Add/Subtract constant: SD unchanged
  • If $Y=X+c$, then ${\sigma }_Y={\sigma }_X$
• Multiply by constant: SD multiplied by |constant|
  • If $Y=cX$, then ${\sigma }_Y=|c|{\sigma }_X$

🧠 Memory Trick

“Shifting doesn’t change spread, scaling does”

📝 Practice Questions

1. Data has SD = 5. Transform: $Y=2X+10$. Find SD of Y.

   • Answer: ${\sigma }_Y=2\times 5=10$ (addition doesn’t affect SD)

2. Data has SD = 3. Transform: $Y=X-7$. Find SD of Y.

   • Answer: ${\sigma }_Y=3$ (unchanged)

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Formula 9: Correlation Coefficient

📐 The Formula

$r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x}{)}^2\sum (y_i-\bar{y}{)}^2}}$

🔍 Anatomy

• Range: $-1\le r\le 1$
• $r=1$: Perfect positive correlation
• $r=-1$: Perfect negative correlation
• $r=0$: No linear correlation

🧠 Memory Trick

“How much do X and Y move together?”

📝 Practice Questions

1. For transformation $Y=2X+5$, find correlation.

   • Answer: $r=1$ (perfect positive, linear relationship)

2. For $Y=-3X+10$, find correlation.

   • Answer: $r=-1$ (perfect negative)

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Week 5: Counting & Permutations

Formula 10: Factorial

📐 The Formula

$n!=n\times (n-1)\times (n-2)\times ...\times 2\times 1$

🔍 Special Cases

• $0!=1$ (by definition)
• $1!=1$

📝 Practice Questions

1. Calculate 5!

   • Answer: $5\times 4\times 3\times 2\times 1=120$

2. Simplify: $\frac{7!}{5!}$

   • Answer: $\frac{7\times 6\times 5!}{5!}=7\times 6=42$

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Formula 11: Permutations

📐 The Formula

${}^n{P}_r=\frac{n!}{(n-r)!}$

🔍 Anatomy

• $n$: Total items
• $r$: Items to arrange
• Meaning: Number of ways to arrange $r$ items from $n$

🧠 Memory Trick

“Order matters!”

• ABC ≠ BAC ≠ CAB (different arrangements)

📝 Practice Questions

1. How many 3-letter arrangements from {A, B, C, D}?

   • Answer: ${}^4{P}_3=\frac{4!}{1!}=24$

2. 5 people, arrange 2 in a row. How many ways?

   • Answer: ${}^5{P}_2=\frac{5!}{3!}=5\times 4=20$

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Formula 12: Permutations with Repetition

📐 The Formula

$\frac{n!}{n_1!n_2!...n_k!}$

🔍 Anatomy

• $n$: Total items
• $n_1,n_2,...$: Frequencies of repeated items

📝 Practice Questions

1. Arrangements of letters in “BOOK”?

   • 4 letters, O repeats twice
   • Answer: $\frac{4!}{2!}=\frac{24}{2}=12$

2. Arrangements of “STATISTICS”?

   • 10 letters: S(3), T(3), I(2), A(1), C(1)
   • Answer: $\frac{10!}{3!\times 3!\times 2!}=50400$

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Formula 13: Circular Permutations

📐 The Formula

$(n-1)!$

🔍 Anatomy

• Why $(n-1)!$?: Fix one position to avoid counting rotations as different

🧠 Memory Trick

“Fix one person, arrange the rest”

📝 Practice Questions

1. 5 people around circular table. How many arrangements?

   • Answer: $(5-1)!=4!=24$

2. 6 people in a circle. How many ways?

   • Answer: $5!=120$

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Week 6: Combinations

Formula 14: Combinations

📐 The Formula

${}^n{C}_r=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$

🔍 Anatomy

• $n$: Total items
• $r$: Items to choose
• Meaning: Number of ways to SELECT $r$ from $n$ (order doesn’t matter)

🧠 Memory Trick

“Order doesn’t matter!”

• {A, B, C} = {B, A, C} = {C, B, A} (same selection)

📝 Practice Questions

1. Choose 3 students from 5. How many ways?

   • Answer: ${}^5{C}_3=\frac{5!}{3!\times 2!}=\frac{5\times 4}{2}=10$

2. Select 2 books from 7. How many ways?

   • Answer: ${}^7{C}_2=\frac{7\times 6}{2}=21$

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Formula 15: Combination Properties

📐 Key Identities

1. ${}^n{C}_r{=}^n{C}_{n-r}$
2. ${}^n{C}_0{=}^n{C}_n=1$
3. ${}^n{C}_1=n$

📝 Practice Questions

1. Verify: ${}^5{C}_2{=}^5{C}_3$

   • Answer: Both = 10 ✓

2. Find ${}^{10}{C}_0$

   • Answer: 1

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Week 7: Probability

Formula 16: Basic Probability

📐 The Formula

$P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

🔍 Range

• $0\le P(A)\le 1$
• $P(A)=0$: Impossible
• $P(A)=1$: Certain

📝 Practice Questions

1. Coin toss. P(Heads)?

   • Answer: $\frac{1}{2}$

2. Roll die. P(even number)?

   • Answer: $\frac{3}{6}=\frac{1}{2}$

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Formula 17: Complement Rule

📐 The Formula

$P(A^c)=1-P(A)$

🧠 Memory Trick

“Probability of NOT happening = 1 minus probability of happening”

📝 Practice Questions

1. P(rain) = 0.3. Find P(no rain).

   • Answer: $1-0.3=0.7$

2. P(passing) = 0.85. Find P(failing).

   • Answer: $1-0.85=0.15$

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Formula 18: Addition Rule (Mutually Exclusive)

📐 The Formula

$P(A\cup B)=P(A)+P(B)\text{(if A and B cannot happen together)}$

📝 Practice Questions

1. P(red card) = 1/2, P(black card) = 1/2. P(red OR black)?
   • Answer: $\frac{1}{2}+\frac{1}{2}=1$ (mutually exclusive)

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Formula 19: Addition Rule (General)

📐 The Formula

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

🧠 Memory Trick

“Inclusion-Exclusion for probability!”

📝 Practice Questions

1. P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2. Find P(A∪B).
   • Answer: $0.5+0.4-0.2=0.7$

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Formula 20: Multiplication Rule (Independent)

📐 The Formula

$P(A\cap B)=P(A)\times P(B)\text{(if A and B are independent)}$

📝 Practice Questions

1. Flip coin twice. P(two heads)?

   • Answer: $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

2. Roll die twice. P(two sixes)?

   • Answer: $\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$

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Week 8: Conditional Probability

Formula 21: Conditional Probability

📐 The Formula

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

🔍 Anatomy

• $P(A|B)$: Probability of A GIVEN B has occurred
• Numerator: Both happen
• Denominator: B happens (new “universe”)

📝 Practice Questions

1. P(A∩B) = 0.2, P(B) = 0.5. Find P(A|B).
   • Answer: $\frac{0.2}{0.5}=0.4$

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Formula 22: Bayes’ Theorem

📐 The Formula

$P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}$

🔍 Anatomy

• $P(A)$: Prior probability
• $P(B|A)$: Likelihood
• $P(B)$: Evidence

🧠 Memory Trick

“Flip the condition using this formula”

📝 Practice Questions

1. Medical test: P(Disease) = 0.01, P(+|Disease) = 0.95, P(+|No Disease) = 0.05. Find P(Disease|+).
   • P(+) = P(+|D)P(D) + P(+|No D)P(No D) = 0.95(0.01) + 0.05(0.99) = 0.059
   • Answer: $\frac{0.95\times 0.01}{0.059}=0.161$ (16.1%)

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Formula 23: Independence Test

📐 The Formula

Events A and B are independent if: $P(A\cap B)=P(A)\times P(B)$

OR equivalently: $P(A|B)=P(A)$

📝 Practice Questions

1. P(A) = 0.3, P(B) = 0.4, P(A∩B) = 0.12. Are they independent?
   • Check: $0.3\times 0.4=0.12$ ✓
   • Answer: Yes, independent

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🎯 Quick Formula Reference Card

Central Tendency

• Mean: $\bar{x}=\frac{\sum x}{n}$
• Weighted: $\bar{x}=\frac{\sum fx}{\sum f}$
• Combined: $\bar{x}=\frac{n_1{\bar{x}}_1+n_2{\bar{x}}_2}{n_1+n_2}$

Dispersion

• Variance: ${\sigma }^2=\frac{\sum (x-\bar{x}{)}^2}{n}$
• SD: $\sigma =\sqrt{\text{Variance}}$
• Transform: $Y=aX+b$ → ${\sigma }_Y=|a|{\sigma }_X$

Counting

• Permutation: ${}^n{P}_r=\frac{n!}{(n-r)!}$
• Combination: ${}^n{C}_r=\frac{n!}{r!(n-r)!}$
• Circular: $(n-1)!$

Probability

• Basic: $P(A)=\frac{\text{favorable}}{\text{total}}$
• Complement: $P(A^c)=1-P(A)$
• Union: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• Independent: $P(A\cap B)=P(A)P(B)$
• Conditional: $P(A|B)=\frac{P(A\cap B)}{P(B)}$
• Bayes: $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$

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End of Stats 1 Formula Dissection Guide ✅