Stats1_Week3_Central_Tendency

Stats 1 Week 3: Measures of Central Tendency & Position

0. Prerequisites

NOTE

What you need to know:

• Summation ($\sum$): Adding up a list of numbers.
• Algebra: Solving for $x$ in equations like $\frac{10+x}{2}=8$.
• Sorting: Arranging numbers from smallest to largest (Crucial for Median!).

Quick Refresher

• $\bar{x}$: Sample Mean.
• $\mu$: Population Mean.
• $n$: Sample size.
• $N$: Population size.

---

1. Core Concepts

1.1 Measures of Central Tendency (The “Center”)

1. Mean (Average): $\frac{\sum x_i}{n}$.
   • Sensitive to Outliers: One big number pulls the mean towards it.
2. Median (Middle): The middle value when sorted.
   • Odd $n$: The exact middle number.
   • Even $n$: Average of the two middle numbers.
   • Robust: Not affected by outliers.
3. Mode (Most Frequent): The value that appears most often.
   • Can be Unimodal (1), Bimodal (2), or Multimodal.

1.2 Measures of Position

1. Percentiles ($P_k$): Value below which $k\%$ of data falls.
   • Formula: Index $i=\frac{k}{100}\times n$.
   • If $i$ is integer: Average of $i$-th and $(i+1)$-th values.
   • If $i$ is decimal: Round UP to next integer.
2. Quartiles: Splits data into 4 chunks.
   • $Q1=P_{25}$ (25th Percentile).
   • $Q2=P_{50}$ (Median).
   • $Q3=P_{75}$ (75th Percentile).

---

2. Pattern Analysis & Goated Solutions

Pattern 1: Finding Missing Frequency ($x$)

Context: “Data: 2, 6, 11. Frequencies: $x+6,x+2,x-3$. Mean is 5.63. Find $x$.”

TIP

Mental Algorithm:

1. Table: Make columns for $x_i$ and $f_i$.
2. Product: Calculate $f_i\cdot x_i$.
3. Sums: Find $\sum f_i$ (Total Count) and $\sum f_i{x}_i$ (Total Sum).
4. Equation: Set $\frac{\text{Total Sum}}{\text{Total Count}}=\text{Given Mean}$.
5. Solve: Solve for $x$.

Example (Detailed Solution)

Problem: Values 2, 6. Frequencies $x,x$. Mean = 4. Solution:

1. Sums:
   • Count = $x+x=2x$.
   • Sum = $2(x)+6(x)=8x$.
2. Equation: $\frac{8x}{2x}=4$.
   • $4=4$. (This example is trivial, but shows the logic. Usually you get an equation like $\frac{10x+5}{2x}=4$).

Pattern 2: Correcting the Mean (Wrong Data Entry)

Context: “Mean of 10 observations is 20. One value 15 was wrongly noted as 5. Find correct mean.”

TIP

Mental Algorithm:

1. Old Sum: $\text{Old Mean}\times n$.
2. Adjust Sum: $\text{Old Sum}-\text{Wrong Value}+\text{Correct Value}$.
3. New Mean: $\frac{\text{New Sum}}{n}$.

Example (Detailed Solution)

Problem: $n=10$, Mean=20. Wrong=5, Correct=15. Solution:

1. Old Sum: $20\times 10=200$.
2. Adjust: $200-5+15=210$.
3. New Mean: $210/10=21$. Answer: 21.

Pattern 3: Effect of Linear Transformation ($y=ax+b$)

Context: “Mean of $x$ is 10. Median is 8. If $y=2x+3$, find new Mean and Median.”

TIP

Mental Algorithm:

• Mean/Median/Mode: They follow the exact same change.
  • $\text{New Mean}=2(\text{Old Mean})+3$.
  • $\text{New Median}=2(\text{Old Median})+3$.
• Note: This does NOT apply to Standard Deviation (Week 4).

Example (Detailed Solution)

Problem: Mean=10. Transform $y=2x+3$. Solution:

1. Plug in: $2(10)+3$.
2. Calc: $20+3=23$. Answer: New Mean = 23.

Pattern 4: Finding Percentiles

Context: “Data: 10, 20, 30, 40, 50. Find 30th Percentile ($P_{30}$).”

TIP

Mental Algorithm:

1. Sort: Ensure data is sorted (Crucial!).
2. Index: $i=\frac{k}{100}\times n$.
3. Decision:
   • Integer?: Take average of $i$-th and $(i+1)$-th.
   • Decimal?: Round UP to next integer. Take that value.

Example (Detailed Solution)

Problem: Data: 10, 20, 30, 40, 50 ($n=5$). Find $P_{30}$. Solution:

1. Index: $0.30\times 5=1.5$.
2. Decimal: Round UP to 2.
3. Value: 2nd value is 20. Answer: 20.

---

3. Practice Exercises

1. Correction: Mean of 5 items is 10. Added 2 to every item. New Mean?
   • Hint: $10+2=12$.
2. Median: Data: 5, 2, 9. Median?
   • Hint: Sort first! 2, 5, 9. Median is 5.
3. Percentile: $n=100$. Index for $P_{25}$?
   • Hint: $0.25\times 100=25$ (Integer). Average of 25th and 26th.

---

🧠 Level Up: Advanced Practice

Question 1: Finding ‘x’ from Mean

Problem: Frequencies are $x+6,x+2,x-3,x$ for values 2, 6, 11, 14. Mean is 5.63. Find $x$. Logic:

1. Total Frequency ($N$): $(x+6)+(x+2)+(x-3)+x=4x+5$.
2. Sum of Values ($\sum fx$):
   • $2(x+6)+6(x+2)+11(x-3)+14(x)$
   • $=2x+12+6x+12+11x-33+14x$
   • $=33x-9$.
3. Equation: $\frac{33x-9}{4x+5}=5.63$.
4. Solve:
   • $33x-9=5.63(4x+5)=22.52x+28.15$.
   • $10.48x=37.15$.
   • $x\approx 3.54$.
5. Constraint: Frequency must be integer? “Enter next highest integer” → 4. Answer: 4.

Question 2: Correcting the Mean

Problem: Mean of 6 obs is 19. One obs 11 was wrongly noted as 7. Correct Mean? Logic:

1. Old Sum: $19\times 6=114$.
2. Correction: Remove wrong (7), Add correct (11). Net change $+4$.
3. New Sum: $114+4=118$.
4. New Mean: $118/6=19.67$. Answer: 19.67.

Question 3: Combined Mean

Problem: Section A (15 students, avg 32), Section B (25 students, avg $x$). Combined avg 34. Logic:

1. Weighted Avg: $\frac{15(32)+25(x)}{15+25}=34$.
2. Solve:
   • $480+25x=34(40)=1360$.
   • $25x=1360-480=880$.
   • $x=880/25=35.2$. Answer: 35.2.