Stats 1 Week 4: Dispersion & Correlation

0. Prerequisites

NOTE

What you need to know:

• Square Roots: $\sqrt{16}=4$.
• Graphing: Plotting $(x,y)$ points.
• Linear Equation: $y=mx+c$ (Slope and Intercept).

Quick Refresher

• Dispersion: Spread of data. (Is it clustered or scattered?).
• Correlation ($r$): Strength of linear relationship between two variables.
• Variance (${\sigma }^2$): Average squared distance from Mean.
• Standard Deviation ($\sigma$): Square root of Variance.

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1. Core Concepts

1.1 Measures of Dispersion

1. Range: Max - Min. (Very sensitive to outliers).
2. Inter-Quartile Range (IQR): $Q3-Q1$. (Robust against outliers).
   • Outlier Rule: Any value outside $[Q1-1.5\times IQR,Q3+1.5\times IQR]$.
3. Standard Deviation (SD):
   • Population SD ($\sigma$): Divide by $N$.
   • Sample SD ($s$): Divide by $n-1$ (Bessel’s Correction).
   • Why $n-1$? To make it an “unbiased estimator” (better guess) of the population.

1.2 Correlation ($r$)

• Range: $-1\le r\le 1$.
• Interpretation:
  • $r=1$: Perfect Positive (Line going up).
  • $r=-1$: Perfect Negative (Line going down).
  • $r=0$: No Linear relationship (Could still be curved!).
• Effect of Scale:
  • Adding constant ($x+a$): No change in $r$.
  • Multiplying positive constant ($ax$): No change in $r$.
  • Multiplying negative constant ($-ax$): Sign flips (becomes $-r$).

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2. Pattern Analysis & Goated Solutions

Pattern 1: Effect of Operations on Variance/SD

Context: “Variance of data is 25. If we multiply every number by 4 and add 7, what is new Variance?”

TIP

Mental Algorithm:

1. Addition (+7): IGNORE. Shifting data doesn’t change spread.
2. Multiplication ($\times 4$):
   • New SD = Old SD $\times |4|$.
   • New Variance = Old Variance $\times 4^2$.

Example (Detailed Solution)

Problem: Var = 25. Op: $4x+7$. Solution:

1. Ignore +7.
2. Scale Factor: 4.
3. New Var: $25\times 4^2=25\times 16=400$. Answer: 400.

Pattern 2: Finding Outliers

Context: “Data: 8, 17, 15, 19, 21, 25, 23, 35. Find outliers.”

TIP

Mental Algorithm:

1. Sort: 8, 15, 17, 19, 21, 23, 25, 35.
2. Find Q1, Q3:
   • $n=8$. Q1 is avg of 2nd/3rd (15, 17) $\to$ 16.
   • Q3 is avg of 6th/7th (23, 25) $\to$ 24.
3. Calculate IQR: $24-16=8$.
4. Calculate Fences:
   • Lower: $16-1.5(8)=16-12=4$.
   • Upper: $24+1.5(8)=24+12=36$.
5. Check: Any value $<4$ or $>36$?
   • Min is 8 (Safe). Max is 35 (Safe).
   • Result: 0 Outliers.

Pattern 3: Correlation Changes

Context: “$r(X,Y)=0.45$. New variables: $U=X+0.1$, $V=2Y$. Find $r(U,V)$.”

TIP

Mental Algorithm:

1. Check Signs: Are multipliers positive or negative?
   • $X$ mult by 1 (Pos). $Y$ mult by 2 (Pos).
2. Result: Signs match $\to$ $r$ stays same.
   • If one was negative $\to$ $r$ flips sign.

Example (Detailed Solution)

Problem: $r=0.45$. $U=X+0.1$, $V=2Y$. Solution:

1. Shift (+0.1): Ignore.
2. Scale (2): Positive.
3. Result: $r$ remains 0.45. Answer: 0.45.

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3. Practice Exercises

1. Variance: If SD is 5, what is Variance?
   • Hint: $5^2=25$.
2. Scaling: SD is 10. Multiply data by -3. New SD?
   • Hint: SD is always positive. $10\times |-3|=30$.
3. Correlation: $r(x,y)=0.8$. Find $r(x,-y)$.
   • Hint: $y$ flipped sign. $r=-0.8$.

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🧠 Level Up: Advanced Practice

Question 1: Correlation under Transformation

Problem: Salespersons A and B. $B=2A+1000$. Correlation coefficient ($r$)? Logic:

1. Linear Relation: $Y=aX+b$.
2. Property: If $a>0$, $r=1$. If $a<0$, $r=-1$.
3. Here: $a=2$ (Positive). Answer: $r=1$. (Perfect positive correlation).

Question 2: Variance Shift

Problem: Variance of dataset is 50. If we add 4 to every number, what is new variance? Logic:

1. Shift Property: Adding a constant ($+c$) shifts the distribution but does not change the spread.
2. Result: Variance remains same. Answer: 50. Note: If we multiplied by 2, variance would become $2^2\times 50=200$.

Question 3: Interpreting Scatter Plots

Problem: Daily screen time vs Sleep duration. Negative trend. Interpretation:

• As screen time increases, sleep duration decreases.
• Correlation: Negative ($r<0$).
• Causation?: Correlation does not imply causation, but there is an association.