Variance, Standard Deviation, and Standardization

📹 Video Overview

📊 What We're Learning

• How to measure spread/dispersion in data

• Variance and Standard Deviation

• How to compare different datasets

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🎯 Core Concept: Measuring Spread

The Big Idea: Average alone isn't enough! Two datasets can have the same average but look totally different.

Example to Remember:

• Dataset 1: 9, 9, 9, 9, 9 (average = 9)

• Dataset 2: 1, 4, 9, 12, 19 (average = 9)

Both have the same average, but Dataset 2 is way more spread out!

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📐 Variance (s²)

What It Measures

How far, on average, each data point is from the mean (but squared).

Formula

$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

Memory Hack: "The Three Steps Dance"

1. Subtract: Find how far each value is from the mean (xᵢ - x̄)

2. Square: Make all differences positive (xᵢ - x̄)²

3. Average: Add them up and divide by n

Why Square?

• Gets rid of negative values (otherwise they'd cancel out!)

• Emphasizes extreme values (outliers get more weight)

Quick Example

Dataset: 1, 4, 9, 12, 19

• Mean (x̄) = 9

• Variance (s²) = 39.6

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📏 Standard Deviation (s)

What It Is

Just the square root of variance → brings us back to original units!

Formula

$s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$

Memory Hack

"Variance's little brother" → Same thing but in normal units (not squared)

Why Use It?

• Easier to interpret (same units as your data)

• If measuring height in cm, SD is in cm (not cm²)

Quick Example

• Variance = 39.6

• Standard Deviation = √39.6 = 6.29

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📊 Frequency Table Version

When data is in a frequency table:

$s^2 = \frac{\sum_{i=1}^{k} f_i(x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}$

Memory Hack

Same formula, but now multiply by frequency (fᵢ) because some values appear more times!

Example: Coffee Consumption

Cups (x)	Frequency (f)
0	10
1	16
2	12
3	27

Each value gets weighted by how often it appears.

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🔄 Linear Transformations (The Magic Rules!)

Rule 1: Adding/Subtracting a Constant (± a)

What happens: NOTHING to variance or SD!

New variance = Old variance
New SD = Old SD

Memory Hack: "Shifting doesn't change spread"

Think of it like moving all your data points together → the spread between them stays the same!

Example

• Original scores: 70, 80, 90

• Add 10 to each: 80, 90, 100

• The spread is identical!

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Rule 2: Multiplying/Dividing by a Constant (× b or ÷ b)

What happens:

• Variance gets multiplied by b²

• SD gets multiplied by b

$s'^2 = b^2 \times s^2$

$s' = b \times s$

Memory Hack: "Stretching stretches the spread"

• If you double all values → spread doubles

• But variance squares it → 2² = 4

Example: Grade Adjustment

Scores: 91, 77, 61, 83, 88, 71, 89

• Mean = 80

• Lecturer adds 5% to each grade

• Multiply by 1.05 (which is b)

• New variance = (1.05)² × old variance = 1.1025 × old variance

• New SD = 1.05 × old SD

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🔍 Coefficient of Variation (CV)

What It Is

Relative measure of spread → lets you compare different datasets even with different units!

Formula

$CV = \frac{s}{\bar{x}}$

Memory Hack: "SD per unit of average"

Shows how big the SD is relative to the mean.

When to Use It

Comparing apples to oranges!

Example: Height vs Weight

1. Height: Mean = 174 cm, SD = 14 cm

   • CV = 14/174 = 0.080 = 8%

2. Weight: Mean = 60 kg, SD = 9 kg

   • CV = 9/60 = 0.150 = 15%

Interpretation: Students are more homogeneous (similar) in height than weight!

The Rule

Lower CV = More homogeneous (less spread)

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⭐ Z-Score (Standard Score)

What It Is

Tells you how many standard deviations a value is from the mean.

Formula

$z = \frac{x_i - \bar{x}}{s}$

Memory Hack: "Distance in SD units"

• z = 0 → exactly at average

• z = 1 → one SD above average

• z = -2 → two SDs below average

Reverse Formula

If you know z-score and want to find the original value:

$x_i = \bar{x} + (z \times s)$

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🎓 Z-Score: Practical Example

The Question

Student got 85 in Statistics and 80 in Economics. Where did they do better?

Statistics Course

• Mean = 82, SD = 5

• Student's score = 85

• z = (85 - 82) / 5 = 0.6

Economics Course

• Mean = 70, SD = 3

• Student's score = 80

• z = (80 - 70) / 3 = 3.33

Answer: Student excelled more in Economics! (3.33 SDs above average vs only 0.6)

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🎯 Z-Score Properties (Super Important!)

No matter what your original data looks like:

$\text{Mean of all z-scores} = 0$

$\text{SD of all z-scores} = 1$

Why This Matters

Z-scores standardize everything → you can compare different variables!

Comparison Table

Student	Height	Weight	IQ
Evyatar	175	70	100
Iti	160	70	130
Ariel	165	60	90
Mean	165	60	100
SD	10	12	12

Calculate z-scores to compare!

• Evyatar height: z = (175-165)/10 = 1.0

• Iti IQ: z = (130-100)/12 = 2.5 ← Most outstanding!

• Ariel weight: z = (60-60)/12 = 0.0 (exactly average)

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🧠 Memory Hacks Summary

Concept	Memory Trick
Variance	"Average squared distance"
SD	"Variance's little brother in normal units"
Adding constant	"Shifting doesn't change spread"
Multiplying constant	"Stretching stretches the spread"
CV	"SD per unit of average"
Z-score	"Distance in SD units"
Z properties	"Always 0 and 1 after standardizing"

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🎯 True/False Practice Problems

Problem 1

"If distribution is negatively skewed, the z-score of the 5th decile will be negative."

Answer: FALSE!

• 5th decile = median = middle value

• In most distributions, median is near the mean

• Z-score of mean = 0 (or close to it)

• Skewness affects the mean position, but the median's z-score isn't necessarily negative

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Problem 2

"After a 7% salary increase for everyone, Daniela's z-score (originally 2.5) will increase."

Answer: FALSE!

• Everyone gets the same percentage increase

• This is multiplication by constant (× 1.07)

• Z-score stays the same! (both numerator and denominator are multiplied by same amount)

• Her relative position doesn't change

Why?

Original: $z = \frac{x_i - \bar{x}}{s}$

After: $z = \frac{1.07x_i - 1.07\bar{x}}{1.07s} = \frac{1.07(x_i - \bar{x})}{1.07s} = \frac{x_i - \bar{x}}{s}$

The 1.07 cancels out!

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📝 Quick Reference Formulas

Variance: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

Standard Deviation: $s = \sqrt{s^2}$

Frequency Table Variance: $s^2 = \frac{\sum_{i=1}^{k} f_i(x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}$

Coefficient of Variation: $CV = \frac{s}{\bar{x}}$

Z-Score: $z = \frac{x_i - \bar{x}}{s}$

Reverse Z-Score: $x_i = \bar{x} + (z \times s)$

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🎓 Final Tips

1. Always check units: SD should match your data units

2. Use CV for comparisons: Different units? Use CV!

3. Z-scores are universal: Best for comparing across different scales

4. Remember the transformation rules: Save time on exams!

5. Negative z-score = below average: Positive = above average

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Good luck! 🍀