Location and Dispersion Metrics

📹 Video Overview

🎯 What We're Learning Today

Main Topics:

1. Review: Distribution shapes and central measures

2. Location Metrics: Quartiles, Deciles, Percentiles

3. Dispersion Metrics: How spread out is the data?

   • Range & Interquartile Range

   • Variance & Standard Deviation

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Part 1: Quick Review - Distribution Shapes

The 5 Main Distribution Types:

Quick Reference Table:

Distribution Type	Relationship	Visual
Normal (Bell)	Mode = Median = Mean	Perfect symmetry 🔔
Dual Peak	Two Modes, Med = Mean	Two humps 🐫
Uniform	No clear mode, Med = Mean	Flat line 📏
Positive Skew (Right)	Mode < Median < Mean	Tail → right 📈
Negative Skew (Left)	Mean < Median < Mode	Tail → left 📉

💡 Memory hack: "Mean follows the tail like a puppy!"

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🧩 Classic Exam Question (Must Know!)

Q: "University graduate wages are positively skewed. Therefore, the percentage earning above average is greater than the percentage earning below average."

TRUE or FALSE?

Answer: FALSE!

Why?

1. Positively skewed = tail extends to the right (high earners)

2. Mean gets pulled UP by extreme high values

3. Mean > Median

4. Median splits data 50-50:

   • 50% below median

   • 50% above median

5. Since Mean > Median:

   • MORE than 50% earn BELOW the mean

   • LESS than 50% earn ABOVE the mean

💡 Memory hack: "In positive skew, most people are below average because the rich people pull the average up!"

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Part 2: Location Metrics

🎯 What Are Location Metrics?

Question they answer: "Where does a specific percentage of the data fall?"

Think of them as dividing lines in your data!

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📊 Quartiles (Q₁, Q₂, Q₃, Q₄)

Definition: Values that divide data into 4 equal parts (25% each)

The Four Quartiles:

Quartile	Name	Meaning
Q₁	First/Lower Quartile	25% of data ≤ Q₁ 75% of data ≥ Q₁
Q₂	Second Quartile	= MEDIAN! 50% below, 50% above
Q₃	Third/Upper Quartile	75% of data ≤ Q₃ 25% of data ≥ Q₃
Q₄	Fourth Quartile	= MAXIMUM value 100% ≤ Q₄

💡 Memory hack: Q₁, Q₂, Q₃, Q₄ = 25%, 50%, 75%, 100%

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🔢 Calculating Quartiles - Discrete Variable

Example: Daily coffee consumption (200 patients)

Cups (x)	f(x)	F(x)
0	10	10
1	16	26
2	12	38
3	27	65
4	55	120
5	48	168
6	15	183
7	17	200

Step-by-Step Process:

Step 1: Calculate cumulative frequency F(x) ✓ (already done)

Step 2: Calculate the positions

For Q₁ (First Quartile):

$\frac{n}{4} = \frac{200}{4} = 50$

For Q₃ (Third Quartile):

$\frac{3n}{4} = \frac{3 \times 200}{4} = 150$

Step 3: Find where F(x) first exceeds these values

For Q₁ = 50:

• F(3) = 65 > 50 ✓ (first time!)

• F(2) = 38 < 50

• Q₁ = 3 cups

For Q₃ = 150:

• F(5) = 168 > 150 ✓ (first time!)

• F(4) = 120 < 150

• Q₃ = 5 cups

💡 Memory hack: "Keep climbing the F(x) stairs until you pass the target!"

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🔢 Calculating Quartiles - Continuous Variable

Example: Test scores (117 students)

Scores (x)	f(x)	F(x)
40-60	5	5
60-70	31	36
70-75	25	61
75-85	42	103
85-100	14	117

Calculate positions:

• Q₁ position: n/4 = 117/4 = 29.25

• Q₃ position: 3n/4 = 3(117)/4 = 87.75

Find the classes:

Q₁:

• F(60-70) = 36 > 29.25 ✓

• Q₁ is in class 60-70

Q₃:

• F(75-85) = 103 > 87.75 ✓

• Q₃ is in class 75-85

💡 Memory hack: For continuous, just identify the CLASS, not exact value (unless you interpolate).

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📊 Deciles (P₁₀, P₂₀, ..., P₉₀)

Definition: Values that divide data into 10 equal parts (10% each)

Key Deciles:

Decile	Symbol	Meaning
First Decile	P₁₀	10% below, 90% above
Second Decile	P₂₀	20% below, 80% above
Ninth Decile	P₉₀	90% below, 10% above

Connection to Quartiles:

• P₂₅ = Q₁

• P₅₀ = Q₂ = Median

• P₇₅ = Q₃

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📊 Percentiles (Pz)

Definition: For any percentage z, Pz is the value where:

• z% of data is ≤ Pz

• (100-z)% of data is ≥ Pz

🔢 Calculating Percentiles - Discrete Variable

Example: Find P₆₅ (65th percentile) for coffee data

Step 1: F(x) already calculated ✓

Step 2: Calculate position

$n \times \frac{z}{100} = 200 \times \frac{65}{100} = 200 \times 0.65 = 130$

Step 3: Find where F(x) first exceeds 130

Cups (x)	f(x)	F(x)
4	55	120
5	48	168

P₆₅ = 5 cups

💡 Memory hack: Formula is n × (z/100) or just n × z%

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🔢 Calculating Percentiles - Continuous Variable

Example: Find P₃₅ (35th percentile) for test scores (n = 117)

Step 1: Calculate position

$\frac{35}{100} \times 117 = 0.35 \times 117 = 40.95$

Step 2: Find the class

Scores (x)	f(x)	F(x)
40-60	5	5
60-70	31	36
70-75	25	61

P₃₅ is in class 70-75

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📏 Part 3: Dispersion Metrics

🎯 What is Dispersion?

Question: How SPREAD OUT is the data?

Two datasets, same mean = 9:

• Dataset 1: 9, 9, 9, 9, 9 (no spread!)

• Dataset 2: 1, 4, 9, 12, 19 (lots of spread!)

💡 Memory hack: Dispersion = "How scattered is the data?"

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📊 Measure 1: Range (R)

Definition: Difference between max and min

Formula:

$R = x_{max} - x_{min}$

Example: Coffee data

• xₘₐₓ = 7 cups

• xₘᵢₙ = 0 cups

• R = 7 - 0 = 7 cups

🔍 Characteristics:

✓ Very simple to calculate

✓ Easy to understand

✗ Affected by extreme outliers (one extreme value changes everything!)

✗ Ignores all middle values

💡 Memory hack: Range looks at the "edges" only, ignores the "middle"

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📊 Measure 2: Interquartile Range (IQR or Q)

Definition: Distance between Q₃ and Q₁ (covers the middle 50% of data)

Formula:

$IQR = Q = Q_3 - Q_1$

Example: Coffee data

• Q₃ = 5 cups

• Q₁ = 3 cups

• IQR = 5 - 3 = 2 cups

🔍 Characteristics:

✓ Not affected by extremes (only looks at middle 50%)

✓ Better than range for skewed data

✗ Ignores outer 50% of data

💡 Memory hack: IQR = "The middle stretch" where most normal people are

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🧩 Special Cases - Range vs IQR

Case 1: Range = 0, what about IQR?

Data: 50, 70, 70, 70, 70, 70, 70, 70, 70, 90

• Range = 90 - 50 = 40

• Q₁ = 70, Q₃ = 70

• IQR = 0 (even though range ≠ 0!)

Why? Most data is at 70, so middle 50% has no spread.

Case 2: IQR = 0, what about Range?

Can be 0 or greater!

If all middle values are the same but extremes differ:

• IQR = 0

• Range > 0

💡 Key insight: IQR focuses on middle, Range looks at extremes!

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📊 IQR and Distribution Shape

Normal Distribution:

$Q_3 - Q_2 = Q_2 - Q_1$

(Symmetric - equal distances)

Positive Skew:

$Q_3 - Q_2 > Q_2 - Q_1$

(Upper half more spread than lower half)

Negative Skew:

$Q_2 - Q_1 > Q_3 - Q_2$

(Lower half more spread than upper half)

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📊 Measure 3: Variance (s²)

Definition: Average of squared deviations from the mean

Why squared? So negative deviations don't cancel positive ones!

📐 Formula (Raw Data):

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

Plain English:

1. Find mean (x̄)

2. For each value: (xᵢ - x̄)²

3. Sum all squared differences

4. Divide by n

💡 Memory hack: "Square the differences so negatives don't cancel!"

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Example 1: No Spread

Data: 9, 9, 9, 9, 9

Step 1: Mean

$\bar{x} = \frac{9+9+9+9+9}{5} = 9$

Step 2: Calculate (xᵢ - x̄)²

xᵢ	xᵢ - x̄	(xᵢ - x̄)²
9	0	0
9	0	0
9	0	0
9	0	0
9	0	0
Sum		0

Step 3: Variance

$s^2 = \frac{0}{5} = 0$

Interpretation: No spread = variance is 0!

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Example 2: With Spread

Data: 1, 4, 9, 12, 19

Step 1: Mean

$\bar{x} = \frac{1+4+9+12+19}{5} = \frac{45}{5} = 9$

Step 2: Calculate (xᵢ - x̄)²

xᵢ	xᵢ - x̄	(xᵢ - x̄)²
1	-8	64
4	-5	25
9	0	0
12	+3	9
19	+10	100
Sum		198

Step 3: Variance

$s^2 = \frac{198}{5} = 39.6$

Interpretation: Lots of spread = high variance!

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📐 Formula (Frequency Table):

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

Plain English: Weight each squared difference by its frequency!

💡 Memory hack: "If a value appears 5 times, its deviation counts 5 times!"

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📊 Measure 4: Standard Deviation (s)

Definition: Square root of variance

Formula:

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

Why do we need it?

• Variance is in "squared units" (e.g., cups²)

• Standard deviation is in original units (e.g., cups)

• Easier to interpret!

Example: From Example 2

• s² = 39.6

• s = √39.6 ≈ 6.29

💡 Memory hack: "Standard deviation = variance in units I can understand!"

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🔍 Characteristics of Variance & Standard Deviation:

✓ Uses ALL data points

✓ Most widely used dispersion measure

✓ Foundation for advanced statistics

✗ Heavily affected by outliers (because we square deviations!)

✗ Always ≥ 0 (can be 0 only if all values identical)

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🔄 Linear Transformations & Dispersion

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

What happens?

If we add/subtract the same amount to every value:

• Mean changes: x̄' = x̄ ± a

• Variance UNCHANGED: s'² = s²

• Standard deviation UNCHANGED: s' = s

Why? Adding a constant shifts everything together - doesn't change spread!

Proof:

$s'^2 = \frac{\sum_{i=1}^{n} [(x_i \pm a) - (\bar{x} \pm a)]^2}{n} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} = s^2$

💡 Memory hack: "Shift everyone together = spread stays same!"

Visual:

Original: [1, 2, 3, 4, 5]  → spread = 2

Add 10:   [11, 12, 13, 14, 15] → spread still = 2

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

What happens?

If we multiply/divide every value by the same amount:

• Mean changes: x̄' = b × x̄

• Variance changes: s'² = b² × s²

• Standard deviation changes: s' = b × s (or |b| × s)

Proof:

$s'^2 = \frac{\sum_{i=1}^{n} [bx_i - b\bar{x}]^2}{n} = \frac{b^2\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} = b^2 s^2$

$s' = \sqrt{b^2 s^2} = |b| \cdot s$

💡 Memory hack:

• Variance gets multiplied by b²

• Standard deviation gets multiplied by b

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📋 Transformation Summary:

Transformation	Mean	Variance	Std Dev
x + a	x̄ + a	s²	s
x - a	x̄ - a	s²	s
b × x	b × x̄	b² × s²	b × s
x ÷ b	x̄ ÷ b	s² ÷ b²	s ÷ b
a + bx	a + bx̄	b² × s²	b × s

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🧮 Example: Grade Transformation

Original grades (7 students): 91, 77, 65, 83, 88, 71, 98

Given:

• Mean: x̄ = 80

• Variance: s² = 120 (assume)

• Standard deviation: s = √120 ≈ 10.95

Scenario 1: Add 2 points to everyone

New statistics:

• Mean: x̄' = 80 + 2 = 82

• Variance: s'² = 120 (unchanged!)

• Std Dev: s' = 10.95 (unchanged!)

Scenario 2: Multiply all grades by 1.05 (5% bonus)

New statistics:

• Mean: x̄' = 1.05 × 80 = 84

• Variance: s'² = (1.05)² × 120 = 1.1025 × 120 = 132.3

• Std Dev: s' = 1.05 × 10.95 = 11.50

💡 Memory hack: "Add = shift only, Multiply = stretch everything!"

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📊 Quick Comparison Chart

Measure	Formula	Units	Affected by Outliers?	Interpretation
Range	xₘₐₓ - xₘᵢₙ	Same as data	YES (very!)	Total spread
IQR	Q₃ - Q₁	Same as data	NO	Middle 50% spread
Variance	Σ(xᵢ-x̄)²/n	Squared units	YES	Average squared deviation
Std Dev	√Variance	Same as data	YES	Typical deviation from mean

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🎯 Decision Guide: Which Measure?

General rules:

• Symmetric data, no outliers: Standard deviation

• Skewed or outliers: IQR

• Quick overview: Range

• Academic/scientific: Almost always variance/std dev

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🎓 Key Formulas Summary

Location Metrics:

Metric	Position Formula	Finding Method
Q₁	n/4	First F(x) > n/4
Q₂	n/2	Median
Q₃	3n/4	First F(x) > 3n/4
Pz	n × z/100	First F(x) > n×z%

Dispersion Metrics:

Metric	Formula
Range	R = xₘₐₓ - xₘᵢₙ
IQR	Q = Q₃ - Q₁
Variance	$s^2 = \frac{\sum(x_i - \bar{x})^2}{n}$
Std Dev	$s = \sqrt{s^2}$
Variance (freq)	$s^2 = \frac{\sum f_i(x_i - \bar{x})^2}{\sum f_i}$

Transformations:

Transform	Variance	Std Dev
±a	No change	No change
×b	×b²	×b

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💡 Master Memory Hacks

1. Quartiles = Quarter marks at 25%, 50%, 75%, 100%

2. Percentiles = Percent-tiles - the z% mark

3. Range = Edge to edge (min to max)

4. IQR = The middle crowd (ignores extremes)

5. Variance = Squared differences (so negatives don't cancel)

6. Std Dev = Variance in real units (take √ to fix squaring)

7. Adding = shifts, no spread change

8. Multiplying = stretches the spread

9. Mean chases tail, outliers affect it

10. IQR ignores tail, robust to outliers

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🎯 Quick Reference Mind Map

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⚠️ Common Exam Mistakes

❌ Confusing Q₃ - Q₂ with Q (should be Q₃ - Q₁)

❌ Forgetting to square the deviations in variance

❌ Thinking adding a constant changes variance (it doesn't!)

❌ Using Range when outliers are present

❌ Forgetting variance is multiplied by b², not just b

❌ Not squaring b in transformation (s'² = b² × s², not b × s²)

❌ Thinking positive skew means more above average (it's the opposite!)

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🏆 Pro Exam Tips

1. Quartiles? Calculate n/4 and 3n/4, then find in F(x)

2. Percentiles? Calculate n × z%, then find in F(x)

3. Variance question? Check if it's asking for raw or transformed data

4. Linear transformation?

   • Adding/subtracting: dispersion unchanged

   • Multiplying: variance ×b², std dev ×b

5. Outliers present? Use IQR, not range or std dev

6. Skewed distribution? Remember mean ≠ median ≠ mode relationship

7. Always check units: variance in squared units, std dev in original units

Final tip: Distribution shape tells you about outliers AND about which measures to trust!