Statistics Basics

Learn to understand and interpret data, calculate meaningful averages, grasp probability, and make data-driven decisions in business and life.

Measures of Central Tendency

These tell you what's "typical" or "average" in a dataset.

Mean (Average)

Definition: Sum of all values divided by count

Formula: Mean = Sum of all values ÷ Number of values

Example: Test scores: 85, 90, 78, 92, 85

• Sum: 85 + 90 + 78 + 92 + 85 = 430
• Count: 5
• Mean: 430 ÷ 5 = 86

Life Application: Average monthly expense

• Expenses: $2,400, $2,100, $2,600, $2,200, $2,300, $2,500
• Mean: $14,100 ÷ 6 = $2,350/month

Business Application: Average revenue per customer

• Total revenue: $125,000
• Customers: 250
• Mean: $125,000 ÷ 250 = $500/customer

Limitation: Sensitive to extreme values (outliers)

Example: Salaries: $45k, $48k, $50k, $52k, $300k

• Mean: $99k (not representative of typical salary)

Median (Middle Value)

Definition: The middle value when data is ordered

Finding Median:

1. Order values from least to greatest
2. Odd count: Pick middle value
3. Even count: Average the two middle values

Example (odd count): 15, 22, 18, 30, 25

• Ordered: 15, 18, 22, 25, 30
• Median: 22

Example (even count): 10, 15, 20, 25, 30, 35

• Ordered: 10, 15, 20, 25, 30, 35
• Median: (20 + 25) ÷ 2 = 22.5

Advantage: Not affected by outliers

Example: Same salaries as before: $45k, $48k, $50k, $52k, $300k

• Median: $50k (much more representative)

Real Estate Application: Home prices in neighborhood

• Prices: $280k, $295k, $310k, $305k, $1.2M (outlier)
• Median: $305k (better represents typical home)
• Mean: $478k (skewed by mansion)

Mode (Most Frequent)

Definition: Value that appears most often

Example: Shoe sizes sold: 7, 8, 8, 8, 9, 9, 10, 11

• Mode: 8 (appears 3 times)

Business Application: Most popular product size

• Helps with inventory decisions

Multiple Modes: Dataset can have more than one mode

• Example: 5, 5, 5, 8, 10, 10, 10, 12
• Modes: 5 and 10 (bimodal)

No Mode: All values appear equally

• Example: 2, 4, 6, 8, 10 (no mode)

When to Use Which?

Measures of Spread

These show how spread out or varied the data is.

Range

Definition: Difference between highest and lowest values

Formula: Range = Maximum − Minimum

Example: Test scores: 65, 78, 82, 91, 95

• Range: 95 − 65 = 30

Business Application: Salary ranges

• Entry: $45k, Senior: $95k
• Range: $50k

Limitation: Only uses two values, ignores everything in between

Quartiles and Interquartile Range (IQR)

Quartiles divide ordered data into four equal parts:

• Q1: 25th percentile (1/4 of data below)
• Q2: 50th percentile (median)
• Q3: 75th percentile (3/4 of data below)

IQR (Interquartile Range): Q3 − Q1

• Shows spread of middle 50% of data
• Less affected by outliers than range

Example: Data: 10, 12, 15, 18, 20, 23, 25, 28, 30

• Q1 (between 12 and 15): 13.5
• Q2 (median): 20
• Q3 (between 25 and 28): 26.5
• IQR: 26.5 − 13.5 = 13

Standard Deviation (Conceptual)

Definition: Average distance of values from the mean

Low standard deviation: Values clustered close to mean
High standard deviation: Values spread out widely

Example: Two classes, both with mean score of 80

Class A: 78, 79, 80, 81, 82 (low spread)
Class B: 50, 70, 80, 90, 100 (high spread)

Class A has lower standard deviation (more consistent).

Business Application:

• Consistent sales: Low standard deviation (predictable)
• Volatile sales: High standard deviation (unpredictable)

Probability Basics

Probability: Likelihood of an event occurring, from 0 (impossible) to 1 (certain)

Formula: Probability = (Favorable outcomes) ÷ (Total possible outcomes)

Often expressed as:

• Decimal: 0.25
• Fraction: 1/4
• Percentage: 25%

Simple Probability

Example: Rolling a die, probability of getting 4

• Favorable: 1 (only one 4)
• Total: 6 (six sides)
• Probability: 1/6 ≈ 0.167 or 16.7%

Example: Drawing ace from standard deck

• Favorable: 4 (four aces)
• Total: 52 cards
• Probability: 4/52 = 1/13 ≈ 7.7%

Complementary Probability

Concept: Probability of event NOT happening

Formula: P(not A) = 1 − P(A)

Example: Probability it rains tomorrow is 30%

• Probability it doesn't rain: 1 − 0.30 = 0.70 or 70%

Independent Events (AND)

Definition: Events that don't affect each other

Rule: Multiply probabilities

Example: Flip coin twice, probability of two heads

• First flip: P(H) = 1/2
• Second flip: P(H) = 1/2
• Both heads: 1/2 × 1/2 = 1/4 = 25%

Business Example: Three independent sales calls

• Each has 20% chance of success
• Probability all three succeed: 0.20 × 0.20 × 0.20 = 0.008 = 0.8%

Dependent Events

Definition: First event affects the second

Example: Draw two cards without replacement

• First card is ace: 4/52
• Second card is also ace: 3/51 (only 3 aces left in 51 cards)
• Both aces: (4/52) × (3/51) ≈ 0.0045 or 0.45%

Mutually Exclusive Events (OR)

Definition: Events that can't happen simultaneously

Rule: Add probabilities

Example: Roll die, probability of 3 OR 5

• P(3) = 1/6
• P(5) = 1/6
• P(3 or 5) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 33.3%

Data Interpretation

Reading Tables

Sales Data Example:

Questions to Ask:

• What's the average? Mean units: (150+180+165)/3 = 165
• What's the trend? Increasing then slight decrease
• What's the best month? February

Understanding Percentages in Data

Survey Result: "60% of 500 respondents prefer option A"

• Number who prefer A: 500 × 0.60 = 300 people
• Number who don't: 500 − 300 = 200 people

Correlation vs Causation

Correlation: Two things change together
Causation: One thing causes the other

Example: Ice cream sales and drowning incidents both increase in summer

• Correlated: Yes (both increase together)
• Causal: No (ice cream doesn't cause drowning)
• Real cause: Hot weather affects both

Business Trap: Don't assume correlation means causation

• Website traffic and sales correlated? Maybe, but need to test
• Marketing spend and revenue correlated? Possible causation, but verify

Common Statistical Pitfalls

Small Sample Size

Problem: 3 people surveyed, 2 prefer product A = "67% prefer A" Reality: Sample too small for reliable conclusion

Better: Survey at least 100+ for reasonable confidence

Cherry-Picking Data

Problem: "Sales increased 50% in December!" Reality: Comparing to lowest month (November) instead of fair comparison

Better: Compare to same month last year or use annual average

Misleading Graphs

Problem: Y-axis doesn't start at zero, making changes look dramatic

Example: Stock price graph

• Starts at $98 instead of $0
• Change from $98 to $102 looks huge
• Actually only 4% increase

Better: Use appropriate scale and clearly label axes

Survivorship Bias

Problem: Only looking at successful examples

Example: "Successful entrepreneurs dropped out of college" Reality: Only counting successes, ignoring thousands who dropped out and failed

Better: Look at complete dataset, including failures

Practical Business Statistics

Conversion Rate

Formula: Conversion Rate = (Conversions ÷ Visitors) × 100

Example:

• Website visitors: 5,000
• Purchases: 150
• Conversion rate: (150 ÷ 5,000) × 100 = 3%

Customer Retention Rate

Formula: Retention = [(End − New) ÷ Start] × 100

Example:

• Start of year: 500 customers
• End of year: 480 customers
• New customers: 100
• Retained: [(480 − 100) ÷ 500] × 100 = 76%

Net Promoter Score (NPS)

Scale: 0-10, "How likely to recommend?"

• 9-10: Promoters
• 7-8: Passives
• 0-6: Detractors

Formula: NPS = % Promoters − % Detractors

Example:

• 100 responses: 50 promoters, 20 passives, 30 detractors
• NPS: 50% − 30% = 20 (score can be −100 to +100)

Practice Problems

Central Tendency

1. Find mean, median, mode: 12, 15, 18, 15, 21, 24, 15
2. Which measure best represents typical salary: $40k, $42k, $45k, $48k, $50k, $200k?

Spread

3. Find range and IQR for: 10, 15, 18, 22, 25, 30, 35

Probability

4. Flip coin three times. Probability of exactly two heads?
5. Draw one card from deck. Probability it's red OR a king?

Data Interpretation

6. Website had 2,000 visitors, 60 made purchases. What's the conversion rate?
7. Class A: mean score 75, std dev high. Class B: mean score 75, std dev low. Which is more consistent?

Solutions

1. Mean: 17.14, Median: 15, Mode: 15
2. Median ($47k); mean ($70.8k) is skewed by outlier
3. Range: 25 (35−10), IQR: 13 (26.5−13.5)
4. 3/8 or 37.5% (HHT, HTH, THH: 3 out of 8 outcomes)
5. 28/52 or 53.8% (26 red + 2 additional black kings)
6. 3% (60÷2000 × 100)
7. Class B (lower standard deviation means more consistent scores)

Key Takeaways

✓ Mean vs Median: use median when outliers are present
✓ Probability basics: multiply for AND, add for OR (mutually exclusive)
✓ Sample size matters: larger samples are more reliable
✓ Correlation ≠ causation: don't confuse the two
✓ Context is critical: numbers without context mislead
✓ Visual representation: graphs can help or deceive

Real-World Applications

• Business Decisions: Analyze sales data, customer metrics
• Investing: Understand market statistics and risk
• Health: Interpret medical statistics and probabilities
• News: Critically evaluate surveys and studies
• Marketing: A/B testing, conversion optimization
• Personal: Understand warranties, odds, risk assessments

Next Steps

Move to Chapter 06: Financial Mathematics to learn about interest calculations, loans, investments, and the time value of money. These skills are essential for wealth building and financial planning.