Measures of Central Tendency & Dispersion: Mean, Median, Mode, Range, Variance & Standard Deviation

Key Concepts & Definitions

1. Measures of Central Tendency

These are statistical values that represent the center or typical value of a dataset. They help simplify large datasets into a single representative value.

Types of Central Tendency

mindmap
  root((Measures of Central Tendency))
    Mean
    Median
    Mode

1.1 Arithmetic Mean (Average)

• Definition: The sum of all values divided by the number of values.
• Formula: $$\text{Mean}=\frac{\sum X}{N}$$ where $X$ = individual values, $N$ = total number of values.

1.2 Median

• Definition: The middle value when data is arranged in ascending or descending order.
• For odd number of values: Middle value.
• For even number of values: Average of the two middle values.

1.3 Mode

• Definition: The most frequently occurring value in a dataset.
• Can be unimodal, bimodal, or multimodal.

Visual Comparison of Mean, Median, and Mode

Observation:

• In a symmetric distribution, mean = median = mode.
• In a skewed distribution, they differ.

2. Measures of Dispersion

These indicate how spread out the data is around the central value.

Types of Dispersion

mindmap
  root((Measures of Dispersion))
    Range
    Variance
    Standard Deviation
    Quartile Deviation

2.1 Range

• Definition: Difference between the highest and lowest values.
• Formula: $$\text{Range}=\text{Max Value}-\text{Min Value}$$

2.2 Variance

• Definition: Average of the squared differences from the mean.
• Formula: $${\sigma }^2=\frac{\sum (X-\bar{X}{)}^2}{N}$$ where $\bar{X}$ = mean, $N$ = number of values.

2.3 Standard Deviation

• Definition: Square root of variance, measures dispersion in original units.
• Formula: $$\sigma =\sqrt{\frac{\sum (X-\bar{X}{)}^2}{N}}$$

Visualizing Variance & Standard Deviation

Key Points:

• 68% of data lies within ±1 standard deviation.
• 95% of data lies within ±2 standard deviations.
• 99.7% of data lies within ±3 standard deviations.

3. Worked Examples

Example 1: Calculating Mean, Median, and Mode

Data: Monthly incomes (Rs.) of 10 employees: 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000, 55000, 60000.

Step 1: Calculate Mean

$$\text{Mean}=\frac{15000+20000+25000+30000+35000+40000+45000+50000+55000+60000}{10}=37500$$

Step 2: Find Median

• Ordered data: Already sorted.
• Middle values: 5th (35000) and 6th (40000).
• Median = $\frac{35000+40000}{2}=37500$.

Step 3: Identify Mode

• No repeating values → No mode.

Example 2: Calculating Range, Variance, and Standard Deviation

Data: Same as above.

Step 1: Calculate Range

$$\text{Range}=60000-15000=45000$$

Step 2: Calculate Variance

$${\sigma }^2=\frac{(15000-37500{)}^2+(20000-37500{)}^2+\cdots +(60000-37500{)}^2}{10}=2,250,000$$

Step 3: Calculate Standard Deviation

$$\sigma =\sqrt{2,250,000}=1500$$

4. Real-World Applications

In the Real World

1. eSewa & Khalti (Digital Payments)

   • Mean & Median: Used to analyze average transaction amounts to set payment limits.
   • Standard Deviation: Helps detect fraud by identifying unusually high/low transactions.

2. Daraz (E-commerce)

   • Range: Determines price variations in product listings.
   • Variance: Used in inventory management to predict demand fluctuations.

3. Nepal Rastra Bank (NRB) & Loan Interest Rates

   • Mean: Average interest rate offered to borrowers.
   • Standard Deviation: Measures risk in loan portfolios (higher SD = higher risk).

Example: Loan Interest Rate Analysis

Scenario: A bank offers loans with interest rates: 8%, 9%, 10%, 11%, 12%.

Step 1: Calculate Mean Interest Rate

$$\text{Mean}=\frac{8+9+10+11+12}{5}=10\%$$

Step 2: Calculate Standard Deviation

$$\sigma =\sqrt{\frac{(8-10{)}^2+(9-10{)}^2+(10-10{)}^2+(11-10{)}^2+(12-10{)}^2}{5}}=1.58\%$$

Interpretation:

• Most loans are around 10%.
• ±1.58% covers most variations, helping the bank set risk-adjusted rates.

5. Comparison Table: Measures of Central Tendency & Dispersion

6. Exam Tip

• For TU/PU exams, expect numerical problems (e.g., given data, calculate mean/median).
• Theoretical questions may ask:
  • Differences between mean, median, and mode.
  • When to use range vs. standard deviation.
• Always show calculations step-by-step—partial credit is given for correct formulas.
• Real-world applications (e.g., business decisions, risk analysis) are often tested—relate answers to companies like Ncell, Daraz, or banks.

Final Summary

• Central Tendency: Mean (most used), Median (robust to outliers), Mode (categorical data).
• Dispersion: Range (simple), Variance/Standard Deviation (precise).
• Business Use: Helps in pricing, risk assessment, quality control, and decision-making.

Practice: Solve past exam questions on income distribution tables and financial data analysis to master this unit!

In the real world

• eSewa & Khalti (Digital Payments) use mean transaction amounts to set daily payment limits (e.g., Rs. 50,000 average per user). Standard deviation helps detect fraud by flagging transactions outside ±2σ (e.g., Rs. 150,000 when mean is Rs. 50,000 and σ = Rs. 20,000).
• Daraz (E-commerce) analyzes price range (e.g., Rs. 500–Rs. 5,000) and variance in demand to adjust inventory (e.g., high σ in electronics sales signals stock fluctuations).
• Nepal Rastra Bank (NRB) uses mean interest rates (e.g., 8–12%) and standard deviation (e.g., σ = 1.58%) to assess loan portfolio risk—higher σ indicates unstable borrower groups.