Mean, median, and mode are three measures of central tendency used to describe the center of a data set, but each provides different insights. **Mean (Arithmetic Average)** = Sum of all values ÷ Number of values.

Formula: Mean = (x₁ + x₂ + x₃ + ... + xₙ) ÷ n.

Example: Data set {2, 4, 6, 8, 10}.

Mean = (2+4+6+8+10) ÷ 5 = 30 ÷ 5 = **6**.

Best for: Evenly distributed data without extreme outliers.

Sensitive to outliers: {2, 4, 6, 8, 100} → Mean = 24 (skewed by 100, not representative of most values). **Median (Middle Value)** = The middle number when data is sorted from smallest to largest.

For odd-sized sets: Middle value directly.

For even-sized sets: Average of two middle values.

Example 1 (odd): {3, 7, 2, 9, 5} → Sort: {2, 3, 5, 7, 9} → Median = **5** (3rd of 5 values).

Example 2 (even): {3, 7, 2, 9, 5, 11} → Sort: {2, 3, 5, 7, 9, 11} → Median = (5+7) ÷ 2 = **6** (average of 3rd and 4th values).

Best for: Skewed distributions or data with outliers.

Resistant to outliers: {2, 4, 6, 8, 100} → Median = 6 (unaffected by 100). **Mode (Most Frequent Value)** = The value(s) that appear most often in the data set.

Can have: No mode (all values occur once), One mode (unimodal), Two modes (bimodal), Multiple modes (multimodal).

Example 1: {2, 3, 3, 5, 7, 7, 7, 9} → Mode = **7** (appears 3 times, most frequent).

Example 2: {1, 2, 2, 3, 4, 4, 5} → Modes = **2 and 4** (both appear twice, bimodal).

Example 3: {1, 2, 3, 4, 5} → **No mode** (all appear once).

Best for: Categorical data or finding most common value.

Real-world application: Shoe sizes sold (mode shows most popular size). **When to use each**: (1) Use **mean** for normally distributed quantitative data (test scores, heights, temperatures) when outliers are not present.

Example: Average class test score 75% represents typical performance. (2) Use **median** for income, home prices, or any data with extreme values.

Example: Median household income $70,000 better represents "typical" than mean $95,000 (skewed by billionaires). (3) Use **mode** for categorical data or identifying most common occurrences.

Example: Mode of customer complaints = "delivery delay" shows most frequent issue. **Key differences summary**: Mean affected by every value (including outliers), Median only considers position (middle value), Mode only considers frequency (most common).

Dataset {1, 1, 1, 2, 3, 100}: Mean = 18 (skewed by 100), Median = 1.5 (middle of sorted values), Mode = 1 (most frequent).

All three give different insights into the same data. **2025 applications**: Data science (choosing appropriate metric for ML models), Business analytics (customer behavior analysis - mean spending vs median spending vs mode purchase), Academic research (reporting statistical results correctly), Quality control (manufacturing defects - mean shows average, mode shows most common type).

Step-by-step calculation guide for mean, median, and mode with examples. **Calculating Mean (Average)**: Step 1: Add all numbers in the data set.

Step 2: Count how many numbers there are.

Step 3: Divide the sum by the count.

Example: Calculate mean of test scores {78, 85, 92, 88, 95, 82, 90}.

Step 1: Sum = 78 + 85 + 92 + 88 + 95 + 82 + 90 = **610**.

Step 2: Count = 7 students.

Step 3: Mean = 610 ÷ 7 = **87.14%** (rounded to 2 decimals).

Interpretation: Average test score is 87.14%. **Calculating Median (Middle Value)**: Step 1: Arrange data in ascending order (smallest to largest).

Step 2: Find the middle position.

Step 3a: If odd number of values → Middle value is the median.

Step 3b: If even number of values → Average the two middle values.

Example 1 (odd count): Same test scores {78, 85, 92, 88, 95, 82, 90}.

Step 1: Sort → {78, 82, 85, **88**, 90, 92, 95} (7 values).

Step 2: Middle position = (7+1) ÷ 2 = 4th position.

Step 3a: Median = **88%** (the 4th value).

Example 2 (even count): Monthly sales {$12K, $15K, $18K, $22K, $25K, $31K}.

Step 1: Already sorted → {12, 15, **18, 22**, 25, 31} (6 values).

Step 2: Middle positions = 3rd and 4th (6 ÷ 2 = 3, so 3rd and 4th).

Step 3b: Median = (18 + 22) ÷ 2 = **$20K**. **Calculating Mode (Most Frequent)**: Step 1: Count how many times each value appears (create frequency table).

Step 2: Identify the value(s) with highest frequency.

Step 3: Report mode(s) - can be one, multiple, or none.

Example 1 (unimodal): Customer ages {25, 28, 30, 30, 30, 32, 35, 35, 40}.

Step 1: Frequency: 25(1), 28(1), 30(3), 32(1), 35(2), 40(1).

Step 2: Highest frequency = 3 (age 30 appears 3 times).

Step 3: Mode = **30 years** (most common customer age).

Example 2 (bimodal): Product ratings {1, 2, 3, 3, 3, 4, 4, 5, 5, 5}.

Frequency: 1(1), 2(1), 3(3), 4(2), 5(3).

Modes = **3 and 5** (both appear 3 times).

Example 3 (no mode): Daily temperatures {72°, 75°, 78°, 81°, 84°} - all appear once, **no mode**. **Complex example - All three measures**: Dataset: Monthly website traffic {2.5K, 3.1K, 2.8K, 15.0K, 3.3K, 2.9K, 3.0K, 3.0K} visitors. (1) **Mean calculation**: Sum = 2.5 + 3.1 + 2.8 + 15.0 + 3.3 + 2.9 + 3.0 + 3.0 = 35.6K.

Count = 8 months.

Mean = 35.6 ÷ 8 = **4.45K visitors/month**.

Issue: Skewed by one viral month (15K), not representative of typical traffic. (2) **Median calculation**: Sort: {2.5, 2.8, 2.9, **3.0, 3.0**, 3.1, 3.3, 15.0} (8 values, even count).

Middle positions: 4th and 5th values = 3.0 and 3.0.

Median = (3.0 + 3.0) ÷ 2 = **3.0K visitors/month**.

Better representation: Typical month gets 3K visitors (ignores viral outlier). (3) **Mode calculation**: Frequency: 2.5(1), 2.8(1), 2.9(1), 3.0(2), 3.1(1), 3.3(1), 15.0(1).

Mode = **3.0K** (appears twice, most frequent).

Insight: 3K is the most common traffic level. **Summary for this dataset**: Mean = 4.45K (inflated by outlier), Median = 3.0K (typical month), Mode = 3.0K (most common).

Median and mode agree → 3K is representative monthly traffic. **Handling special cases**: (1) Decimals: {2.5, 3.7, 4.1, 5.3, 6.8} → Mean = 22.4 ÷ 5 = 4.48, Median = 4.1, No mode. (2) Negative numbers: {-5, -2, 0, 3, 4} → Mean = 0 ÷ 5 = 0, Median = 0, No mode. (3) Large datasets: Use calculator for {1, 2, 3, ..., 100} → Mean = 50.5, Median = 50.5, No mode (but if grouped: modes appear in frequency distribution). (4) Weighted data: Sales {$100(3 times), $200(5 times), $300(2 times)} → Mean = [(100×3) + (200×5) + (300×2)] ÷ 10 = $190, Median = $200 (6th value in sorted list), Mode = $200 (appears 5 times). **Verification tips**: Mean × Count should equal Sum (check calculation).

Median should be within the data range (not outside min/max).

Mode must actually exist in the dataset (not a calculated average).

All three should make logical sense for the data context. **2025 practical applications**: Business dashboards (display mean revenue, median customer value, mode product category).

Research papers (report all three for complete statistical picture).

Data cleaning (outliers show when mean ≠ median significantly).

Performance analysis (employee productivity - mean output vs median vs mode task type).