Measures of Central Tendency are statistical values that describe the center or typical value of a dataset. They help summarize a large set of numbers with a single representative value.

Three Main Measures of Central Tendency

1. Mean (Average)

• The sum of all values divided by the total number of values.
• Formula:
  • Mean=∑X / N​
  • where $\sum X$ is the sum of all values and
  • N is the total number of values.

🔹 Example:
For the dataset 3, 5, 7, 10, 15:

Mean = [3 + 5 + 7 + 10 + 15] /5 = 40 / 5 = 8.

✅ Best used when: The data has no extreme values (outliers).

2. Median (Middle Value)

• The middle value when data is arranged in ascending order.
• If the number of observations is odd, the median is the middle value.
• If the number of observations is even, the median is the average of the two middle values.

🔹 Example:

• Odd dataset: 3, 5, 7, 10, 15 → Median = 7
• Even dataset: 3, 5, 7, 10, 12, 15 → Median = [7 + 10] / 2 = 8.5

✅ Best used when: The data has outliers or is skewed.

3. Mode (Most Frequent Value)

• The value that appears most frequently in the dataset.
• A dataset can have:
  • One mode (Unimodal)
  • Multiple modes (Bimodal, Multimodal)
  • No mode (if all values appear the same number of times)

🔹 Example:
For 3, 5, 7, 7, 10, 15 → Mode = 7 (appears twice).

✅ Best used when: Identifying the most common category or value in a dataset (e.g., mode of shoe sizes in a store).

4. Relationship Between Mean, Median, and Mode

The relationship depends on the distribution of data:

1. Symmetrical Distribution (Normal Distribution)

   • Mean = Median = Mode
   • Example: Heights of people in a population.

2. Right-Skewed Distribution (Positively Skewed)

   • Mean > Median > Mode
   • Example: Income distribution (a few very high salaries raise the mean).

3. Left-Skewed Distribution (Negatively Skewed)

   • Mode > Median > Mean
   • Example: Exam scores (a few very low scores lower the mean).

5. Empirical Formula for Mean, Median, and Mode

The empirical formula that describes the approximate relationship between Mean, Median, and Mode is:

Mean − Mode ≈ 3 ( Mean − Median )

🔹 Rearranged Form:

Mode ≈ 3 × Median − 2 × Mean

When to Use This Formula?

• This formula is used for moderately skewed distributions (not highly skewed or perfectly symmetrical).
• It helps estimate the Mode when Mean and Median are known.