Measures of Central Tendency

When someone asks "What is the average score?" or "What is the typical income?", they are asking about central tendency. The three main measures -- mean, median, and mode -- each tell you something different about where the "center" of your data lies.

Mean (Arithmetic Average)

Ungrouped Data: Mean = Sum of all values / Number of values

Example: Marks of 5 students: 45, 60, 72, 58, 65. Mean = 300/5 = 60

Grouped Data: Mean = Sum of (frequency x midpoint) / Total frequency

Example:

| Marks | Frequency | Midpoint | f x midpoint | |-------|-----------|----------|--------------| | 10-20 | 3 | 15 | 45 | | 20-30 | 5 | 25 | 125 | | 30-40 | 7 | 35 | 245 | | 40-50 | 5 | 45 | 225 | | Total | 20 | | 640 |

Mean = 640/20 = 32

Median (Middle Value)

The median divides ordered data into two equal halves.

Ungrouped Data: Arrange in order. If n is odd, median is the middle value. If n is even, average the two middle values.

Example (odd): 3, 7, 9, 12, 15 -- Median = 9

Example (even): 4, 8, 12, 16 -- Median = (8 + 12)/2 = 10

Grouped Data: Median = L + ((n/2 - cf) / f) x h, where L = lower boundary of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

Mode (Most Frequent Value)

The mode is the value that appears most often.

Example: 3, 5, 5, 7, 5, 9, 3 -- Mode = 5 (appears 3 times)

When to Use Which?

• Mean: Best for data without extreme values
• Median: Best when there are outliers (e.g., income data)
• Mode: Best for categorical data

Example: Monthly incomes (NPR thousands): 15, 18, 20, 22, 200. Mean = 55 (misleading!), Median = 20 (better represents typical income).

Did You Know? Nepal's per capita income is often reported as the median rather than the mean because a small number of very wealthy individuals would push the mean much higher than what the typical Nepali earns.

Key Takeaways

• Mean = total divided by count; sensitive to extreme values
• Median = middle value when ordered; resistant to outliers
• Mode = most frequent value; useful for categorical data
• For grouped data, use the appropriate formulas with class boundaries