Mean, Median, Mode & Dispersion
Scenario
Your startup's app shows an 'average' salary of 20 LPA, but 9 out of 10 devs make 5 LPA while the founder takes 1.5 Cr. This is why standard 'averages' lie, and why you need median and dispersion to see the real picture.
Mental model
Mean is the balancing point, median is the middle guy, mode is the popular choice, and dispersion is how wildly the crowd is scattered.
Statistics summarize large datasets into single numbers. Central tendency tells you where the middle is, and dispersion (variance/standard deviation) tells you how reliable that middle is.
Formula
Mean = Sum / N. Median: sort data; if N is odd, median = middle value; if N is even, median = average of two middle values. Mode = most frequent value. Variance = Sum of (xi - mean)^2 / N. SD = sqrt(Variance). Shift rule: adding k to all values shifts mean/median/mode by k but leaves SD unchanged. Scale rule: multiplying all values by k scales mean/median/mode AND SD by k, and Variance by k^2. Empirical relation: Mode = 3 x Median - 2 x Mean (for moderately skewed data).
Standard Deviation (SD) measures volatility or spread. A high SD means data points are far from the mean. Variance is SD squared - always in squared units. Adding a constant to all data only shifts the center, it does not change the spread.
Worked example
Problem: Find the mean, median, and mode of: 4, 7, 3, 7, 9, 2, 7, 5.
- Sort: 2, 3, 4, 5, 7, 7, 7, 9.
- Mean = Sum / N = (2+3+4+5+7+7+7+9) / 8 = 44 / 8 = 5.5.
- Median: N=8 (even). Middle two values = 5 and 7. Median = (5+7)/2 = 6.
- Mode = 7 (appears 3 times, more than any other value).
- Sanity-check: Mean (5.5) < Median (6) < Mode (7). This is a left-skewed pattern, consistent with the lower outlier (2).
Answer: Mean = 5.5, Median = 6, Mode = 7.
Problem: A dataset has mean 10 and SD 4. If every value is multiplied by 3, what are the new mean, median shift, and variance?
- Scale rule: when multiplied by 3, mean becomes 10 x 3 = 30.
- SD scales by 3: new SD = 4 x 3 = 12.
- Variance scales by 3^2 = 9: new Variance = 4^2 x 9 = 16 x 9 = 144.
- Sanity-check: new Variance = new SD^2 = 12^2 = 144. Matches.
Answer: New mean = 30, new SD = 12, new Variance = 144.
Shortcuts
- Shift and Scale: Add 'k' to all numbers -> mean/median/mode shift by 'k', but SD stays the same. Multiply by 'k' -> EVERYTHING gets multiplied by 'k' (Variance by k^2). (Original SD = 5. Add 10 to all values -> New SD is still 5.)
- The Empirical Formula: Mode = 3 x Median - 2 x Mean (for moderately skewed distributions). (Mean = 10, Median = 12. Mode = 3(12) - 2(10) = 16.)
Common mistakes
- Forgetting to Sort for Median: You cannot pick the middle element of an unsorted array. Always sort in ascending order first.
- Variance vs Standard Deviation: Variance is in squared units. If asked for standard deviation, take the square root of variance. Multiplying by k squares the variance but only scales SD by k.
Glossary
- median
- The middle number in a list when they are ordered from smallest to largest.
- mode
- The number that appears most frequently in a list.
- variance
- A measurement of how spread out or scattered numbers are from the average.
Recall questions
- If the standard deviation of a dataset is 4, what is the variance?
- Data: 2, 4, 4, 6, 8. What is the mode?
- If you add 5 to every number in a dataset, what happens to its standard deviation?
Questions & answers
The mean of 5 observations is 10. If a new observation 22 is added, what is the new mean?
12
Approach: Old total = 50. New total = 72. New mean = 72 / 6 = 12.
The standard deviation of 10 values is 3. If each value is multiplied by 4, what is the new variance?
144
Approach: New SD = 3 x 4 = 12. New Variance = 12^2 = 144. (Multiplying by k scales SD by k and Variance by k^2.)
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