We live in the age of Big Data. Statistics is the art of extracting meaning from raw numbers. It helps us find signals in the noise.
Before any fancy machine learning model runs, someone asks the two most basic questions in statistics: where is the center of this data, and how spread out is it? Answer those two and you already understand most of what a dataset can tell you.
Measures of Central Tendency
- Mean (Average): The sum divided by the count. It's the "center of gravity" of the data. Warning: It is easily skewed by outliers (e.g., Bill Gates walks into a bar, and suddenly everyone is a millionaire on average).
- Median (Middle): The value separating the higher half from the lower half. It is robust against outliers. This is why we use it for home prices and salaries.
- Mode (Common): The most frequent value. This is the only average you can use for non-numerical data (like "What is the average favorite color?").
Worked Example: One Data Set, Three Answers
Suppose seven students score the following marks on a quiz (out of 20):
First, with the typo left in:
- Mean: (4 + 12 + 13 + 14 + 14 + 17 + 96) / 7 = 170 / 7 ≈ 24.3 — higher than 6 of the 7 actual scores! One outlier dragged the "average" above almost everyone.
- Median: Sort the data, take the 4th of 7 values: 14. Completely unaffected by the outlier.
- Mode: 14 appears twice, so the mode is 14.
After fixing the typo (96 → 16), the mean becomes 90 / 7 ≈ 12.9 while the median stays 14. Moral: when the mean and median disagree sharply, go hunting for outliers or skew before trusting either number.
Variance and Standard Deviation
Knowing the average isn't enough. You need to know the spread. If Class A scores 50, 50, 50 and Class B scores 0, 50, 100, they have the same average (50). But Class B has high variance.
Standard Deviation (σ) tells you how far, on average, a data point is from the mean. To compute it: find each value's distance from the mean, square those distances, average the squares (that's the variance), then take the square root. For Class B: distances are 50, 0, 50; squares are 2500, 0, 2500; variance is 5000/3 ≈ 1667; σ ≈ 40.8. For Class A, σ = 0 — no spread at all.
Common Mistakes to Avoid
- Reporting the mean for skewed data. Income, house prices, and hospital wait times are heavily skewed — the median is the honest summary. Using the mean quietly inflates the picture.
- Forgetting to sort before finding the median. The median is the middle of the sorted list, not the middle of the list as given. With an even count, average the two middle values.
- Confusing variance with standard deviation. Variance is in squared units (dollars², cm²) and isn't directly comparable to your data. Take the square root to get back to real units.
- Comparing spreads of different scales. A σ of 5 is huge for quiz scores out of 10 but tiny for salaries. Always judge standard deviation relative to the mean and the measurement scale.
Frequently Asked Questions
When should I use the median instead of the mean?
Use the median whenever your data is skewed or contains outliers — incomes, home prices, response times. Use the mean when data is roughly symmetric, or when you genuinely need the arithmetic total to matter (like averaging grades that sum to a final mark).
Can a data set have more than one mode, or none at all?
Yes to both. If two values tie for most frequent, the data is bimodal — often a clue that two different groups are mixed together (e.g., heights of men and women combined). If every value appears exactly once, there is no mode.
Why do we square the distances when computing variance?
Squaring does two jobs: it makes all distances positive so they can't cancel out, and it weights far-away points more heavily, making variance sensitive to extreme values. (Taking absolute values instead gives a different, less common measure called mean absolute deviation.)