Standard Deviation

Imagine two archers who each shoot five arrows.

The first archer scores 8, 8, 8, 8, and 8, giving an average score of 8. The second archer scores 10, 10, 8, 6, and 6, which also averages out to 8. Even though their average scores are identical, their performances look completely different.

The first archer hit the 8-point ring every single time, showing rock-solid consistency. The second archer hit two bullseyes, but also had wild misses landing on the 6-point ring. Standard deviation is simply a single number that captures how spread out data points are around their average.

When data points cluster tightly like the first archer's shots, the standard deviation is small. When they swing wildly like the second archer's, the standard deviation is large.

To calculate the standard deviation, you first measure how far each data point sits from the average. This distance is called a 'deviation'—simply your score minus the average score.

However, scores above the average yield positive (+) deviations, while scores below the average produce negative (-) deviations. If you simply add all these deviations together, they cancel each other out and always equal exactly zero.

To solve this, mathematicians square each deviation to eliminate the negative signs, then calculate their average. This resulting number is called the 'variance.' But squaring changes the unit—turning points into points squared—which distorts our sense of scale and makes it hard to compare with original scores. Taking the square root restores the original scale and units, giving us the standard deviation.

Standard deviation does more than measure spread—it serves as a benchmark for calculating probabilities. Many real-world measurements, such as human height or exam scores, naturally form a symmetrical, bell-shaped curve known as a 'normal distribution.'

In a normal distribution, roughly 68% of all data falls within one standard deviation above and below the mean. Expand that window to two standard deviations, and it covers about 95%. Thanks to this rule, whenever data follows a bell curve, you can estimate what percentile you land in using only your score, the average, and the standard deviation.

For instance, suppose an exam follows a normal distribution with an average of 70 and a standard deviation of 10. Scoring an 80 places you in roughly the top 16%. But if another exam has an average of 70 with a standard deviation of 5, that same score of 80 sits two full standard deviations above average—landing you in the elite top 2.5%.