Standard Deviation Introduction

Pre-Algebra · Axiom Academy

The mean tells you the center of your data — standard deviation tells you how far, typically, the data strays from it. Start with five test scores: 70, 75, 80, 85, 90 . The mean is their balance point — the spot where the values on the left exactly counterweight the values on the right. Add the values, divide by how many there are 2. Deviation: Distance From the Mean A deviation is how far one value sits from the mean. Below the mean it's negative, above it's positive. Watch each score's distance to the mean line of 80 draw in: −10 70 is 10 points below the mean −5 75 is 5 points below the mean +5 85 is 5 points above the mean +10 90 is 10 points above the mean 3. Standard Deviation: The Typical Distance To stop the signs from cancelling, we square each deviation (a square is never negative), average those squares, then take the square root to return to the original units. The result is the standard deviation — one number for the typical distance from the mean. Square each deviation: the squared deviations are 100, 25, 0, 25, 100 — all positive, so nothing cancels. Average, then take the root: the average squared deviation is , and . So scores sit, typically, about 7 points from the mean. Now compare two classes with the same mean of 80 but very different spreads: Every score is within 10 points of the mean. Some scores land 30 points from the mean.

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