Normal Distribution Preview

Algebra 2 · Axiom Academy

LESSON The Normal Distribution How a coin-flip histogram becomes the bell curve — and the 68–95–99.7 rule that reads it. 1. From Coin Flips to the Bell Flip a fair coin n times and count the heads. Repeat, and the counts pile up into a binomial bar chart — tall in the middle, short at the extremes. Watch what happens as n grows: the bars get finer and their tops trace out one continuous, symmetric bell. That limiting shape is the normal distribution. Probability of exactly k heads in n flips Every normal curve has the same anatomy: it is symmetric with a single peak at the mean , and it bends (its inflection points) exactly one standard deviation out, at . The spread sets the width — a bigger makes the curve wider and flatter, a smaller makes it taller and narrower. Two numbers fix the whole curve: center , spread The peak sits at , and the two sides are mirror images. Shifting slides the whole curve left or right without changing its shape. Larger spreads the area wider and lowers the peak; smaller concentrates it. The curve always bends at . Because every normal curve has the same shape, the fraction of area within a fixed number of standard deviations of the mean is always the same . This is the empirical rule — the one fact you need to read a bell curve at a glance. Suppose exam scores are normal with mean and standard deviation . What fraction of scores lands between and ? Split the region at the mean and add the two half-bands — watch the area accumulate.

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