Normal Distribution
In short: A normal distribution describes data that pile up around the average and taper off evenly on both sides, forming a smooth, mirror-image bell whose far tails are rarely visited.
Algebra 2StatisticsS-ID.A.4
What is Normal Distribution?
A normal distribution is the famous bell curve. Heights of adults, weights of bagged flour and scores on standardized tests often pile up in the middle and thin out smoothly on both sides.
Two numbers fix the whole curve: the mean \(\mu\), which sets the center, and the standard deviation \(\sigma\), which sets the spread. The curve is symmetric about \(\mu\), its mean, median and mode coincide, its total area is 1, and it bends from curving down to curving up at \(\mu – \sigma\) and \(\mu + \sigma\).
How to spot it
- A problem says that data are approximately normal or follow a bell-shaped curve with a stated mean and standard deviation.
- A histogram is symmetric, with its tallest bar in the middle and short bars at both ends.
- You are asked what percent of values fall within some number of standard deviations of the mean.
- A quality-control story, like bottle fill or loaf weight, gives a target mean and a small standard deviation.
The method
- Identify the mean \(\mu\) and the standard deviation \(\sigma\) from the description.
- Mark \(\mu\), \(\mu \pm \sigma\), \(\mu \pm 2\sigma\) and \(\mu \pm 3\sigma\) on a number line.
- Convert each boundary value to a z-score if exact areas are needed.
- Use the 68-95-99.7 pattern or a table to find the share of data between two boundaries.
Normal Distribution: solved example
Problem
A bakery’s loaves have masses that follow a normal distribution with mean 640 g and standard deviation 12 g. Describe the curve: its center, the points where it changes bending, and the interval that holds almost all loaves.
Solution
The curve is centered at \(\mu = 640\) g, and half of all loaves weigh more than that. It changes bending at \(640 \pm 12\), which is 628 g and 652 g.
Practically every loaf (roughly 997 of 1,000) sits no farther than three spreads from the center: \(640 \pm 36\). That window is 604 g to 676 g.
Try it
Question 1
A bell curve is centered at 50 and has spread 5. What fraction of its values exceeds 50?
Show the answer
The curve is symmetric about the mean, so exactly half, 0.5, lie above 50.
Question 2
For the same curve (center 50, spread 5), which two endpoints trap nearly all of the data?
Show the answer
Go three spreads each way: \(50 \pm 3(5)\) gives 35 and 65.
Normal Distribution vs. Empirical Rule
The distribution is the curve itself: a model for how a whole population of values is spread. The 68-95-99.7 pattern is a set of three quick facts about areas under that curve. You need the curve first, and then the shortcut tells you what fraction lies within one, two or three standard deviations.
Keep going
Lessons
Worksheets and quizzes
- The Normal Distribution – Worksheet
- The Normal Distribution – Quiz
- z-Scores and the Normal Distribution – Worksheet
- z-Scores and the Normal Distribution – Quiz
Related words
- Empirical Rule (68-95-99.7)
- Z-Score
- Standard Deviation
- Mean, Median and Mode
- Histogram
- Margin of Error
Whole dictionary: A–Z · Algebra 2 · Statistics