The Normal Distribution
“The normal distribution is 68-95-99.7 — the further from the mean, the sharply rarer.”
The formula
P(μ−σ ≤ X ≤ μ+σ) ≈ 0.68How to read it: in a normal distribution, about 68% of the data falls within one standard deviation σ of the mean μ
- μ
- — the mean — the center of the distribution (peak position)
- σ
- — the standard deviation — how spread out the data is (width)
- 68%
- — the fraction of data lying within ±1σ of the mean
The hook
Heights, test scores, measurement errors — countless data from nature and society pile up, as if by agreement, into the same 'bell shape'. That's the normal distribution.
In plain words
A bell-shaped distribution, symmetric about its mean. Data clusters most densely near the mean and thins out sharply as you move away.
The intuition
When many small independent causes stack up by chance, the results bunch into a bell shape. The center (mean μ) is the peak, and the spread (standard deviation σ) is the bell's width. A small σ is tall and narrow, a large σ is broad and flat — μ sets the position, σ sets the spread.
How it's built
The key is the '68-95-99.7 rule'. About 68% of data lies within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ. Remember just those three numbers and you can estimate most of any normal distribution.
Example
A test with mean 60 and standard deviation 10: about 68% of students score 50–70 (±1σ), and about 95% score 40–80 (±2σ).
Common mistake
Not all data is normal. Right-skewed data like income is not bell-shaped. And ±2σ covers 95%, not 100% — 5% still lies outside.
Where it's used
Grading on a curve (standard scores), control limits in quality management, margins of error in polls, financial risk measurement — the basic yardstick of statistical inference.
Where it came from
Gauss refined this curve while studying the distribution of measurement errors, so it's also called the 'Gaussian distribution'. The old German 10-mark note even pictured this bell curve.
Prerequisites
Quick check
In a normal distribution, about what fraction of data lies within ±1 standard deviation of the mean?
- 50%
- 68%✓
- 95%
- 99.7%
Practice
In a normal distribution, about what percent of data lies within ±2 standard deviations?
Answer: 95
- From the 68-95-99.7 rule, ±2σ
- about 95%
Takeaway: ±1σ→68%, ±2σ→95%, ±3σ→99.7%.
For a normal distribution with mean 100 and standard deviation 15, about what percent lies between 85 and 115?
Answer: 68
- 85 to 115 is 100 ± 15, i.e. mean ±1σ
- about 68%
Takeaway: A ±1σ interval is immediately 68%.
For a normal distribution with mean 50 and standard deviation 5, about what percent lies between 40 and 60?
Answer: 95
- 40 to 60 is 50 ± 10 = mean ±2σ
- about 95%
Takeaway: First figure out how many σ the interval spans.
Explain what standardizing with z=(x−μ)/σ means for a normal distribution.
Answer: undefined
- It measures how far x is from the mean (subtract μ)
- in units of the standard deviation σ (divide by σ)
- z tells you 'how many σ from the mean'
Takeaway: Standardizing compares different distributions with one common ruler.