A normal distribution is symmetric and bell-shaped, with its mean, median and mode all at the centre.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it appears everywhere
Quantities built from many small independent effects tend toward this shape: heights, measurement errors, test scores.
Two numbers describe it
The mean fixes the centre and the standard deviation fixes the width. Nothing else is needed.
The empirical rule
About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
Reading the tails
If 95% lies within two standard deviations, 5% lies outside, split evenly at 2.5% in each tail.
Not everything is normal
Income is strongly skewed, so the rule does not apply. Check the shape before using it.
The bell curve
Symmetric about the mean, with most values close to it and fewer in the tails. It is determined entirely by two numbers, the mean and the standard deviation, which is unusually economical.
The 68-95-99.7 rule
About 68% of values fall within one standard deviation of the mean, 95% within two, 99.7% within three. Three numbers give a usable reading of any normal distribution without a table.
Why it appears so often
Quantities produced by many small independent influences added together tend to be normal — heights, measurement errors, and sample means. The central limit theorem is the formal statement of this.
It is not universal
Incomes are strongly right-skewed and waiting times are not normal. Assuming normality without checking is a real error, and a histogram of the data is the check.
Step 2: Try It Yourself
Tap and try it out.
Low
Mid-low
Mid-high
High
Step 3: Watch an Example
One step at a time.
Watch Diego Use the Empirical Rule
Heights are normal with mean 170 cm and standard deviation 8 cm. Diego wants the proportion above 186 cm.
- Step 1
186 is 16 above the mean, which is exactly two standard deviations.
Step 4: Your Turn
Practice makes it stick.
The Middle
Problem 1 of 2
What percentage lies within one standard deviation of the mean?
%
The Tail
Problem 2 of 2
What percentage lies more than two standard deviations above the mean?
%
Read the Curve
1 of 8
What percentage lies within two standard deviations?
2 of 8
What percentage lies within three standard deviations?
3 of 8
Mean 170, standard deviation 8. What value is two above the mean?
4 of 8
Mean 100, standard deviation 15. What value is one below the mean?
5 of 8
What percentage lies above the mean in a normal distribution?
6 of 8
What percentage lies outside one standard deviation?
7 of 8
Sort each quantity by whether a normal model is reasonable.
Tap something to move it.
- Empty
- Empty
8 of 8
Mean 50, standard deviation 5. What value is three above the mean?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What percentage lies within one standard deviation of the mean?
Question 2 of 2
Why does 2.5% lie above two standard deviations rather than 5%?
What You Learned
- A normal distribution is symmetric and bell-shaped.
- Its mean sets the centre and its standard deviation sets the width.
- About 68%, 95% and 99.7% lie within one, two and three standard deviations.