Continuous variables and normal curves
A normal distribution models some continuous measurements. Begin by separating a measurement from a count, then decide whether a symmetric bell is a sensible model.
A continuous variable can take any value in an interval
A random variable records a numerical result. A discrete variable records separate values, usually counts. A continuous variable records a measurement and can take every value between two boundaries.
Swipe left or right to read every column →
| Variable | Type | Possible values |
|---|---|---|
| number of late buses | discrete | 0, 1, 2, ... |
| waiting time in minutes | continuous | any non-negative time |
Key idea
A normal model is symmetric, single-peaked and continuous
Read the bell as an area model
total area = 1
mean = median = mode
- The total area under the curve is 1.
- The curve is symmetric about , so its mean, median and mode are equal.
- The tails approach the axis but never stop. The mathematical model allows every real value, although distant values have tiny probability.
Here is the mean and is the standard deviation. The second parameter in the notation is variance, .
Common mistake
The mean moves the curve; the standard deviation changes its spread
same σ, different μ
equal shape · shifted centre
same μ, different σ
smaller σ: narrower and higher · larger σ: wider and lower
- Changing shifts the whole curve left or right without changing its shape.
- Increasing makes the curve wider and lower. Decreasing it makes the curve narrower and higher.
- Every normal curve still has total area 1.
Key idea
The mass of a component is modelled by . State whether is discrete or continuous, give its mean and standard deviation, and state . On a labelled sketch, mark 72 and shade . Describe the change if the mean stays 72 but the variance becomes 64. Would a strongly right-skewed mass distribution be well modelled by a normal curve?
Show worked answer
Mass is measured, so is continuous. The second normal parameter is variance.
On the sketch, 72 is at the centre and the left half is shaded. If the variance becomes 64, then , so the curve keeps the same centre but becomes wider and lower. A strongly right-skewed distribution is not symmetric, so a normal model is unsuitable.