Many small influences.
One familiar shape.
Repeated length measurements can be a little high or a little low. Small effects from positioning, temperature and reading the instrument may combine. Effects pushing up and down can partly cancel. Many results then cluster near a centre, while extreme combined errors are less common.
Move the centre. Change the spread.
X ∼ N(100, 9)Read the picture.
- μ sets the centre. A larger mean moves the entire curve to the right. If the variance stays fixed, its shape stays the same.
- σ² controls the spread. A larger variance makes the curve wider and its peak lower. If the mean stays fixed, the centre stays in place.
- The total area stays 1. A taller peak does not mean a larger total probability.
The dashed vertical marks are μ − σ and μ + σ. The solid vertical mark is the mean μ. Both axes stay fixed while you compare curves.
Probability means area.
For a continuous random variable, P(a < X < b) is the area under the curve between a and b. The height of the curve at x is a density, not P(X = x).
So P(X < a) = P(X ≤ a), and P(X > a) = P(X ≥ a). A single endpoint has zero area.
Two facts we will use repeatedly.
Area to the right of μ = 0.5
The curve is symmetric about μ. Matching regions on opposite sides of the centre have equal areas.
First identify the centre and the spread. Then ask which area you need.
Different measurements.
The same Z curve.
A Z-score tells us how many standard deviations a value is above or below its mean. This lets us use one probability table for every normal distribution.
P(X > 106)
Original unitsP(Z > 2)
Standard deviation unitsSubtract the mean. Divide by the SD.
- A value at the mean has Z = 0.
- A value above the mean has positive Z.
- A value below the mean has negative Z.
- Because σ > 0, the inequality direction stays the same.
The shaded probability is unchanged. The horizontal scale is expressed in different units.
The remaining task is to read the standard normal table correctly.
One right-tail table.
Many probabilities.
Every entry in this table is T(a) = P(Z > a), for a ≥ 0. Draw the event first. Use symmetry, complements and subtraction to express its area using these right tails.
For 1.23, the row is 1.2 and the column is 0.03. Add the row label and column label to recover a.
The highlighted row and column meet at the required entry. Click that cell, or focus it and press Enter. Arrow keys move between cells. A displayed 0.0000 is rounded; the actual normal tail can still be positive.
This answer uses the displayed 4-decimal table entries. A calculator using more digits can differ in the last decimal place.
Keep these pictures in mind.
| Event, with a > 0 | Use the table as | Reason |
|---|---|---|
| P(Z > a) | T(a) | The table gives it directly. |
| P(Z < a) | 1 − T(a) | Complement. |
| P(Z < −a) | T(a) | Reflect in zero. |
| P(Z > −a) | 1 − T(a) | Reflect the missing left tail, then subtract. |
| P(0 < Z < a) | 0.5 − T(a) | Start with the right half. |
| P(−a < Z < a) | 1 − 2T(a) | Remove the two equal tails. |
Practise positive and negative limits before mixing them in one interval.
Read. Rewrite.
Then calculate.
Use these right-tail entries: T(0.52) = 0.3015, T(1.23) = 0.1093 and T(2.00) = 0.0228. Each question checks one small step. Results stay on this page.
The table is a tool for reading areas, not a replacement for the picture.