Standard Normal and z-Tables

Key idea

Look up probabilities for z.

In depth

Any normal variable can be standardised as z = (x − μ)/σ, giving mean 0 and SD 1. Tables or software give cumulative probabilities Φ(z), so one table serves every normal distribution.

Worked example

P(X < 85) with μ = 70, σ = 10: z = 1.5, Φ(1.5) ≈ 0.933.

Real-world uses

• Finding the chance a battery lasts longer than a given time
• Reading standard statistics tables

Common mistake

Forgetting that tables usually give the area to the left of z.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related advanced topics in Probability & Statistics

• Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
• Box Plots – Five-number summary visualised.
• Outliers – Values beyond 1.5 IQR from the quartiles.
• Variance – Average squared deviation from the mean.
• Standard Deviation – Square root of variance; typical spread.
• Sample vs Population Variance – Divide by n−1 for sample variance.