The standard normal table is a tool for finding probabilities from a normal distribution after values have been converted to z-scores. It connects a point on the horizontal axis of a bell curve to the area under the curve, which represents probability. This matters because many measurements, test scores, errors, and sampling results are modeled with normal distributions.
A z-table lets you avoid calculus while still finding useful probabilities quickly.
Key Facts
- Standardization formula: z = (x - μ) / σ
- The standard normal distribution has mean μ = 0 and standard deviation σ = 1.
- Total area under the normal curve is 1, so total probability is 1.
- For many z-tables, the table entry gives P(Z < z), the area to the left of z.
- Right-tail probability: P(Z > z) = 1 - P(Z < z)
- Between two z-scores: P(a < Z < b) = P(Z < b) - P(Z < a)
Vocabulary
- Standard normal distribution
- A normal distribution with mean 0 and standard deviation 1.
- Z-score
- A number that tells how many standard deviations a data value is above or below the mean.
- Z-table
- A table that lists areas or probabilities for values of the standard normal variable Z.
- Cumulative probability
- The probability that a random variable is less than or equal to a given value.
- Tail area
- The probability in one end of a distribution, either above or below a chosen cutoff.
Common Mistakes to Avoid
- Using the raw value x directly in the z-table is wrong because the table only works for standardized z-scores. Always compute z = (x - μ) / σ first unless the value is already a z-score.
- Reading the wrong row or column is wrong because a z-table splits the z-score into two parts. For z = 1.23, use row 1.2 and column 0.03.
- Confusing left-tail and right-tail areas is wrong because many z-tables give only P(Z < z). For P(Z > z), subtract the table value from 1.
- Forgetting to subtract for a middle area is wrong because P(a < Z < b) is not found from one table entry. Find the two left-tail areas and compute P(Z < b) - P(Z < a).
Practice Questions
- 1 A test score has mean μ = 70 and standard deviation σ = 8. Find the z-score for a score of x = 82, then describe whether the score is above or below average.
- 2 Using a z-table where entries give P(Z < z), find P(Z < 1.25). Then find P(Z > 1.25).
- 3 A z-table shows P(Z < 0.84) = 0.7995 and P(Z < -0.84) = 0.2005. Explain why these two probabilities are related by symmetry of the standard normal curve.