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.
Understanding Statistics: The Standard Normal Table
A normal table works because it puts measurements from very different settings onto one common scale. A height, a reaction time, and an exam mark may use different units, yet each can be described by its distance from the average in standard deviation units. A z-score tells both the direction and the size of that distance.
A positive z-score lies above the average. A negative z-score lies below it.
A z-score of two means the value is two standard deviations above the average, regardless of whether the original measurement was measured in centimetres, seconds, or marks. This makes comparisons fairer and gives the table a much wider use than one set of data.
Most standard normal tables are read using a row and a column. The row usually gives the ones digit and tenths digit of the z-score. The column gives the hundredths digit.
For example, for a z-score of one point two three, find the row for one point two and the column for zero point zero three. Their intersection might show zero point eight nine zero seven. In a common table, this means about eighty nine point zero seven percent of values lie below that score.
Students should always read the heading and notes on their own table first. Some tables give the area to the left, while others give only the area from the mean to the z-score. Using the wrong type of table can produce a sensible-looking but incorrect answer.
The bell curve is symmetric around its centre, and this symmetry helps with negative values. The area below a z-score of negative one point two three equals the area above a z-score of positive one point two three. If the left area for positive one point two three is zero point eight nine zero seven, the left area for negative one point two three is zero point one zero nine three.
For values above a cutoff, take the left area away from the whole area of one. For values within an interval, find the left area at each endpoint and subtract the smaller area from the larger one. Drawing a quick curve, marking the cutoff points, and shading the required region prevents many direction errors.
Tables give approximate probabilities, not guarantees about individual outcomes. They are most useful when the data are reasonably close to a bell shape. Look at a graph of the data when possible.
Strong skew, several peaks, extreme outliers, or a hard upper or lower limit can make a normal model less reliable. Real examples include checking whether a manufactured part falls within an acceptable range, estimating unusually high or low test results, and studying sampling variation in repeated surveys. When learning this topic, pay close attention to the wording of probability statements.
Words such as below, above, at most, at least, and between determine which area must be shaded and calculated. Rounding the z-score only as instructed matters too, since a small rounding change can alter the table value.
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.