What a Cumulative Standard Normal Table Shows
A cumulative standard normal table is a lookup tool: give it a z-score, and it reports the standard normal area to the left of that score. The table does not report the z-score itself or the height of the normal curve at that point. Reading its row and column correctly is therefore as important as recognizing what the resulting number means.
The word cumulative means that the area is accumulated from the far left of the distribution up to the specified z-score. In terms of the standard normal density \(\phi(t)=\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\), the table value is
Because a normal variable is continuous, the probability of taking any one exact value is zero. Thus \(P(Z\le z)\) and \(P(Z<z)\) have the same value. The table entry represents the area under the standard normal curve to the left of \(z\), not the density at \(z\).
How Rows and Columns Encode a Z-Score
Most cumulative standard normal tables display z-scores to two decimal places. The row gives the z-score through its tenths place, and the column supplies its hundredths place. For a positive score such as \(1.23\), the row is \(1.2\) and the column is \(0.03\). Their labels identify the full score \(1.23\); the cell where they meet gives \(\Phi(1.23)\).
A small excerpt of a typical table illustrates the arrangement. The entries are cumulative probabilities, not z-scores.
| Row z | Column .00 | Column .01 | Column .02 | Column .03 | Column .04 |
|---|---|---|---|---|---|
| 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 |
| 1.3 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 |
For this layout, the entry in row \(1.2\) and column \(.03\) corresponds to \(z=1.23\), and the number in that cell is \(0.8907\). The table gives an approximation because the printed entries are rounded.
Worked Example: Looking Up a Positive Z-Score
Suppose a measurement has already been converted to a z-score of \(1.23\). Use a cumulative standard normal table to find and interpret its table value.
Locate the cell. Split the score into its tenths and hundredths places: \(1.23=1.2+0.03\). Use row \(1.2\) and column \(.03\). Their intersection is \(0.8907\).
Write the meaning of the entry.
In the standard normal model, about \(0.8907\), or \(89.07\%\), of the distribution lies at or below a z-score of \(1.23\). The entry is an area accumulated to the left, not a statement that the z-score is \(0.8907\).
Reading Negative Z-Scores
A table that includes negative scores may list signed row labels, such as \(-0.7\), and columns \(.00\), \(.01\), and so on. In that layout, row \(-0.7\) and column \(.04\) identify \(z=-0.74\): the column appends the hundredths digit to the magnitude while retaining the negative sign. The row and column labels together specify the score; do not treat them as ordinary signed numbers to add.
Worked Example: Using a Table With Negative Rows
Suppose a standardized measurement has z-score \(-0.74\), and the table includes negative z-scores. Find the cumulative table value.
Locate the cell. The tenths part is \(-0.7\), and the hundredths digit is \(4\). Use row \(-0.7\) and column \(.04\). The cell contains \(0.2296\).
Interpret the entry.
Thus about \(22.96\%\) of the standard normal distribution lies at or below \(-0.74\). Since this score is below the center \(0\), a cumulative area less than one-half is consistent with the curve’s symmetry.
Not every table prints negative z-scores. A positive-only table can still be used because the standard normal density is symmetric about zero. For any positive \(a\), the area to the left of \(-a\) equals the area to the right of \(a\), which is the total area \(1\) minus the area to the left of \(a\). Therefore,
This relationship is useful for locating a negative score when the available table lists only nonnegative rows and columns. It changes how the cell is found, not what the final quantity means: the result is still the area to the left of the original negative z-score.
Worked Example: Finding a Negative Score in a Positive-Only Table
A positive-only cumulative table is available, and the z-score is \(-1.35\). Find the left-tail area.
Look up the matching positive score. In row \(1.3\) and column \(.05\), the table gives \(\Phi(1.35)=0.9115\).
Use symmetry and calculate. The cumulative area at \(-1.35\) is the remainder after the positive score’s left-tail area:
So about \(8.85\%\) of the standard normal distribution lies at or below \(-1.35\). The subtraction is needed because the positive-only table did not directly display the negative score.
A Short Justification for the Negative-Score Lookup
The symmetry relationship follows from the standard normal density being an even function: \(\phi(-t)=\phi(t)\). The total area under the density is \(1\), and the area to the left of \(-a\) matches the area to the right of \(a\). The area to the right of \(a\) is the total area minus the area to its left. Hence, for \(a>0\),
In the second line, the change of variable \(u=-t\) maps the interval from \(-\infty\) to \(-a\) onto the interval from \(a\) to \(\infty\), and symmetry gives \(\phi(-u)=\phi(u)\). The next line uses the fact that the entire standard normal density has area \(1\). This explains why a positive-only table can supply the needed value.
Checking What the Table Entry Means
A reliable table lookup has two separate parts: identify the cell and interpret its contents. The cell is determined by the supplied z-score and the table’s layout. The number in that cell is then read as \(\Phi(z)\), the proportion of standard normal area at or below that score. Keeping these parts distinct helps avoid confusing the location on the horizontal axis with the area under the curve.
Table formats can differ. Some include negative rows; others provide only nonnegative scores. Some headings display the hundredths as \(.00\) through \(.09\), while others use \(0.00\) through \(0.09\). Check the headings and any key printed with the particular table rather than assuming every table is arranged identically. In a signed table, also verify how negative rows and columns are combined.
Entries are typically rounded to four decimal places, so they are approximations. If a z-score is supplied to two decimal places, locate those digits directly. If it has more decimal places than the table supports, do not quietly change the score without considering the precision required; follow the stated directions or use a calculation method with adequate precision.
Common Mistakes
- Reading the entry as a z-score. A cell value such as \(0.8907\) is an area, not the score \(z\). The row and column identify the score; the intersection supplies its cumulative probability.
- Calling the entry a density height. A normal density height is not a probability. The cumulative table entry is an area under the curve from the far left through \(z\).
- Reversing the negative-score digits. In a signed table, row \(-0.7\) and column \(.04\) typically represent \(-0.74\), not \(-0.66\). Read the table’s convention carefully.
- Using a positive-only entry directly for a negative score. A positive-only table does not directly give \(\Phi(-a)\) when you look up \(a\). Use symmetry: \(\Phi(-a)=1-\Phi(a)\).
- Forgetting the direction of accumulation. “Cumulative” here means area to the left. Do not interpret the displayed value as area to the right.
Check Your Understanding
Answer each question by identifying the table lookup and stating what its entry means.
- For \(z=0.86\), which row and column would you use in a table with tenths-place rows and hundredths-place columns?
- A signed table gives \(0.3085\) at row \(-0.5\), column \(.00\). Express this entry using \(\Phi\) and describe the area it represents.
- A positive-only table reports \(\Phi(0.90)=0.8159\). Use symmetry to find the cumulative value for \(z=-0.90\).
- Explain why a cumulative table entry is not the height of the standard normal curve at the corresponding z-score.
- A learner says that the entry for \(z=1.00\) is the z-score itself. Identify the error and explain what the entry actually represents.