Reading Mean-to-Z Standard Normal Tables

What a Mean-to-Z Table Reports

Standard normal tables use \(Z\sim N(0,1)\), a normal random variable with mean 0 and standard deviation 1. A mean-to-z table reports the area under the standard normal curve between its center, 0, and a specified positive z-score. This is a different quantity from the cumulative table entry: the two tables describe areas measured from different starting points.

Before using a table, check its title or a note at the top. A mean-to-z table may be labelled with wording such as “area from 0 to z.” Its entries are not probabilities to the left of the z-score. Once the table convention is clear, the lookup is straightforward: use the z-score’s first decimal place for the row and its second decimal place for the column.

The entries range from 0 at \(z=0\) toward 0.5000 as \(z\) increases. The curve has half of its total area on either side of 0, so the area from 0 to a positive cutoff can never exceed 0.5. A value of, for example, 0.3000 means that 30% of the standard normal distribution lies between the mean and that positive z-score.

Finding a Positive Z-Score in the Table

A common layout lists z-scores in increments of 0.01. The row gives the ones and tenths digits, and the column gives the hundredths digit. For example, to look up \(z=1.23\), use the row labelled 1.2 and the column labelled 0.03. The entry at their intersection is the area from 0 to 1.23.

The row-and-column arrangement is only a way to locate the value. The table entry itself is a rounded approximation to the area under the curve, not a density height. In particular, it is not the value of the normal density at \(z\). If a requested z-score falls between the values printed in the table, use the nearest available hundredth unless the problem or table instructions specify interpolation.

Using Symmetry for Negative Z-Scores

Mean-to-z tables are usually printed for nonnegative z-scores. To find an area between a negative z-score and zero, use the positive version of its magnitude. The standard normal curve is symmetric about zero, so its area from \(-a\) to 0 equals its area from 0 to \(a\), for any \(a>0\). Thus the same table entry gives the corresponding area on either side of the mean.

The absolute value is used only to find the matching table entry. It does not mean the original z-score has changed sign: a negative z-score still lies to the left of the mean. Be precise about which interval the entry represents. For \(z=-1.37\), for example, the matching area is between \(-1.37\) and 0, not between 0 and \(-1.37\) in reversed order.

Why the Symmetry Lookup Works

The standard normal density has the same height at \(x\) and \(-x\). If \(\phi(x)\) denotes that density, then \(\phi(-x)=\phi(x)\). The area from \(-a\) to 0 can be rewritten by changing the variable of integration from \(x\) to \(u=-x\): the left-hand area becomes the area from 0 to \(a\).

The second line uses the substitution \(u=-x\), which reverses the integration limits and introduces a minus sign that is accounted for when the limits are put in increasing order. The third line follows from the evenness of the standard normal density. This proves why a positive table entry can also be used for the matching interval on the negative side.

How This Differs From a Cumulative Table

A cumulative standard normal table, as used in the earlier material on reading cumulative standard normal tables, reports the area to the left of a z-score. In this convention, the entry at \(z\) is \(\Phi(z)=P(Z<z)\). A mean-to-z table instead reports only the area between 0 and a positive z-score. The number in the two tables can therefore differ substantially even when both tables use the same z-score.

For a positive z-score, the cumulative area to the left includes all the area to the left of the mean, which is 0.5, as well as the area from the mean to the z-score. So the cumulative-table value equals 0.5 plus the mean-to-z entry. This relationship helps identify which convention a table is using; it does not change what either table’s entry means.

For a negative score, a mean-to-z table still gives the area between that score and zero by symmetry. A cumulative table’s entry, however, is defined as the area to the left of the score. These remain different regions. Before comparing or using values, identify the interval represented by each table rather than relying on the table’s numerical layout alone.

Common Mistakes and Checks

The most common error is to treat a mean-to-z entry as though it were a cumulative probability. For instance, reading 0.3907 at \(z=1.23\) and claiming that 39.07% of the distribution lies to the left of 1.23 uses the wrong interval. The correction is to say that 39.07% lies between 0 and 1.23; a cumulative table would include the area to the left of zero as well.

A second mistake is to look up a negative score as if it were a negative table entry. The areas in the table are nonnegative. Use the magnitude of the score to find the row and column, then state the matching interval on the correct side of zero. For example, the entry for 1.37 represents the interval from 0 to 1.37, and by symmetry it also gives the area from \(-1.37\) to 0.

Use these checks when reading a mean-to-z table:

• Check the convention. Confirm that the table reports area between 0 and z, rather than area to the left of z.
• Check the lookup. For \(z=1.23\), the row is 1.2 and the column is 0.03.
• Check the range. A mean-to-z entry must be between 0 and 0.5.
• Check the interval. For a negative cutoff, describe the area as between the cutoff and 0.
• Check the interpretation. An entry is an area under the curve, not a density height or the probability that a particular continuous variable equals one exact value.