Z-Score [standard Score]

A z-score is a statistical measure that describes the position of a raw score in terms of its distance from the mean, measured in standard deviation units. A positive z-score indicates that the value lies above the mean, while a negative z-score indicates that the value lies below the mean.

It is also known as a standard score because it allows scores on different variables to be compared by standardizing the distribution.

A standard normal distribution (SND) is a normal distribution with a fixed mean and spread. As Figure 1 shows, its mean is 0 and its standard deviation (SD) is 1.

Why Are Z-Scores Important?

It is useful to standardize the values (raw scores) of a normal distribution by converting them into z-scores because:

1. Probability estimation: Z-scores can be used to estimate the probability of a particular data point occurring within a normal distribution. By converting z-scores to percentiles or using a standard normal distribution table, you can determine the likelihood of a value being above or below a certain threshold.
2. Hypothesis testing: Compare a sample statistic’s z-score with critical values to decide whether to reject or fail to reject the null hypothesis, the claim of no difference or effect. Failing to reject does not prove it true.
3. Comparing datasets: Z-scores allow you to compare data points from different datasets by standardizing the values. This is useful when the datasets have different scales or units.
4. Identifying outliers: Z-scores help identify outliers, which are data points significantly different from the rest of the dataset. Typically, data points with z-scores greater than 3 or less than -3 are considered potential outliers and may warrant further investigation.

Z-Scores in Psychological Testing

Many psychological tests report scores on rescaled z-scales. Each is a linear transformation of the z-score, chosen to avoid negative numbers and decimals.

• IQ-style scores: 100 + 15z. A score of 130 is z = +2, the 97.7th percentile.
• T-scores: 50 + 10z. These are not the same as the t statistic from a t-test.
• Admission-test scores: 500 + 100z in an older convention.
• Wechsler subtests: Scaled to a mean of 10 and a standard deviation of 3.

A percentile rank is the cumulative area written as a percentage. It is not the percentage of items answered correctly. A person can answer 55% of items correctly and still sit at the 95th percentile on a hard test.

Z-scores also put different measures side by side. A student scores 118 on a verbal test (mean 100, standard deviation 15). The same student scores 62 on a memory test (mean 50, standard deviation 8).

The z-scores are +1.20 and +1.50, about the 88th and 93rd percentiles. The memory score is stronger. That holds even though 62 is the smaller number.

Clinicians use the same logic to judge whether a client’s change exceeds measurement error. Jacobson and Truax (1991) proposed a reliable change index: the change score divided by the standard error of the difference between two scores.

A value beyond ±1.96 is unlikely to reflect measurement error alone at the 0.05 level. Such a change is called reliable.

On an anxiety scale with a baseline standard deviation of 7.0 and a reliability of 0.88, a 9-point improvement gives an index of 2.62. That counts as reliable. A 5-point improvement gives 1.46 and does not.

How To Calculate

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation.

As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

The population mean and standard deviation are usually unknown. In that case, calculate the standard score from the sample mean (x̄) and sample standard deviation (s), which estimate the population values.

To calculate a z-score, follow these steps:

1. Identify the individual score (x) you want to convert to a z-score.
2. Determine the mean (μ or mu) of the dataset. The mean is the average of all the scores.
3. Calculate the standard deviation (σ or sigma) of the dataset. The standard deviation measures how spread out the scores are from the mean.
4. Subtract the mean (μ) from the individual score (x). This will give you the difference between the score and the mean.
5. Divide the difference you calculated in step 4 by the standard deviation (σ). The result is the z-score.

A set of z-scores always has a mean of exactly 0 and a standard deviation of exactly 1, whatever the raw scores were. Subtracting the mean centers the scores, and dividing by the standard deviation rescales their spread.

The shape does not change. A bell-shaped distribution stays bell-shaped, and a skewed one stays skewed. Standardizing changes the unit of measurement, not the distribution.

Statistical software usually divides by N − 1 when it standardizes a sample. The population formula divides by N instead, so the two choices give slightly different z-scores. State which one you used.

• If you have a one-tailed test, you will look for the area to the left (for a left-tailed test) or right (for a right-tailed test) of your z-score.
• If you have a two-tailed test, you will look for the area in both tails combined.

The significance level (α) is the probability threshold for rejecting the null hypothesis. Common significance levels are 0.01, 0.05, and 0.10.

The critical values are the z-scores that correspond to the chosen significance level. These values can be found using a standard normal distribution table or calculator.

The table below lists the most common critical values. An observed z-score beyond the critical value is statistically significant at that level.

Significance level (α)	Two-tailed critical value	One-tailed critical value
0.10	±1.645	1.282
0.05	±1.960	1.645
0.01	±2.576	2.326
0.001	±3.291	3.090

These cut-offs are conventions, not laws of nature. Cowles and Davis (1982) traced the 0.05 level to early twentieth-century statements about when a deviation was too large to blame on chance. Fisher later restated those conventions.

A Z-score table shows the percentage of values (usually a decimal figure) to the left of a given Z-score on a standard normal distribution.

Check Which Layout Your Z-Table Uses

A z-table (standard normal table) is a printed or on-screen grid of areas under the standard normal curve, indexed by z-score. Z-tables are not all the same, and reading the wrong layout gives an answer that looks plausible but is wrong. Three layouts are common.

• Cumulative table: Each entry is the area to the left of z. The entry for z = 0 is 0.5000, and entries rise toward 1.
• Mean-to-z table: Each entry is the area between the mean and z. The entry for z = 0 is 0.0000, and entries rise toward 0.5.
• Tail table: Each entry is the area beyond z in one tail. The entry for z = 0 is 0.5000, and entries fall toward 0.

To identify an unfamiliar table, look up z = 1.00. A cumulative table gives 0.8413, a mean-to-z table gives 0.3413, and a tail table gives 0.1587. Looking up z = 0 is not enough, because cumulative and tail tables both give 0.5000 there.

The steps below assume a cumulative table, the most common layout and the one statistical software uses.

1. Identify the parts of the z-score:

• Round your z-score to two decimal places, because z-tables print two decimals. A z-score of 1.0934 becomes 1.09.
• Split it into a row label and a column label. The whole number plus the first decimal gives the row (1.2 for z = 1.24), and the second decimal gives the column (0.04).
• Rounding to one decimal place can change the answer. In a cumulative table, z = 1.09 gives 0.8621 but z = 1.1 gives 0.8643, a gap that can matter near a significance threshold.

2. Find the corresponding probability in the z-score table:

• Z-score tables print the row labels (the whole number and first decimal) down the left column and the column labels (the second decimal) across the top row
• Locate the row for your z-score in the leftmost column (1.2 for z = 1.24)
• Move across the row until you reach the column that matches the second decimal (0.04)
• The value where the row and column meet is the area under the curve (the proportion of scores in that stretch of the distribution) associated with your z-score
• In a cumulative table, z = 1.24 gives 0.8925, so about 89% of scores fall below it

3. Interpret the probability:

• For a left-tailed test, the probability you found in the table is your p-value
• For a right-tailed test, subtract the probability you found from 1 to get your p-value
• For a two-tailed test, double the tail area beyond the absolute value of your z-score. For a positive z-score, that is 2 × (1 − table value). For a negative z-score, it is 2 × the table value
• Compare the probability to your chosen alpha level (0.05 or 0.01). If the probability is less than the alpha level, the result is considered statistically significant

A result is statistically significant when its p-value falls below the significance level chosen in advance, usually 0.05. The p-value is the probability of a result at least this extreme if the null hypothesis, and every other assumption behind the test, were true.

Even academic psychologists misread it (Badenes-Ribera et al., 2016). The p-value is not the probability that the null hypothesis is true. Nor is it the probability that chance alone produced the result, or that the effect is real (Greenland et al., 2016; Wasserstein & Lazar, 2016).

A small p-value says the data sit awkwardly with the null model. It does not say the effect is large. Significance also depends on sample size.

Worked Example: A Two-Tailed Z-Test

A wellbeing scale has a population mean of 50 and a standard deviation of 10. Twenty-five participants complete a four-week program and score a mean of 54.5. Does this group differ from the population mean?

1. State the hypotheses: H0 is μ = 50 and H1 is μ ≠ 50. Set α = 0.05 before seeing the data.
2. Calculate the standard error (the typical gap between a sample mean and the population mean): 10 / √25 = 2.0. Then z = (54.5 − 50) / 2.0 = +2.25.
3. Look up the area. The area to the left of 2.25 is 0.9878, so the right tail is 0.0122.
4. Double the tail for a two-tailed test: p = 2 × 0.0122 = 0.0244.
5. Decide. The z of 2.25 is beyond the critical value of ±1.96, so reject the null hypothesis.

At α = 0.01 the same data fail to reject the null hypothesis, because 2.25 does not reach ±2.576. A result is only significant relative to a threshold fixed in advance.

Notice the divisor.

A sample mean is divided by the standard error, not by the standard deviation of individual scores. Dividing by 10 would give z = 0.45 and wrongly suggest nothing unusual.

Report an effect size as well. Here d = (54.5 − 50) / 10 = 0.45, a small-to-medium effect by the usual benchmarks. The same 4.5-point difference in 100 participants gives z = 4.5 and a tiny p-value, yet d is still 0.45.

Common Mistakes When Using Z-Scores and Z-Tables

Most errors come from the wrong tail, the wrong layout or the wrong divisor. Sketch the bell curve and shade the area the question asks about before you touch the table.

• Wrong tail: A cumulative table always gives the area to the left. For “the proportion who scored higher,” subtract the entry from 1.
• Wrong layout: A mean-to-z table read as cumulative is off by exactly 0.5. Check with z = 1.00 first.
• Negative z-scores: There is no negative area. Look up the absolute value, then decide whether you need the entry or 1 minus it.
• Two-tailed p-values: Double the single tail beyond the absolute value of z. Doubling the cumulative entry of a positive z gives a value above 1.
• Z as a probability: A z-score is a position on the axis, not a probability. The probability is the area linked to that position.
• Wrong divisor: For a sample mean, divide by the standard error (σ / √n), not by the standard deviation σ.
• “Accepting” the null: A p-value above α means you fail to reject the null hypothesis. It does not show the null hypothesis is true.

Critical Evaluation

The arithmetic of a z-score is simple and reliable. The weaknesses lie in how it is used. Four stand out.

The Normal Distribution Assumption

The z-score formula works on any distribution. The z-table does not. Reading a percentile or p-value from it treats the tails of your data as if they were the tails of a normal distribution.

Standardizing never changes a distribution’s shape, so a skewed variable stays skewed. Real psychological measures are often far from normal, as Micceri (1989) showed.

With very large samples a normality test detects trivial departures, so the shapes found matter more than the significance count. The warning is sharpest in the tails, where percentile conversions are most sensitive. Percentiles counted directly from the ranked scores need no assumption about shape.

Outliers and Small Samples

A z above 3 is a prompt, not proof. Even in perfectly normal data, about 2.7 scores per 1,000 lie beyond ±3.

Extreme scores also distort the mean and standard deviation used to judge them. One outlier can inflate the standard deviation enough to hide its own z-score. This is called masking. Leys et al. (2013) therefore recommended the median and the median absolute deviation instead.

Small samples add a hard ceiling. With n scores and the sample standard deviation, no z-score can exceed (n − 1) / √n in size. The ceiling is 1.79 for five scores, 2.85 for ten and 4.25 for twenty.

In nine zeros and one 100, the extreme score has a z of 2.85. It never reaches a cut-off of three.

Estimated Standard Deviations

The z-table assumes the standard deviation is known. When it is estimated from a small sample, the normal curve understates the uncertainty. Student (1908) worked out the correction.

Using z-table areas for a small-sample statistic therefore gives p-values that are too small.

In the worked example above, suppose the standard deviation of 10 had been estimated from the 25 scores. The two-tailed p-value would then rise from 0.0244 to 0.034.

Reference Groups and Norms

A z-score only says where someone stands compared with a specific group. Change the reference group and the same raw score earns a different z-score. This is why norm groups matter.

Behavioral-science samples are drawn heavily from Western, educated, industrialized, rich and democratic societies, which prove to be unusual outliers across many measured domains (Henrich et al., 2010). A norm group built from such samples inherits that narrowness.

Someone assessed against a group that does not resemble them is placed on the wrong ruler. Norms also drift over time, as the Flynn effect shows.

This matters when cut-offs feed decisions about people. A cut-off such as an IQ about two standard deviations below the mean turns a gradual difference into a sharp category.

Contemporary Research

Recent meta-research asks whether the normal curve is the right ruler for real psychological data. It consistently answers: often not.

An analysis of 693 small-sample distributions of cognitive and other psychological measures found only 5.5% close to normal values of skewness and kurtosis (Blanca et al., 2013). Larger collections point the same way.

These are descriptive studies. They show how common non-normality is, not how often it changes a conclusion.

How Common Is Non-Normal Data?

Cain et al. (2017) used real data and a simulation anchored in observed values. The data came from authors in two journals, so they are not a random draw of all psychology.

The simulations concern inferential tests, not z-table percentiles.

Modelled Norms Instead of Table Lookups

Test norms are conventionally built by ranking scores within age bands and converting them to standard scores. Regression-based continuous norming instead models raw scores as a smooth function of percentile and age.

Lenhard and Lenhard (2021) compared the two in a simulation. They cross-validated the norms on a representative sample of 840,000 cases across 90,000 repetitions.

Both approaches improved with sample size. Modelled norms reached optimal results with much smaller samples. A simulation is weaker evidence than clinical data, but it knows the true answer.

Do Researchers Misread P-Values?

This is a convenience sample with a 7% response rate. Respondents may have been more statistically knowledgeable than non-respondents, so the figures may understate the problem.

The survey also measures quiz answers, not how people interpret p-values in their own analyses. Expert guidance already warns against these misreadings (Greenland et al., 2016; Wasserstein & Lazar, 2016). That guidance is a statement of consensus, not new data.

Together, the studies point one way. A correct table lookup can still mislead, either because the data are not normal or because the p-value is misread.

References

Badenes-Ribera, L., Frias-Navarro, D., Iotti, B., Bonilla-Campos, A., & Longobardi, C. (2016). Misconceptions of the p-value among Chilean and Italian academic psychologists. Frontiers in Psychology, 7, Article 1247. https://doi.org/10.3389/fpsyg.2016.01247

Blanca, M. J., Arnau, J., López-Montiel, D., Bono, R., & Bendayan, R. (2013). Skewness and kurtosis in real data samples. Methodology, 9(2), 78–84. https://doi.org/10.1027/1614-2241/a000057

Cain, M. K., Zhang, Z., & Yuan, K.-H. (2017). Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence, influence and estimation. Behavior Research Methods, 49(5), 1716–1735. https://doi.org/10.3758/s13428-016-0814-1

Cowles, M., & Davis, C. (1982). On the origins of the .05 level of statistical significance. American Psychologist, 37(5), 553–558. https://doi.org/10.1037/0003-066X.37.5.553

Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350. https://doi.org/10.1007/s10654-016-0149-3

Henrich, J., Heine, S. J., & Norenzayan, A. (2010). The weirdest people in the world? Behavioral and Brain Sciences, 33(2–3), 61–83. https://doi.org/10.1017/S0140525X0999152X

Jacobson, N. S., & Truax, P. (1991). Clinical significance: A statistical approach to defining meaningful change in psychotherapy research. Journal of Consulting and Clinical Psychology, 59(1), 12–19. https://doi.org/10.1037/0022-006X.59.1.12

Lenhard, W., & Lenhard, A. (2021). Improvement of norm score quality via regression-based continuous norming. Educational and Psychological Measurement, 81(2), 229–261. https://doi.org/10.1177/0013164420928457

Leys, C., Ley, C., Klein, O., Bernard, P., & Licata, L. (2013). Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. Journal of Experimental Social Psychology, 49(4), 764–766. https://doi.org/10.1016/j.jesp.2013.03.013

Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105(1), 156–166. https://doi.org/10.1037/0033-2909.105.1.156

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.2307/2331554

Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108

Further Information

• How to Use a Z-Table (Standard Normal Table) to Calculate the Percentage of Scores Above or Below the Z-Score
• Z-Score Table (for positive or negative scores)
• Statistics for Psychology Book Download