Z-Score: Definition, Formula, Calculation & Interpretation (2024)

A z-score is a statistical measure that describes the position of a raw scorein 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 normally shaped distribution with a mean of 0 and a standard deviation (SD) of 1 (see Fig. 1).

Why Are Z-Scores Important?

It is useful to standardize the values (raw scores) of anormal distributionby 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: Z-scores are used in hypothesis testing to determine the significance of results. By comparing the z-score of a sample statistic to critical values, you can decide whether to reject or fail to reject a null hypothesis.
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.

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.

When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean (x̄) and sample standard deviation (s) as estimates of 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.

Interpretation

The value of the z-score tells you how many standard deviations you are away from the mean. A larger absolute value indicates a greater distance from the mean.

• Positive z-score: If a z-score is positive, it indicates that the data point is above the mean. For example, a z-score of 1.5 means the data point is 1.5 standard deviations above the mean.
• Negative z-score: If a z-score is negative, it indicates that the data point is below the mean. For example, a z-score of -2 means the data point is 2 standard deviations below the mean.
• Zero z-score: A z-score of zero indicates that the data point is equal to the mean.

Another way to interpret z-scores is by creating a standard normal distribution, also known as the z-score distribution, or probability distribution (see Fig. 3).

Probability Estimation

When working with z-scores, the data is assumed to follow a standard normal distribution with a mean of 0 and a standard deviation of 1. This allows for the use of standard normal distribution tables or calculators to determine probabilities.

The z-score tells us how many standard deviations a data point is from the mean. Once we know the z-score, we can estimate the probability of a data point falling within a specific range or being above or below a certain value.

In a standard normal distribution, there’s a handy rule called the empirical rule, or the 68-95-99.7 rule. This rule states that:

• Approximately 68% of the data falls within one standard deviation of the mean (z-scores between -1 and 1).
• Around 95% of the data falls within two standard deviations of the mean (z-scores between -2 and 2).
• Nearly 99.7% of the data falls within three standard deviations of the mean (z-scores between -3 and 3).

Figure 3 shows the proportion of a standard normal distribution in percentages. As you can see, there’s a 95% probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean.

See Also

How to calculate Z-scores (formula review) (article) | Khan Academy

Using the standard normal distribution, researchers can calculate the probability of randomly obtaining a score from the sample. For example, there’s a 68% chance of randomly selecting a score between -1 and +1 standard deviations from the mean.

Hypothesis Testing

Using a z-score table lets you quickly determine the probability associated with a specific value in a dataset, helping you make decisions and draw conclusions based on your data.

• 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.

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.

1. Identify the parts of the z-score:

• The z-score consists of a whole number and decimal parts
• For example, if your z-score is 1.24, the whole number part is 1, and the decimal part is 0.24

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

• Z-score tables are usually organized with the whole number part of the z-score in the leftmost column and the decimal part across the top row
• Locate the whole number part of your z-score in the leftmost column
• Move across the row until you find the column that matches the decimal part of your z-score
• The value at the intersection of the row and column is the probability (area under the curve) associated with your z-score

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, if your z-score is positive, double the probability you found to get your p-value; if your z-score is negative, subtract the probability from 1 and then double the result to get your p-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

In statistical analysis, if there is less than a 5% chance of randomly selecting a particular raw score, it is considered a statistically significant result. This means the result is unlikely to have occurred by chance alone and is more likely to be a real effect or difference.

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