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:
- 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.
- 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.
- 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.
- 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:
- Identify the individual score (x) you want to convert to a z-score.
- Determine the mean (μ or mu) of the dataset. The mean is the average of all the scores.
- Calculate the standard deviation (σ or sigma) of the dataset. The standard deviation measures how spread out the scores are from the mean.
- Subtract the mean (μ) from the individual score (x). This will give you the difference between the score and the mean.
- 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.
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.
Practice Problems for Z-Scores
Calculate the z-scores for the following:
Sample Questions
- Scores on a psychological well-being scale range from 1 to 10, with an average score of 6 and a standard deviation of 2. What is the z-score for a person who scored 4?
- On a measure of anxiety, a group of participants show a mean score of 35 with a standard deviation of 5. What is the z-score corresponding to a score of 30?
- A depression inventory has an average score of 50 with a standard deviation of 10. What is the z-score corresponding to a score of 70?
- In a study on sleep, participants report an average of 7 hours of sleep per night, with a standard deviation of 1 hour. What is the z-score for a person reporting 5 hours of sleep?
- On a memory test, the average score is 100, with a standard deviation of 15. What is the z-score corresponding to a score of 85?
- A happiness scale has an average score of 75 with a standard deviation of 10. What is the z-score corresponding to a score of 95?
- An intelligence test has a mean score of 100 with a standard deviation of 15. What is the z-score that corresponds to a score of 130?
Answers for Sample Questions
Double-check your answers with these solutions. Remember, for each problem, you subtract the average from your value, then divide by how much values typically vary (the standard deviation).
- Z-score = (4 – 6)/2 = -1
- Z-score = (30 – 35)/5 = -1
- Z-score = (70 – 50)/10 = 2
- Z-score = (5 – 7)/1 = -2
- Z-score = (85 – 100)/15 = -1
- Z-score = (95 – 75)/10 = 2
- Z-score = (130 – 100)/15 = 2
Calculating a Raw Score
Sometimes, we know a z-score and want to find the corresponding raw score. The formula for calculating a z-score in a sample into a raw score is given below:
X = (z)(SD) + mean
As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean.
Check your answer makes sense: If we have a negative z-score, the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean.
Calculating a Z-Score using Excel
To calculate the z-score of a specific value, x, first, you must calculate the mean of the sample by using the AVERAGE formula.
For example, if the range of scores in your sample begins at cell A1 and ends at cell A20, the formula =AVERAGE(A1:A20) returns the average of those numbers.
Next, you must calculate the standard deviation of the sample by using the STDEV.S formula. For example, if the range of scores in your sample begins at cell A1 and ends at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers.
Now to calculate the z-score, type the following formula in an empty cell: = (x – mean) / [standard deviation].
To make things easier, instead of writing the mean and SD values in the formula, you could use the cell values corresponding to these values. For example, = (A12 – B1) / [C1].
Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), type the following into a blank cell: = NORMSDIST( and input the z-score you calculated).
To find the probability of LARGER z-score, which is the probability of observing a value greater than x (the area under the curve to the RIGHT of x), type: =1 – NORMSDIST (and input the z-score you calculated).
Frequently Asked Questions
Can z-scores be used with any type of data, regardless of distribution?
Z-scores are commonly used to standardize and compare data across different distributions. They are most appropriate for data that follows a roughly symmetric and bell-shaped distribution.
However, they can still provide useful insights for other types of data, as long as certain assumptions are met. Yet, for highly skewed or non-normal distributions, alternative methods may be more appropriate.
It’s important to consider the characteristics of the data and the goals of the analysis when determining whether z-scores are suitable or if other approaches should be considered.
How can understanding z-scores contribute to better research and statistical analysis in psychology?
Understanding z-scores enhances research and statistical analysis in psychology. Z-scores standardize data for meaningful comparisons, identify outliers, and assess likelihood.
They aid in interpreting practical significance, applying statistical tests, and making accurate conclusions. Z-scores provide a common metric, facilitating communication of findings.
By using z-scores, researchers improve rigor, objectivity, and clarity in their work, leading to better understanding and knowledge in psychology.
Can a z-score be used to determine the likelihood of an event occurring?
No, a z-score itself cannot directly determine the likelihood of an event occurring. However, it provides information about the relative position of a data point within a distribution.
By converting data to z-scores, researchers can assess how unusual or extreme a value is compared to the rest of the distribution. This can help estimate the probability or likelihood of obtaining a particular score or more extreme values.
So, while z-scores provide insights into the relative rarity of an event, they do not directly determine the likelihood of the event occurring on their own.
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
