Z-Score Calculator
This z-score calculator computes the z-score and probabilities associated with a normal distribution. It also provides a figure depicting the z-score in a standard normal distribution. Additionally, the calculator can convert between z-score and p-value. To use this calculator, simply select the desired calculation from the tabs above the calculator, enter the values, and click the 'Calculate' button.
The z-score, also known as a standard score, quantifies how many standard deviations an element is from the mean of its distribution. Essentially, it converts data from different scales to a common scale with a mean of zero and a standard deviation of one, thereby standardizing the data. This standardization allows for the direct comparison of scores from different normal distributions and is useful for making statistical comparisons and calculations.
Formula for calculating the z-score
The z-score is calculated using the following formula:
| Z = |
|
μ is the mean of the data set,
σ is the standard deviation of the data set.
This formula adjusts the score x by subtracting the mean μ and dividing the result by the standard deviation σ. The outcome is a score (z-score) that tells us how far and in what direction the original value lies from the mean, measured in units of standard deviation.
Example 1: Consider a set of test scores with a mean (μ) of 75 and a standard deviation (σ) of 10. Given that a student scores an 85 on this test, the z-score is calculated as follows:
| Z = |
| = 1 |
The z-score indicates that the student's score is 1 standard deviation above the mean. If the score data follow a normal distribution, the probability associated with a z-score of 1 is 0.841, which means that approximately 84.1% of students scored below an 85, while 15.9% of students scored above an 85.
Example 2: Imagine a study in which the mean BMI is 22 with a standard deviation of 2. Given that an individual in the study has a BMI of 18, the corresponding z-score is calculated as follows:
| Z = |
| = -2 |
This result indicates that the individual's BMI is 2 standard deviations below the mean. If the BMI data follows a normal distribution, the probability associated with a z-score of -2 is 0.023, which means that approximately 2.3% of people have a BMI lower than 18, while about 97.7% of people have a BMI higher than 18.
Interpreting the z-score
The z-score tells us both the direction (above or below) and the degree to which each entry in a dataset diverges from the mean, expressed in terms of standard deviation. The list below describes how to interpret various z-scores:
| Z = 0: | The score is identical to the mean. |
| Z > 0: | The score is above the mean. |
| Z < 0: | The score is below the mean. |
| |Z| > 1: | The score is more than one standard deviation away from the mean. |
| |Z| > 2: | The score is more than two standard deviations away from the mean, which is considered unusual. |
| |Z| > 3: | The score is more than three standard deviations away from the mean, so the data point is considered an outlier. |
The larger the z-score, the farther the data point is from the mean; the smaller a z-score, the closer it is to the mean.
Z-scores are especially useful in situations where comparison between different data sets (which means different normal distributions) is necessary. For instance, in standardized testing, z-scores are used to compare the performance of students from different classrooms or schools. In finance, z-scores help assess the risk of investment portfolios by comparing the volatility of various assets against the market or sector averages.
What is a p-value?
A p-value is a statistical value that tells us how likely it is for us to obtain the observed data, or more extreme data, assuming that the null hypothesis (a claim that the effect being studied does not exist) is true. It does not tell us the probability that the null hypothesis is true, given the data; it only tells us the probability of the data. There is a lot to unpack here, and the following is an attempt to illustrate what a p-value is and how it is used.
Interpreting p-value
Because it represents a probability, a p-value ranges between 0 and 1, with a smaller p-value indicating that the data is further away from the range predicted by the null hypothesis. A p-value of 0.05 or smaller indicates that the observed data is surprising, and therefore indicates statistical significance. A p-value larger than 0.05 indicates the observed data is not surprising, as it falls within the expected range. A p-value smaller or greater than 0.05 does not necessarily indicate that there is or isn't an effect. A larger sample size may be necessary to detect an effect, and statistical significance provides an indication that an effect may merit further study.
P-value example
As an example, consider an experiment where we have a coin and we want to determine whether it is a fair coin. In this case we form the null hypothesis that if the coin is fair and we flip it a large number (the more the better) of times, we expect an approximately equal incidence of heads and tails (50/50). Say we flipped the coin one hundred times and found that it landed on tails 49 times and heads 51 times. In this case we would not expect that anything is awry, and were we to test it, we would find a relatively high p-value. Consider instead that we found the coin to land on tails 10 times and on heads 90 times. In this case we might suspect that the coin is not fair. Assuming that the coin is fair, the likelihood that it lands on tails only 10 times in 100 tosses is extremely low—the p-value would be low, certainly below 0.05, likely lower. Let's say we calculated a p-value of 0.03. It is important to note that this means that there is a 3% chance that a value as extreme or more extreme than 10 tails in 100 tosses would occur given the assumption that the null hypothesis (the coin is fair) is true. It does not mean that the alternative hypothesis (that the coin is unfair) is 97% likely. It could be the case that the coin is unfair, but the p-value only describes the likelihood of the data; it says nothing about the theory itself.
The benefit of a p-value is that given repeated testing, if we find a significant effect (p < 0.05) consistently, we can be confident that the effect is not just noise or random variation. It gives us a basis for rejecting the null hypothesis and exploring further to potentially discover a new theory.