## # Z Score Calculator (More Math Tools) Calculating Z-Score Population Mean Standard Deviation Raw Score Z-Score: 1.5. Percentile: 93 Calculation Detail Z = (Raw Score - Population Mean) / Standard Deviation Z = (7 - 2.5) / 3

### How to use the Z Score Calculator

This z score calculator is intended to help you quickly convert your raw measurements of a sample's population mean into a Z score and probability estimate.

For a normally distributed variable with known parameters, enter the population mean and standard deviation. This defines the expected range of results given a normal distribution (aka the normal curve). Next, enter the raw score from your sample. Hit the calculate button. The calculator will calculate the z score based on comparing the sample mean with the population mean and population standard deviation. It will generate a standard score and probability (p value), which can be used to assess the confidence level. These z score calculation results are valid if the underlying source of the data follows the normal distribution curve.

### How Do I interpret the z score?

The z score formula itself is a ratio of the difference between the population mean and the sample mean, compared with the population standard deviation. The z score value is thus normalized, allowing you to compare this with the expected results of a standard normal distribution.

We use this for hypothesis testing: we're testing the hypothesis that the sample was drawn from a population similar to the one we're comparing it to. Our null hypothesis is that there is no significant difference between the samples and they were (effectively) drawn from similar populations. Our alternative hypothesis, which we will accept in lieu of the null hypothesis if the difference is large enough, is that something has changed.

For a reasonable sized sample, the mathematics of the standard normal distribution indicate the sample mean should be very close to the population mean. The further the distance between the two (in either direction), the greater the chance that the underlying population is different. This is what we're testing for. For small sample size(s), you will want to look at using other approaches, such as a t test. We have a statistics calculator for that as well.

### What are critical value(s) of a z score?

The critical values of a statistical test are the "cut off" points where we would accept the alternative hypothesis for a given confidence level. These are expressed as a probability of incorrectly accepting the alternative hypothesis. By convention, there are some common percentile values that are used for defaults: 1% (99% confidence level), 2.5% (97.5% confidence level), 5% (95% confidence interval), 10%. If you're not using the z score calculator, you can get these from a z score table (back of your statistics book, the z table). These are based on a standardized score, relative to the population variance.

In real world applications, you would balance the cost and risk associated with testing (and acting on false negatives) against the confidence level. Mistakes that are easily fixed with minimal cost may use a 10% confidence level; costly errors (for example, building a bridge) could require higher levels.

### What does a negative z score mean?

This means the raw score for your sample was below the expected mean value of the population. Conversely, a positive z score indicates the sample was above the mean value.

Mathematically this doesn't affect the analysis of a two sided test score.

### What is the altman Z score?

The altman Z score is a concept from finance, where a model was built to predict the bankruptcy public companies. It takes several variables that measure cash flow and liquidity to identify companies that are close to running out of money.

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