# What Are Z-Tests and When Are They Used?

A z-test is a statistical test that determines if two population means are different when a large sample size and known variances are involved.

The z-test is assumed to have normal distribution, and nuisance parameters should be known, like standard deviation, for accurate results. Standard deviation is a measure of the dispersion or variance between a group of values. If values in a group are close to its mean, you have a low standard deviation. If the group’s values fall over a broader range, you have a high deviation.

This type of test also tests hypotheses where the z-statistic follows normal distributions, and it’s ideal for samples of more than 30, since, according to the central limit theorem, when the number of samples grows, these samples are thought to be close to normally distributed. This theorem assumes that all samples are the same size, no matter the population distribution shape.

Normal distribution is a probability distribution that looks symmetrical and has more data near the mean of a group of values than it does farther away from that mean. This makes the data look like a bell curve when shown graphically.

## What is a Z-Score?

A z-score or z-statistic is the number indicating how many standard deviations above or below the mean population a z-test’s score is. It can be positive, negative, or zero. Basically, the z-score is a measurement of a number’s relationship to a group of values’ mean. If you have 0 as a z-score, it means the data point’s score is the same as the mean score. If you have 1 as a z-score, it means the value is one standard deviation above the mean. A -1 z-score indicates one standard deviation below the mean.

## Z-Tests in Commercial Real Estate

When you’re looking to invest in commercial real estate, you need to be confident in your decision and the opportunity over the long term. A z-test helps with various financial analyses that can give you that assurance.

## Z-Test Steps

First, state the alternative (what you want to believe is true) and null (a population parameter like the mean being a hypothesized value) hypotheses along with the alpha (significance level representing the probability of getting your results because of chance) and z-score/z-statistic. Then, calculate the test statistic with this formula:

**z = (p−P) / σ **

P = hypothesized value of population proportion in the null hypothesis

P = sample proportion

σ = standard deviation of the sampling distribution

Finally, note the results and conclusion.

The following are examples of z-tests:

- One-sample location test
- Two-sample location test
- Paired difference test
- Maximum likelihood estimate

## T-Test vs. Z-Test

T-tests are quite similar to z-tests, except that they are used when you have a small sample size of under 30. T-tests also assume an unknown standard deviation, whereas z-tests assume this is known. When using a z-test in a case where the population’s standard deviation isn’t known but the sample size is at least 30, the sample variance is assumed to be the same as the population variance.