# Z score and value relationship in numbers

### z-score: Definition

Read and learn for free about the following article: Z-scores review. A z-score measures exactly how many standard deviations above or below the mean a .. But if you have a set of values who's average is some number and then perform a . Z scores and the Empirical Rule. The heights of individuals Note a Z-score tells how many standard deviations a value is above or below the mean. Suppose. (X value) into a z-score or a standard score. • The purpose of the number of standard deviations. • Thus, in a Other Relationships Between z, X, μ, p.,, μ.

Remember that z-scores tell us how far a value is from the mean. When you "standardize" a variable, its mean becomes zero and its standard deviation becomes one. Areas under all normal curves are related. For example, the area percentage to the right of 1.

The term "area" will refer to "area percentage". The fact stated above is the reason we can find an area over an interval for any normal curve by finding the corresponding area under a standard normal curve with a mean of 0 and a standard deviation of 1. These subdivisions are fine for determining percentages as long as we are dealing with values that fall at these exact subdivision locations.

What do we do when the value does not fall at an Empirical Rule subdivision? By using z-scores, we have the ability to locate a percentage or area under a standard normal distribution at any location.

Z-scores allow for the calculation of area percentages also called proportions or probabilities anywhere along a standard normal distribution curve and, consequently along the corresponding normal distribution. The area percentage proportion, probability calculated using a z-score will be a decimal value between 0 and 1, and will appear in a Z-Score Table. Since the normal curve is symmetric about the mean, the area on either sides of the mean is 0. To find a specific area under a normal curve, find the z-score of the data value and use a Z-Score Table to find the area.

A Z-Score Table, is a table that shows the percentage of values or area percentage to the left of a given z-score on a standard normal distribution.

Negative Z-Score Table You need both tables! The label in the row contains the integer part and the first decimal of the z-score. The label for columns contains the second decimal of the z-score.

Each value in the body of the table is a cumulative area. Z-Score Tables come in different formats, determined by where the computations were started. Consider these two most popular formats: One form of the table yields probability or area starting at the mean and going to the right of the mean up to the needed z-score.

### An explanation of z-scores (standardized values)

These tables are usually labeled "cumulative from mean". This table basically works with half of the area under the normal curve, and the user must take this into consideration and make adjustments when using this table. This type of table lists positive z-scores only. What is a Z score What is a p-value Most statistical tests begin by identifying a null hypothesis. The null hypothesis for pattern analysis tools essentially states that there is no spatial pattern among the features, or among the values associated with the features, in the study area -- said another way: The Z score is a test of statistical significance that helps you decide whether or not to reject the null hypothesis.

The p-value is the probability that you have falsely rejected the null hypothesis. Z scores are measures of standard deviation. Both statistics are associated with the standard normal distribution. This distribution relates standard deviations with probabilities and allows significance and confidence to be attached to Z scores and p-values. Very high or a very low negative Z scores, associated with very small p-values, are found in the tails of the normal distribution.

## Z-scores review

When you perform a feature pattern analysis and it yields small p-values and either a very high or a very low negative Z score, this indicates it is very UNLIKELY that the observed pattern is some version of the theoretical spatial random pattern represented by your null hypothesis. In order to reject the null hypothesis, you must make a subjective judgment regarding the degree of risk you are willing to accept for being wrong.

To give an example: If your Z score is between If the Z score falls outside that range for example In this case, it is possible to reject the null hypothesis and proceed with figuring out what might be causing the statistically significant spatial pattern. A key idea here is that the values in the middle of the normal distribution Z scores like 0. When the absolute value of the Z score is large in the tails of the normal distribution and the probabilities are small, you are seeing something unusual and generally very interesting.

For the Hot Spot Analysis tool, for example, "unusual" means either a statistically significant hot spot or a statistically significant cold spot. The Null Hypothesis Many of the statistics in the spatial statistics toolbox are inferential spatial pattern analysis techniques i. Inferential statistics are grounded in probability theory. Probability is a measure of chance, and underlying all statistical tests either directly or indirectly are probability calculations that assess the role of chance on the outcome of your analysis.

Typically, with traditional non-spatial statistics, you work with a random sample and try to determine the probability that your sample data is a good representation is reflective of the population at large.