### PreMBA Analytical Methods

To solve this problem, you would use the Z score formula. Using the The difference is the probability of the random variable x falling between the two values. Correlation and Regression Overview Correlation Regression Analysis of Covariance. The z-score formula for a correlation is useful for conceptualizing a. A Z score is a number of standard deviations a score is above or below the mean. But to find out the exact Z score we turn to the Z score formula: Well this is how we are going to determine if there is a significant difference between groups.

See the table of areas under a standard normal curve which shows the z-score in the left column and the corresponding area in the next column. In actuality, the area represented in the table is only one half of the normal curve, but since the normal curve is symmetrical, the other half can also be estimated from the same table and the 0.

As an example use of the table, a z-score of 1. This indicates that The area beyond that particular z-score to the tail end of the distribution would be the difference between 0.

Now let's look at the lower half of the distribution down to a z-score of This too represents an area of. Sometimes statisticians want to accumulate all of the negative z-score area the left half of the curve and add to that some of the postive z-score area.

All of the negative area equals. Here is an example of how to use the table: The area from This is why 3 SD control limits have a very low chance of false rejections compared to 2 SD limits. The concept of the standard normal distribution will become increasingly important because there are many useful applications.

One useful application is in proficiency testing PTwhere a laboratory analyzes a series of samples to demonstrate that it can provide correct answers. The results from PT surveys often include z-scores. Other laboratories that analyzed this same sample show a mean value of and a standard deviation of 6. Said another way, there is only a 0.

Most likely it represents a measurement error by the laboratory. Because there is so much confusion concerning this topic, it's worthwhile to review the relation between z-score and standard deviations SD.

## Z-Stats / Basic Statistics

The lines between statistical definitions sometimes blur over time. Remember that the mean and standard deviation are the first statistics that are calculated to describe the variation of measurements or distribution of results.

The standard deviation is a term that has the same units as the measurement, therefore it can be used to describe the actual range of measurement results that might be expected.

A z-score can be calculated once the mean and standard deviation are available. Notice that SD is in the denominator of the z-score formula, so SD's and z's are not really the same. The z-score is a ratio and therefore is unitless, whereas the SD is expressed in concentration units.

## Z-scores review

That value lies in the 0. We often use this probability of 0. When the population mean and standard deviation are not known, which is a more typical situation, z-scores cannot be calculated. The t-distributions generally are said to have "heavier tails" as compared to the normal distribution. A t-value can be calculated much like a z-value, e. A t-distribution can be used to help in making the decision that the means of two samples are far enough apart to be considered to be different, i.

When N is less than 30, it will be necessary to look up a critical t-value from a table.

Self-assessment questions What is the probability of tossing three fair coins and getting all heads? As you might expect, a Z score of zero, when looked up in the table, yields a probability of.

This makes sense if you remember the definition of a normal distribution having half of its values distributed on either side of the mean.

**Find Standard Deviation with the Z-Score Formula**

Returning to the example of apparel shipments, you can look up the Z score of 2. The probability can be found by looking first down the left-hand column of the table for the number 2. Since the number after the decimal point is zero, use the first column titled. The probability that Z will be less than or equal to 2.

This number is highlighted in this Z table. This means that the probability that the shipment will arrive within 30 days is. Now let's look at how we can perform a hypothesis test, first, by using the Z score. We will start with an understanding of Z scores. A Z score is a number of standard deviations a score is above or below the mean.

### Standard Score - Definition of the Standard Score (Z-Score)

In the Standard Normal Distribution, the mean is always equal to 0 and the standard deviation is equal to 1. The Z scores help us to describe various aspects of the distribution, such as percentile ranks, percentages of scores between points, etc. In short, it allows us to compare any one score to any other score in a distribution, or across distributions because it is standardized based on the distributions mean and standard deviation. Take a look at the following diagrams of the normal distribution.

This breaks down the percentage of the distribution falling between the Z scores. This is a constant. Thus we can see that almost, but not quite all of the distribution lies between the Z scores of -3 and 3. Normal distribution and Z scores: This 3-part diagram shows the percent of a normal distribution that lies between 1, 2, and 3 standard deviations from the mean: All normal models follow this pattern, so a common name for this property is the These percentages also represent the probability of finding a z score in one of these intervals, so this rule can be useful in answering probability questions such as we find in hypothesis testing.

Here is another view of the Standard Normal Distribution. In this diagram the percentages represent the amount of the distribution between consecutive Z scores -2 and -1, -1 and 0, etc.

The diagram also illustrates how the Z scores fall on the normal distribution. Each Z score represents a unit of standard deviation away from the mean.