z-Score: definition, formula, calculation & interpretation

This task focuses on Z standardization (Z conversion). Define what is a Z score, how Z standardization works, and that this is actually a common normal distribution. Furthermore, we will look at the Z standardization table and what it is used for.

What is z-standardization?

Z standardization is a statistical procedure used to compare data points from different datasets. In this procedure, all data-bathta is converted to a Z-score. Z standardization indicates how many regular deviations have a data specimen from the average dataset.

Example of z-standardization

Suppose you are a doctor and want to measure your patient's blood pressure. To do so, select from 40 patients to measure arterial pressure. From the measured data, you can clearly determine the average significance, the significance of 40 patients on average.

Now, one of the patients knows how high their blood pressure is compared to others. In fact, you answered that his blood pressure was 10mmHg higher than average. 10mmHg higher than average. Here is a question. Is this a lot or less?

It is 10mmHg if other patients are quite densely grouped in a circle with moderate importance. It is. -Art. Is a larger number than the spread, but if other patients are quite wide around the circle of average significance, 10mmHg. Art. 。

The standard deviation indicates how tight the data is. If the data is close to the average value, the common abnormality is small, and if it is scattered closely, the common abnormality is large.

In our data, let's say that the common abnormal value is 20mmHg. art. This means that the patient deviates by 20 % from the average significance difference.

The Z score indicates how far a person is from the average value. In other words, the Z score of a person who has fallen once by normal value from the average value is the same, the Z score of the person who has fallen twice from the average value is the same 2, and the average value three times by normal value. The Z score is the same 3.

Therefore, the Z score of a person whose normal difference is negative 1 i s-1, the Z score of a person with a normal difference i s-2, and the Z score of a person with a normal difference i s-3.

If a person literally has the same meaning as the average, the person deviates from the average with no regular difference and receives 0 points.

Thus, the Z score indicates how normal the measured value is from the average value. As already mentioned, a general abnormal value is only an indicator of the spread of blood pressure characteristics in patients in a circle of the average value.

In one sentence, Z-Score is useful for exceptional or simply a specific measurement value to compare the overall average indicator.

Calculating the z-score

How to calculate the Z score method We want to convert initial data (in our case arterial pressure) to Z-ball.

Here we are looking for the Z standardization formula. Here, Z-BIDIING, of course, is the Z-Bidiing we want to decide, X is an observed significance, in our case it is a human arteriic pressure, μ is the average intimidation of the group, in our case 40. It is the average significance of all human patients, and σ is a normal selection deviation. Normal deviation of 40 patients.

Note: μ and σ are actually considered to be the average and normal deviation of the mother group, but in our case only choices. However, the average and normal specimen deviation can be considered on a specific standard to be defined later.

In fact, suppose that 40 patients in our case had an average significant difference of 130 and the conventional abnormal value of 20. If you use both meanings, Z will be as follows: X-130 ÷ 20

This allows you to exercise the arterial pressure of individual patients to X and determine the significance of Z. Let's do this for the first patient. Assuming that this patient's arterial pressure was 97, we will introduce a prime number 97 for X at this time, and a significant difference Z is equal t o-1 and 65.

Therefore, this person deviates from the normal deviation o f-1 and 65 from the average significance. Now, this can be done for all patients.

Regardless of the measurement unit of the initial data, an educational program has been created to see how a person deviates from the average significance in a normal difference unit.

Of course, we only have a part of a group. However, if the data is well distributed and there is a selection of 30 or more, the percentage of the average arterial pressure of the patient is 110 or less, and the percentage of arterial pressure is higher than 110. Can explain what percentage is.

But how does it work? If the initial data has a conventional diversion, the support of Z value standardization can provide standard normal dispersion.

Standard ordinary dispersion is an image given by an ordinary scream of 0 average significance and standard deviation 1.

Its peculiarity has the potential for all kinds of riotes, independent of average meaning and habitual differences, into customistic habitual dissatisfaction.

The only thing you need to do is to get as many Z scores as possible, as the standard distribution has been obtained.

Such a table is listed in almost every statistical book, and is here as well as a Z distribution, of course, doubts: How do you recite this table?

For example, if the Z score i s-2, the significance of 0, 0228 can be read from this table.

This means that 2, 28%of the value is smaller than the Z score-2. As the sum is always equal to 100 % or 1, the value of 97, 72 % is larger.

As described above, 50%of the value is smaller than the Z score 0, and the 50%of the value is larger than 0. Since the normal distribution is symmetrical, we can literally count the probability of a positive Z score.

If Z's significance is just 1, just fin d-1. However, in this case, there is a footprint that indicates the meaning of how many percent of the value is larger than Z-Sense. In other words, if Z-Sense is equal to 1, the value of 15, 81 % is larger, and the value of 84, 14 % is smaller.

But what if you want to read the Z score o f-1 and 81 in a table? To do so, another column is required. We can rea d-1, 81 Z scores a t-1, 8, 0, 01.

Let's look at the blood pressure pattern again. For example, if you want to recognize the ratio of patients with a lower blood pressure than 123, you can use the Z standardization to convert 123 blood pressure to a Z score. In this case, the Z score i s-0, 35.

You can now stop the Z distribution table and find an equal Z value o f-0 and 35. Here we have 0, 3632 significance. This means that the value of 36, 32 % is lower than th e-0, 35 Z score, higher than 63, 68 %.

Compare different data sets with the z-score

However, Z-standardization has other powerful applications. Z-standardization has the ability to compare the measured values ​​in a different way. Let's give an example.

There are two classes, A class and B class, and say they have studied different arithmetic.

The test is designed in a different way and has the highest score different from different difficulty levels.

To correctly compare the results of the students, use Z standardization.

The average score of the A class was 70 points, and the typical deviation was 10 points. The B-class analysis average score was 140 points, and the typical deviation was 20 points.

Now we want to compare the results of Max from class A, who got 80 points, with Emma from class B, who got 160 points.

For this we plot the z-means of Max and Emma. After this we introduce 160 in X, which also gets a z-mean of 1.

Thus, the z-means of Max and Emma are similar. This means that both students performed best in their class in terms of mean performance and variance. Both, with a mean deviation of exactly 1, are higher than the average of their class.

Assumptions

But what about the assumptions? Can we take the first step of Z-standardization and apply the usual normal distribution table?

Z-standardization itself, i. e. the reorganization of data points into Z-values ​​with the support of this formula, does not inherently contain any strict criteria. It can be done autonomously from the data of the data.

But if the obtained Z-values ​​are to be used in statistical analysis (e. g. testing hypotheses or confidence intervals) in the context of the usual normal distribution, certain assumptions must be made.

The z distribution means that the basic Sobokupa is well distributed, and the mean (μ) and normal anomaly (σ) of the Sobokupa are popular.

However, since we do not have all the Sobokupa, and the mean significance and common anomaly are not usually learned, this application is of course often not done. But fortunately, there is another conjecture.

The z distribution is determined for a well-distributed collection, but the central maximum axiom can be used for a huge collection. This axiom is that when the value of the collection exceeds 30, it actually approaches the normal split. As a result, if the selection is 30 or more, we can apply the normal normal split AS An approximation and take into account the mean and normal sampling anomalies.

When evaluating the conventional anomaly in a sample, the return σ is usually written as S, and the mean significanc e-x dash return MU.

Do not confuse z standardization with z test or t test. If you want to acknowledge that it is actually a t test, watch the right video.

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