Introduction
Tabel z, also known as the standard normal table or the z-score table, is a mathematical table that shows the area under the standard normal curve for different values of z-scores. A z-score, also known as a standard score, is a measure of how many standard deviations away from the mean a value is in a normal distribution. Tabel z is useful for finding the probability of certain values or ranges of values in a normal distribution.
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In this article, you will learn:
What is tabel z and how it is related to normal distribution and z-scores
How to use tabel z to find probabilities for different scenarios
How to standardize any normal distribution using the z-score formula
What are the benefits of using tabel z for statistical analysis
What are some alternatives to using tabel z
What is tabel z?
Tabel z is a table that contains the cumulative probabilities for different values of z-scores in a standard normal distribution. A standard normal distribution is a special type of normal distribution that has a mean of 0 and a standard deviation of 1. A normal distribution is a symmetrical, bell-shaped curve that describes how data values are distributed around the mean.
A z-score is a measure of how many standard deviations away from the mean a value is in a normal distribution. It is calculated by subtracting the mean from the value and dividing by the standard deviation. A positive z-score means that the value is above the mean, while a negative z-score means that the value is below the mean. A zero z-score means that the value is equal to the mean.
Tabel z shows the area under the standard normal curve to the left of a given z-score. This area represents the probability that a value in a standard normal distribution is less than or equal to that z-score. For example, if you look up the value 1.96 in tabel z, you will find that it corresponds to an area of 0.9750. This means that there is a 97.5% chance that a value in a standard normal distribution is less than or equal to 1.96.
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How to use tabel z?
To use tabel z To use tabel z, you need to follow these steps:
Identify the z-score or the range of z-scores that you are interested in.
Look up the corresponding area or areas in tabel z. You can use the row and column labels to find the exact value or the closest value.
Interpret the area or areas as probabilities for the scenario that you are analyzing.
Let's look at some examples of how to use tabel z for different scenarios.
Example 1
Suppose you want to find the probability of a value being less than -1.25 in a standard normal distribution. This means that you want to find the area under the curve to the left of -1.25. To do this, you need to look up -1.25 in tabel z. You can find it in the row labeled -1.2 and the column labeled 0.05. The value in tabel z is 0.1056. This means that there is a 10.56% chance that a value in a standard normal distribution is less than -1.25.
Example 2
Suppose you want to find the probability of a value being between -0.5 and 1.5 in a standard normal distribution. This means that you want to find the area under the curve between -0.5 and 1.5. To do this, you need to look up both -0.5 and 1.5 in tabel z and subtract the smaller area from the larger area. You can find -0.5 in the row labeled -0.5 and the column labeled 0.00. The value in tabel z is 0.3085. You can find 1.5 in the row labeled 1.5 and the column labeled 0.00. The value in tabel z is 0.9332. To find the area between -0.5 and 1.5, you need to subtract 0.3085 from 0.9332. The result is 0.6247. This means that there is a 62.47% chance that a value in a standard normal distribution is between -0.5 and 1.5.
Example 3
Suppose you want to find the probability of a value being greater than 2 in a standard normal distribution. This means that you want to find the area under the curve to the right of 2. To do this, you need to look up 2 in tabel z and subtract the area from 1, since the total area under the curve is 1. You can find 2 in the row labeled 2.0 and the column labeled 0.00. The value in tabel z is 0.9772. To find the area to the right of 2, you need to subtract 0.9772 from 1. The result is 0.0228. This means that there is a 2.28% chance that a value in a standard normal distribution is greater than 2.
How to standardize a normal distribution?
Tabel z only works for standard normal distributions, which have a mean of 0 and a standard deviation of 1. However, most real-world data sets do not have these characteristics. Therefore, to use tabel z for any normal distribution, you need to standardize it first. Standardizing a normal distribution means transforming it into a standard normal distribution by using the z-score formula.
The z-score formula is:
Where:
z is the z-score
x is the value in the original normal distribution
\mu is the mean of the original normal distribution
\sigma is the standard deviation of the original normal distribution
By applying this formula to every value in the original normal distribution, you can obtain a new set of values that have a mean of 0 and a standard deviation of 1. These values are the z-scores that you can use with tabel z to find probabilities.
Example 4
Suppose you have a normal distribution with a mean of 50 and a standard deviation of 10. You want to find the probability of a value being less than 40 in this distribution. To do this, you need to standardize the distribution and use tabel z.
First, you need to find the z-score for 40 using the z-score formula:
Second, you need to look up -1 in tabel z and find the corresponding area. You can find it in the row labeled -1.0 and the column labeled 0.00. The value in tabel z is 0.1587. This means that there is a 15.87% chance that a value in the standardized normal distribution is less than -1.
Third, you need to interpret this result in terms of the original normal distribution. Since -1 is the z-score for 40, this means that there is a 15.87% chance that a value in the original normal distribution is less than 40.
What are the benefits of using tabel z?
Tabel z has some benefits for statistical analysis, such as:
It simplifies calculations by providing pre-computed probabilities for different values of z-scores.
It allows comparisons between different normal distributions by using a common scale of z-scores.
It facilitates hypothesis testing and confidence intervals by providing critical values for different significance levels.
What are some alternatives to using tabel z?
Tabel z is not the only method or tool for finding probabilities in normal distributions. Some alternatives are:
Using calculators or software
Calculators or software can perform calculations faster and more accurately than using tabel z. They can also handle more complex scenarios, such as finding probabilities for non-standard normal distributions or non-normal distributions. Some examples of calculators or software that can find probabilities are:
[Online Z-Score Calculator]
[Microsoft Excel]
[R]
[Python]
Using other types of tables
Tabel z only works for standard normal distributions, which are not always applicable or appropriate for different types of data or tests. Other types of tables can be used for different distributions or tests, such as:
[T-table] for t-distributions, which are used when the sample size is small or the population standard deviation is unknown.
[Chi-square table] for chi-square distributions, which are used for testing independence or goodness-of-fit.
[F-table] for F-distributions, which are used for testing variance or comparing multiple means.
Conclusion
Tabel z is a mathematical table that shows the area under the standard normal curve for different values of z-scores. It is useful for finding probabilities of certain values or ranges of values in a normal distribution. To use tabel z, you need to identify the z-score or the range of z-scores that you are interested in, look up the corresponding area or areas in tabel z, and interpret them as probabilities for your scenario. You also need to standardize any normal distribution using the z-score formula before using tabel z.
Tabel z Tabel z has some limitations and drawbacks, such as:
It requires manual lookup and interpolation, which can be time-consuming and prone to errors.
It only provides probabilities for discrete values of z-scores, which may not match the exact values of interest.
It only works for standard normal distributions, which may not fit the data or assumptions of some tests or situations.
Therefore, it is important to consider other methods or tools for finding probabilities in normal distributions, such as calculators, software, or other types of tables.
FAQs
Here are some frequently asked questions about tabel z:
Q: How do I read tabel z?
A: To read tabel z, you need to locate the row and column that correspond to the z-score that you are interested in. The row label shows the first decimal place of the z-score, while the column label shows the second decimal place. The value in the table shows the area under the standard normal curve to the left of that z-score. For example, if you want to find the area for a z-score of 1.23, you need to look at the row labeled 1.2 and the column labeled 0.03. The value in the table is 0.8907, which means that there is an 89.07% chance that a value in a standard normal distribution is less than or equal to 1.23.
Q: How do I find negative z-scores in tabel z?
A: To find negative z-scores in tabel z, you need to use the same row and column labels as for positive z-scores, but look at the lower half of the table. For example, if you want to find the area for a z-score of -1.23, you need to look at the row labeled -1.2 and the column labeled 0.03. The value in the table is 0.1093, which means that there is a 10.93% chance that a value in a standard normal distribution is less than or equal to -1.23.
Q: How do I find probabilities for non-standard normal distributions?
A: To find probabilities for non-standard normal distributions, you need to standardize them first by using the z-score formula. This formula converts any normal distribution into a standard normal distribution by subtracting the mean from each value and dividing by the standard deviation. Then, you can use tabel z to find probabilities for the resulting z-scores.
Q: How do I find probabilities for values above or between certain z-scores?
A: To find probabilities for values above or between certain z-scores, you need to use some basic rules of probability and arithmetic. For values above a certain z-score, you need to subtract the area to the left of that z-score from 1, since the total area under the curve is 1. For values between two z-scores, you need to subtract the smaller area from the larger area.
Q: Where can I find more information or resources about tabel z?
A: Here are some links that you may find helpful:
[Tabel Z - Wikipedia]
[How to Use Tabel Z - Statology]
[Tabel Z Calculator - Math Is Fun]
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