Download Z Distribution Table: A Complete Guide
A z distribution table, also known as a standard normal table or a z table, is a mathematical tool that helps you find the probability or area under the curve of a normal distribution. It is useful for performing statistical tests, finding confidence intervals, and comparing different data sets.
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In this article, you will learn what a z distribution table is, how to download it from various sources, how to use it for different purposes, and some examples of how to apply it in real-world scenarios.
What is a Z Distribution Table?
Definition and explanation of the z distribution table
A z distribution table is a table that shows the values of Φ, which is the cumulative distribution function of the standard normal distribution. The standard normal distribution, also called the z distribution, is a special case of the normal distribution where the mean is 0 and the standard deviation is 1.
A z distribution table tells you the probability or area under the curve to the left of a given z score. A z score, also known as a standard score, is a measure of how many standard deviations away a value is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation.
A z score can be positive or negative depending on whether the value is above or below the mean. A positive z score means that the value is greater than the mean, while a negative z score means that the value is less than the mean. A z score of 0 means that the value is equal to the mean.
Types and formats of the z distribution table
There are different types and formats of the z distribution table depending on how they present the probabilities or areas. Some common types are:
Cumulative: This type of table gives the probability or area under the curve from negative infinity to a given z score. It tells you how likely it is that a value will be less than or equal to that z score.
Cumulative from mean: This type of table gives the probability or area under the curve from 0 (the mean) to a given z score. It tells you how likely it is that a value will be between 0 and that z score.
Complementary cumulative: This type of table gives the probability or area under the curve from a given z score to positive infinity. It tells you how likely it is that a value will be greater than or equal to that z score.
Inverse: This type of table gives the z score corresponding to a given probability or area under the curve. It tells you what value has that probability or area.
The format of the z distribution table can vary depending on how it displays the values. Some common formats are:
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Decimal: This format shows only two decimal places for both the z scores and the probabilities or areas. It is easy to read and use, but it may not be very accurate or precise.
Fraction: This format shows the probabilities or areas as fractions instead of decimals. It is more accurate and precise, but it may not be very convenient or familiar to use.
Scientific notation: This format shows the probabilities or areas as powers of 10 instead of decimals or fractions. It is very accurate and precise, but it may not be very intuitive or easy to understand.
Uses and applications of the z distribution table
The z distribution table is widely used in statistics, probability, and data analysis for various purposes. Some common uses are:
Performing hypothesis tests: A hypothesis test is a method of testing a claim or assumption about a population parameter (such as the mean or the proportion) based on a sample statistic. The z distribution table helps you find the critical values and the p-values for the test.
Finding confidence intervals: A confidence interval is a range of values that contains the true population parameter with a certain level of confidence. The z distribution table helps you find the margin of error and the confidence limits for the interval.
Comparing different data sets: The z distribution table helps you standardize and normalize different data sets that have different means and standard deviations. This allows you to compare them on the same scale and measure their relative positions and distances.
How to Download Z Distribution Table?
Step 1: Choose a reliable source or website
The first step to download a z distribution table is to choose a reliable source or website that provides the type and format of the table that you need. There are many online sources and websites that offer free and printable z distribution tables, but not all of them are accurate, updated, or user-friendly.
Some examples of reliable sources and websites are:
[Math Is Fun]: This website provides a simple and interactive cumulative z distribution table that allows you to enter a z score and get the corresponding probability or area. It also provides an explanation of how to use the table and some examples.
[Stat Trek]: This website provides four types of z distribution tables: cumulative, cumulative from mean, complementary cumulative, and inverse. It also provides an option to download each table as a PDF file.
[Math Portal]: This website provides two types of z distribution tables: cumulative and inverse. It also provides an option to download each table as an Excel file.
Step 2: Select the type and format of the z distribution table
The second step to download a z distribution table is to select the type and format of the table that suits your purpose and preference. Depending on the source or website that you choose, you may have different options to choose from.
For example, if you choose Stat Trek, you can select from four types of tables: cumulative, cumulative from mean, complementary cumulative, and inverse. You can also select from three formats: decimal, fraction, and scientific notation.
Step 3: Download the z distribution table as a PDF or Excel file
The third step to download a z distribution table is to download the table as a PDF or Excel file. Depending on the source or website that you choose, you may have different ways to do this.
For example, if you choose Stat Trek, you can click on the "Download PDF" button below each table to download it as a PDF file. Alternatively, you can right-click on the table image and select "Save image as" to save it as an image file.
Step 4: Save and print the z distribution table if needed
The fourth step to download a z distribution table is to save and print the table if needed. Depending on your device and software, you may have different ways to do this.
For example, if you download a PDF file, you can open it with a PDF reader such as Adobe Acrobat Reader and save it in your preferred location. You can also print it by selecting "Print" from the "File" menu or by pressing Ctrl+P on your keyboard.
How to Use Z Distribution Table?
Step 1: Find the z-score of the data value or statistic
The first step to use a z distribution table is to find the z-score of the data value or statistic that you want to analyze. To do this, you need to know the mean and the standard deviation of the population or sample that you are working with.
The formula for finding the z-score is:
z = (value - mean) / standard deviation
For example, if you want to find the z-score of a test score of 85, and you know that the mean of the test scores is 75 and the standard deviation is 10, you can use the formula as follows:
z = (85 - 75) / 10 z = 1
This means that the test score of 85 is one standard deviation above the mean.
Step 2: Locate the z-score in the z distribution table
The second step to use a z distribution table is to locate the z-score in the table and find the corresponding probability or area under the curve. Depending on the type and format of the table that you are using, you may have different ways to do this.
For example, if you are using a cumulative z distribution table in decimal format, you can look for the row and column that match the z-score and read the value in the cell. The value is the probability or area under the curve from negative infinity to that z-score.
If the z-score is positive, you can look for it directly in the table. If the z-score is negative, you can use the symmetry property of the normal distribution and look for its absolute value in the table. Then, you can subtract the value from 1 to get the probability or area.
For example, if you want to find the probability or area under the curve for a z-score of 1, you can look for 1.0 in the row and 0.00 in the column and read the value in the cell. The value is 0.8413. This means that there is a 84.13% chance that a value will be less than or equal to 1.
If you want to find the probability or area under the curve for a z-score of -1, you can look for 1.0 in the row and 0.00 in the column and read the value in the cell. The value is 0.8413. Then, you can subtract it from 1 to get 0.1587. This means that there is a 15.87% chance that a value will be less than or equal to -1.
Step 3: Interpret the probability or area corresponding to the z-score
The third step to use a z distribution table is to interpret the probability or area corresponding to the z-score and answer your question or solve your problem. Depending on your purpose and context, you may have different ways to do this.
For example, if you are performing a hypothesis test, you can compare the probability or area with your significance level and decide whether to reject or fail to reject your null hypothesis.
If you are finding a confidence interval, you can use the probability or area to find the critical value and multiply it by your standard error to get your margin of error.
If you are comparing different data sets, you can use the probability or area to find how likely it is that a value from one data set is higher or lower than a value from another data set.
Step 4: Repeat the process for different data values or statistics
The fourth step to use a z distribution table is to repeat the process for different data values or statistics that you want to analyze using the z distribution table. You can use the same steps and methods as described above, or you can use different types and formats of the table depending on your needs and preferences.
Examples of Z Distribution Table
Example 1: Finding the probability of a z-score using a cumulative z distribution table
Suppose you want to find the probability that a randomly selected student from a class has a test score of 85 or less, given that the mean of the test scores is 75 and the standard deviation is 10. You can use the following steps:
Find the z-score of the test score of 85 using the formula: z = (value - mean) / standard deviation. In this case, z = (85 - 75) / 10 = 1.
Locate the z-score of 1 in the cumulative z distribution table and find the corresponding probability or area under the curve. In this case, the probability is 0.8413.
Interpret the probability or area as the answer to your question. In this case, the probability that a randomly selected student has a test score of 85 or less is 0.8413 or 84.13%.
Example 2: Finding the z-score of a probability using an inverse z distribution table
Suppose you want to find the test score that corresponds to the 90th percentile of the test scores, given that the mean of the test scores is 75 and the standard deviation is 10. You can use the following steps:
Find the probability or area under the curve that corresponds to the 90th percentile. In this case, the probability is 0.9.
Locate the probability of 0.9 in the inverse z distribution table and find the corresponding z-score. In this case, the z-score is 1.28.
Find the value that corresponds to the z-score of 1.28 using the formula: value = (z * standard deviation) + mean. In this case, value = (1.28 * 10) + 75 = 87.8.
Interpret the value as the answer to your question. In this case, the test score that corresponds to the 90th percentile of the test scores is 87.8.
Example 3: Finding the probability of a range of z-scores using a cumulative from mean z distribution table
Suppose you want to find the probability that a randomly selected student from a class has a test score between 70 and 80, given that the mean of the test scores is 75 and the standard deviation is 10. You can use the following steps:
Find the z-scores of the test scores of 70 and 80 using the formula: z = (value - mean) / standard deviation. In this case, z = (70 - 75) / 10 = -0.5 and z = (80 - 75) / 10 = 0.5.
Locate the z-scores of -0.5 and 0.5 in the cumulative from mean z distribution table and find the corresponding probabilities or areas under the curve. In this case, the probabilities are 0.1915 and 0.1915.
Find the difference between the two probabilities or areas to get the probability or area under the curve between the two z-scores. In this case, the difference is 0.1915 - 0.1915 = 0.383.
Interpret the probability or area as the answer to your question. In this case, the probability that a randomly selected student has a test score between 70 and 80 is 0.383 or 38.3%.
Conclusion
A z distribution table is a useful tool for finding probabilities or areas under the curve of a normal distribution. It can help you perform statistical tests, find confidence intervals, and compare different data sets.
To download a z distribution table, you need to choose a reliable source or website, select the type and format of the table, download it as a PDF or Excel file, and save and print it if needed.
To use a z distribution table, you need to find the z-score of the data value or statistic, locate it in the table and find the corresponding probability or area, interpret it and answer your question or solve your problem, and repeat the process for different data values or statistics.
FAQs
Q: What is the difference between a z distribution table and a t distribution table?
A: A z distribution table is based on the standard normal distribution, which has a mean of 0 and a standard deviation of 1. A t distribution table is based on the t distribution, which has a mean of 0 but a variable standard deviation that depends on the degrees of freedom. The t distribution is more spread out than the normal distribution and has heavier tails.
Q: How can I find the probability or area under the curve for a non-standard normal distribution?
A: You can use a z distribution table for a non-standard normal distribution by first converting your data values or statistics into z-scores using the characteristics of a normal distribution, such as mean = median = mode, standard deviation = 1/6 of the range, skewness = 0, and kurtosis = 3.
Statistical tests: You can perform a hypothesis test such as the Shapiro-Wilk test, the Kolmogorov-Smirnov test, or the Anderson-Darling test and see if you can reject or fail to reject the null hypothesis that your data follows a normal distribution.
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