The standard distribution table, also known as the z-table, is a statistical tool used to find the probability of a value being less than or equal to a given value in a standard normal distribution. It is a fundamental concept in statistics and is widely used in various fields, including medicine, social sciences, and engineering. In this article, we will provide an easy-to-understand guide to the standard distribution table and its applications.
What is a Standard Normal Distribution?
A standard normal distribution, also known as the z-distribution, is a normal distribution with a mean of 0 and a standard deviation of 1. It is a continuous probability distribution that is symmetric about the mean and has a bell-shaped curve. The standard normal distribution is used as a reference distribution to compare the distribution of a dataset to a normal distribution.
Understanding the Z-Score
The z-score, also known as the standard score, is a measure of how many standard deviations an element is from the mean. It is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. The z-score is used to determine the probability of a value being less than or equal to a given value in a standard normal distribution.
How to Use the Standard Distribution Table
The standard distribution table is used to find the probability of a value being less than or equal to a given value in a standard normal distribution. The table is organized with z-scores on the left-hand side and probabilities on the right-hand side. To use the table, follow these steps:
- Calculate the z-score using the formula: z = (X - μ) / σ
- Look up the z-score in the standard distribution table
- Find the corresponding probability on the right-hand side of the table
For example, suppose we want to find the probability of a value being less than or equal to 1.5 in a standard normal distribution. First, we calculate the z-score: z = (1.5 - 0) / 1 = 1.5. Then, we look up the z-score in the standard distribution table and find the corresponding probability: P(Z ≤ 1.5) = 0.9332.
Interpreting the Results
The probability obtained from the standard distribution table represents the area under the standard normal curve to the left of the given z-score. For example, if we find a probability of 0.9332, it means that 93.32% of the values in the standard normal distribution are less than or equal to the given value.
| Z-Score | Probability |
|---|---|
| 1.0 | 0.8413 |
| 1.5 | 0.9332 |
| 2.0 | 0.9772 |
Applications of the Standard Distribution Table
The standard distribution table has numerous applications in statistics, including hypothesis testing, confidence intervals, and regression analysis. It is also used in various fields, such as medicine, social sciences, and engineering, to make informed decisions and predictions.
Hypothesis Testing
The standard distribution table is used in hypothesis testing to determine the probability of a test statistic being less than or equal to a given value. This probability is then compared to a significance level to determine whether the null hypothesis should be rejected or not.
Confidence Intervals
The standard distribution table is used to construct confidence intervals, which provide a range of values within which a population parameter is likely to lie. The table is used to find the critical values of the z-score, which are then used to construct the confidence interval.
Common Mistakes to Avoid
When using the standard distribution table, there are several common mistakes to avoid, including:
- Incorrectly calculating the z-score
- Failing to look up the correct probability in the table
- Misinterpreting the results
By avoiding these common mistakes and following the steps outlined in this guide, you can use the standard distribution table with confidence and accuracy.
What is the standard distribution table used for?
+The standard distribution table is used to find the probability of a value being less than or equal to a given value in a standard normal distribution.
How do I calculate the z-score?
+The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
What is the difference between a z-score and a probability?
+A z-score is a measure of how many standard deviations an element is from the mean, while a probability represents the area under the standard normal curve to the left of the given z-score.
