Z Score Table is the table for determining the probability of a standard normal variable falling below or above a certain value. Z-score table, also known as a standard normal table or z-score Table, is a mathematical table that provides the area under the curve to the left of a z-score in a standard normal distribution. The standard normal distribution is a bell-shaped curve with a mean of 0 and a standard deviation of 1.
The z-score table is used to find the probability that a random variable from a standard normal distribution will fall below a certain value. In this article, we will learn about the Z Score Table in sufficient detail and also learn how to use the Z Score Table in numerical problems.
Z= X- μ/σ
Where,
- Z is the Z-score.
- X is the value to standardize.
- μ is the mean of the distribution.
- σ is the standard deviation, which measures the spread of the data.
What is a Z-Score Table?
A chart or database containing pre-calculated values assigned to particular Z-scores is known as a Z-score table. Finding the probability that a random variable from a typical normal distribution will go below or above a specific Z-score is made easier.
Fundamentally, a Z-score, sometimes referred to as a standard score, measures how much a data point deviates from a distribution’s mean (average) by the number of standard deviations. The Z-score formula, Z = (X – μ) / σ, can be used to compute this deviation. The Z-score is represented by Z, the value to be standardized by X, the distribution mean by μ, and the spread of the data by σ, which is the standard deviation.
Z-Score Definition
Z score also referred to as a standard score, calculates how many standard deviations a data point deviates from the normal distribution’s mean (average).
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How to Calculate Z-Scores?
To calculate Z-Score using table, we can use the following steps:
Step 1: Identify the data point (X).
Step 2: Find the mean (µ) and standard deviation (σ).
Step 3: Substitute the coefficients in the formula.
Step 4: Calculate the Z-score.
Let’s consider an example for better understanding.
Example: If the class average on a math test is 65 with a standard deviation of 10, a student who scored 75 can calculate his Z-score as follows.
Solution:
Given: X = 75, µ= 65, and σ = 10
Using the formula for Z-Score.
Z = (X – µ)/σ
⇒ Z = (75-65)/10
⇒ Z = 1
Z-Score Table
The z-score table is divided into two sections: negative z-scores and positive z-scores. The negative z-scores are below the mean, while the positive z-scores are above the mean. The rows and columns of the table define the z-score, and the table cells represent the area. For example, the z-score 1.50 corresponds to the area 0.9332, which is the probability that a random variable from a standard normal distribution will fall below 1.50.
Positive Z Score Table
A data point is above the median if its Z-score is positive (greater than 0), with a higher value denoting a larger divergence from the mean.
Negative Z Score Table
A negative Z-score indicates that the data points are nearer the mean.
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How to Use a Z-Score Table?
To use the Z Score Table, use the following steps:
- Step 1: Determine the Z-score associated with the relevant data point. Verify that you possess the distribution’s mean (μ) and standard deviation (σ). The number of standard deviations that a data point deviates from the mean is shown by the Z-score.
- Step 2: Look for a normal distribution table or a Z-score table. Z-scores are usually shown in the left column of these tables, with matching probabilities or percentiles shown in the table’s body.
- Step 3: Look for the Z-score you determined in the table’s left column. Two decimal places are frequently used when listing Z-scores (e.g., 0.00, 0.01, 0.02, etc.). Calculate the likelihood by interpolating between the two closest values if your Z-score is not exactly listed.
Note: The chance of getting a result less than or equal to the one you computed is represented by the corresponding value in the table.
Let’s consider an example for better understanding.
Example: A school has a normally distributed test score with a mean (μ) of 75 and a standard deviation (σ) of 10. A student wants to know the probability of scoring less than 80 on a test.
Solution:
Calculate the Z-score:
Z = 80 −75/10
⇒ Z = 0.5
Look at the Z-scores in the Z-score table to find the corresponding cumulative probability. Let’s say 0.6915.
Thus, the probability of a student scoring less than 80 would be 0.6915, or 69.15%.
How to Interpret z-Score?
We can interpret z-score table as:
- If the Z-score is positive (greater than 0), the data point is above the median. The further the data point is from the mean, the higher the positive Z-score.
- For instance, A Z-score of 2 indicates that the data point is two standard deviations above the mean.
- Data points with Z-scores less than 0 (negative) are said to be in the middle. As the Z-score decreases, the distance between the data point and the mean increases.
- For example, a Z-score of -1.5 indicates that the data point is 1.5 standard deviations from the mean.
Applications of Z Score
Some common application of z score are:
- Z-scores are widely used in a variety of fields, including statistics.
- Z-scores are useful for risk assessment, quality assurance, and outlier discovery since they normalize and compare data. Based on financial statistics, Z-scores, such as the Altman Z-score, are used in finance to forecast a company’s likelihood of bankruptcy. Children’s growth is evaluated in the medical field by comparing measures to reference charts.
- Z-scores are a useful tool for epidemiologists to examine public health trends and health statistics. Z-scores are used in sports to assess an athlete’s performance in comparison to their peers.
- Z-scores also play a role in college admissions, allowing scores on standardized tests to be compared. Z-scores are essential for data analysis, confidence interval calculations, and hypothesis testing.
- Z-scores are also essential for statistical process control, which finds differences in both manufacturing and customer service activities.
- Z-scores are a useful tool since they enable standard comparisons and support decision-making across a variety of disciplines.
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Example of Z Score
Example 1: If the Z-score is 1.5. Find the probability that a randomly selected data point falls below this Z-score.
Solution:
To determine the probability that a randomly selected data point falls below the Z score, we can do the following.
Using a Z-score table or calculator, look for a Z-score of 1.5 and get a corresponding probability, of about 0.9332. This means there is a 93.32% probability that the data point falls below a Z-score of 1.5 in the standard normal distribution.
Example 2: Find the probability that the Z score is greater than -1.2
Solution:
To determine the probability that the Z-score is greater than -1.2.
Using the Z-score table, find the cumulative probability associated with -1.2, which would be 0.1151. Subtract this value from 1 to find the probability of greater than -1.2:
1 − 0.1151 = 0.8849
Thus, the probability that the Z-score is greater than -1.2 is approximately 0.8849, or 88.49%.
Check: What is Statistical Analysis in Data Science?
Practice Questions on Z Score
Q1. A class of 100 students took a math test. The mean score is 75 with a standard deviation of 10. What is the Z-score of a student who scored 85 on the test?
Q2. In applied physics, students measure the time it takes for a ball to fall from a certain height. It is 3 seconds with a standard deviation of 0.5 seconds. If a student measures a fall time of 2.2 seconds, what is the Z-score for this measurement?
Q3. A company is conducting an employee compensation audit. The average salary is 50,000 with a standard deviation of 8,000. What is the Z-score of an employee with a salary of 56,000?
Q4. A doctor is measuring the height of a child to compare it with a group of children of the same age. The height of this group is 120 cm and the standard deviation is 5 cm. If the child is 130 cm tall, what is the Z-score for this measurement?
Q5. In a study of test anxiety among students, the average test anxiety score was 60 with a standard deviation of 10. If a student scores a test anxiety score of 75, what is the Z-score of this score?
Z-Score Table – FAQs
What is a Z-Score?
A z-score is a statistical measure that indicates how many standard deviations an element is from the mean of its dataset.
How Do You Use a Z-Score Table?
A Z-score table, or standard normal table, is used to find the percentage of values to the left of a given Z-score in a standard normal distribution.
What is the Purpose of a Z-Score Table?
The Z-score table is used to calculate the area under the curve in a normal distribution, helping in probability and statistics calculations.
Can Z-Scores Be Negative?
Yes, z-scores can be negative if the data point is below the mean of the dataset.
How Do Z-Scores Relate to Standard Deviation?
Z-scores directly relate to standard deviations, indicating how far and in what direction, an observation is from the mean, measured in units of standard deviation.
What Does a Z-Score of 1 Mean?
A Z-score of 1 means that the data point is one standard deviation above the mean of the dataset.
How Can You Calculate the Area Using a Z-Score Table?
To calculate the area using a Z-score table, locate the Z-score on the table, which will give you the area to the left of the Z-score in a standard normal distribution.
What is the Difference Between Z-Score and T-Score?
Z-scores are used for distributions with a known variance and are based on the standard normal distribution, while T-scores are used when the variance is unknown and the sample size is small.
How to Find Z-Score Without a Table?
You can find a Z-score without a table by using the formula.
Can Z-Score Table Be Used for All Distributions?
No, the Z-score table is specifically designed for the standard normal distribution. For other distributions, different methods or tables might be required.
What does a Positive or Negative Z-Score mean?
A positive Z-score indicates that the data point is above average, while a negative Z-score indicates above average. The magnitude of the Z-score indicates the distance of the data point from the mean.
How to use the Z-Score Table?
Z-scores tables give probabilities associated with particular Z-scores. To use it, find your Z-score in the table and read the corresponding probability.
What is the Significance of Z-scores in Hypothesis Testing?
In hypothesis testing, Z-scores help determine whether an observed data point is significantly different from what would be expected by chance. To assess statistical significance, the Z-score is compared to the critical value.
Can the Z-score be Used with any Data Set?
Z-scores are most beneficial when working with normally distributed data, but can often applied to normal distributions or large data sets due to the central limit theorem.
Are Z-scores the same as T-scores?
Z-scores and T-scores are similar concepts but different populations are used for standardization. The Z-score is based on the standard normal distribution, while the T-score is used in cases where the sample size is small and follows the Student t-distribution.
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