Utilize this z table, frequently referred to as a z score table, standard normal distribution table, or z value chart, to locate a z score.
If you’re unsure about how to find z score probabilities in the z tables, you’ll come across a handy z score probability calculator (to find right and left tail probabilities) and a z score calculator (to compute a z score) right after the tables. You can also scroll down on this page to find both calculators.
Unearth values to the left of the mean using this negative Z score table. Entries for z in the table represent the area under the bell curve left to z. Negative scores in the z-table align with the values that are lesser than the mean.
Discover values to the right of the mean using this z-table. Entries for z in the table illustrate the area under the bell curve left to z. Positive scores in the Z-table align with the values greater than the mean.
If you have an urgent need to swiftly calculate z score probabilities for a specific z score at a given probability level, feel free to utilize a z score probability calculator provided below.
Z Score Probability Calculator
This tool assists you in calculating the probability associated with a specific z-score. To leverage this calculator, adhere to these straightforward steps:
- Input the z-score value into the “Enter z-score” field.
- Input the probability level into the “Enter probability level” field.
- Click on the “Calculate” button.
- The left tail probability and right tail probability associated with the given z-score will be displayed in the “Left tail probability” and “Right tail probability” fields respectively.
Remember that the z-score represents the number of standard deviations a data point is from the mean. The left tail probability indicates the probability of getting a z-score less than the given z-score, and the right tail probability denotes the probability of getting a z-score greater than the given z-score.
Z-Score Probability Calculator Result
Left tail probability:
Right tail probability: Z Score Calculation and Z Table Application Illustration
Let’s look at a practical example of how a z-score is applied and how it can be calculated using a z-table. Picture a crowd of 200 applicants who participated in a math test. George was one of the participants and he scored 700 points (X) out of 1000. The average score was 600 (µ) and the standard deviation was 150 (σ). We want to know how well George performed compared to the other applicants.
We need to standardize his score (that is, compute a z-score corresponding to his actual test score) and use a z-table to determine how well he performed in the test compared to his peers. To calculate the z-score we need to use the following formula:
Therefore: Z score = (700-600) / 150 = 0.67
To figure out George’s performance on the test, we need to find the percentage of his peers who scored higher and lower. That’s when the z-table (also known as the standard normal distribution table) becomes useful. You may have noticed there are two z-tables with negative and positive values. If a z-score calculation results in a negative standardized score, refer to the 1st table; if it’s positive, use the 2nd table. For George’s case, we need to use the 2nd table as his test result aligns with a positive z-score of 0.67.
Finding a corresponding probability is quite simple. Find the first two digits on the y-axis (0.6 in our example). Then, head to the x-axis to find the second decimal number (0.07 in our case). The number is 0.7486. Multiply this number by 100 to get percentages. So 0.7486 x 100 = 74.86%. This implies that almost 75% of the students scored lower than George and only 25% scored higher. 75% out of 200 students is 150. George performed better than 150 students. Z-score Calculator By Z-Table.com Z Score
Frequently Asked Questions
Q: What is a z-score?
A: A z-score is a statistical measure that informs us how many standard deviations a data point is from the mean of a dataset. The z-score is computed by taking the difference between the data point and the mean, and dividing it by the standard deviation.
Q: Why are z-scores useful?
A: Z-scores are beneficial because they enable us to compare data points from different datasets with different scales and measurement units. We can perform meaningful comparisons and identify outliers and extreme values by standardizing the data.
Q: How do you interpret a z-score?
A: A z-score of 0 suggests that the data point is equivalent to the mean. When the z-score is positive, it indicates that the data point is above the mean, and when the z-score is negative, it signifies that the data point is below the mean. Moreover, the magnitude of the z-score quantifies the distance between the data point and the mean in terms of standard deviations.
Q: What is a good z-score?
A: A z-score of +/- 1.96 or greater is considered statistically significant at the 5% significance level (i.e., p < 0.05). This implies that the data point significantly differs from the mean at a 95% confidence level.
Q: How do you calculate a z-score in Excel?
A: A z-score can be calculated in Excel using the formula: = (data point – mean) / standard deviation. For instance, if your data point is in cell A1, and your mean and standard deviation are in cells B1 and C1, respectively, the formula would be: =(A1-B1)/C1.
Q: Can a z-score be negative?
A: Yes, a z-score can be negative if the data point is below the mean. This indicates that the data point is below average and further away from the mean in the negative direction.
Q: How do you use z-scores to identify outliers?
A: Z-scores can help identify outliers by searching for data points that are more than 3 standard deviations away from the mean. These data points are considered to be extreme values and might be due to measurement error or other factors that aren’t representative of the dataset as a whole.
Q: What is the relationship between z-scores and normal distribution?
A: Z-scores are utilized in conjunction with the normal distribution to standardize and compare data across different datasets. The normal distribution is a probability distribution often used to model real-world phenomena, and z-scores enable us to convert any normal distribution into a standard normal distribution with a mean of zero and a standard deviation of one. Z Score Sample Problems and Answers
Z-scores are a robust tool for analyzing data by standardizing the data points to a uniform scale. Here are some common z-score problems with detailed solutions:
Problem 1:
The mean height of a group of students is 65 inches, with a standard deviation of 3 inches. What is the z-score for a student who is 70 inches tall?
Solution: To find the z-score, we use the formula: z = (x – mean) / standard deviation. Plugging in the values, we get:
z = (70 – 65) / 3 = 1.67
The z-score for a student who is 70 inches tall is 1.67, which suggests that this student’s height is 1.67 standard deviations above the mean height of the group.
Problem 2:
A company has 100 employees, with an average salary of $50,000 and a standard deviation of $5,000. What is the z-score for an employee who earns $60,000?
Solution: To determine the z-score, we use the formula: z = (x – mean) / standard deviation. Substituting the values, we get: z = (60,000 – 50,000) / 5,000 = 2 The z-score for an employee who earns $60,000 is 2, which indicates that this employee’s salary is 2 standard deviations above the average salary of the company.
Problem 3:
A survey of 250 people discovered that the average income of the participants was $50,000, with a standard deviation of $10,000. What is the z-score for a participant who earns $70,000?
Solution: To compute the z-score, we use the formula: z = (x – mean) / standard deviation. Substituting the values, we get: z = (70,000 – 50,000) / 10,000 = 2 The z-score for a participant who earns $70,000 is 2, which indicates that this participant’s income is 2 standard deviations above the mean income of the group.
What is a Z Table?
A Z-table, also known as the standard normal table, is a mathematical table used in statistics to find the values of the Z-statistic, which is a standardized statistic.
In simpler terms, a Z-table helps find the probability that a statistic is less than a particular value when the statistic follows a standard normal distribution.
The Z-table is divided into two types – the positive Z-table and the negative Z-table. The positive Z-table lists the cumulative probabilities of a standard normal random variable, Z, up to a given value. The negative Z-table, on the other hand, provides the probabilities for negative values of Z.
Each entry in a Z-table represents the area under the standard normal curve to the left of a particular Z-score. This area corresponds to the probability of observing a value less than or equal to that Z-score. The Z-score itself measures how many standard deviations an element is from the mean.
For instance, a Z-score of 1.96 is associated with a probability of approximately 0.975, which means that about 97.5% of the area under the standard normal curve is to the left of Z = 1.96. This probability corresponds to a percentile rank of 97.5, meaning that roughly 97.5% of the population is expected to have a Z-score less than or equal to 1.96.