A standard normal table(also called the unit normal table or z-score table)is a mathematical table for the values of ϕ, indicating the values of the cumulative distribution function of the normal distribution. Z-Score, also known as the standard score, indicates how many standard deviations an entity is, from the mean.
Since probability tables cannot be printed for every normal distribution, as there is an infinite variety of normal distribution, it is common practice to convert a normal to a standard normal and then use the z-score table to find probabilities.
Z-ScoreFormula
It is a way to compare the results from a test to a “normal” population.
If X is a random variable from a normal distribution with mean (μ) and standard deviation (σ), its Z-score may be calculated by subtracting meanfrom X and dividing the whole by standard deviation.
Where, x = test value
μ is mean and
σ is SD (Standard Deviation)
For the average of a sample from a population ‘n’, the mean is μ and the standard deviation is σ.
How to Interpret z-Score
Here is how to interpret z-scores:
- A z-score of less than 0 represents an element less than the mean.
- A z-score greater than 0 represents an element greater than the mean.
- A z-score equal to 0 represents an element equal to the mean.
- A z-score equal to 1 represents an element, which is 1 standard deviation greater than the mean; a z-score equal to 2 signifies 2 standard deviations greater than the mean; etc.
- A z-score equal to -1 represents an element, which is 1 standard deviation less than the mean; a z-score equal to -2 signifies 2 standard deviations less than the mean; etc.
- If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2 and about 99% have a z-score between -3 and 3.
Example of Z Score
Let us understand the concept with the help of a solved example:
Example: The test scores of students in a class test has a mean of 70 and with a standard deviation of 12. What is the probable percentage of students scored more than 85?
Solution: The z score for the given data is,
z= (85-70)/12=1.25
From the z score table, the fraction of the data within this score is 0.8944.
This means 89.44 % of the students are within the test scores of 85 and hence the percentage of students who are above the test scores of 85 = (100-89.44)% = 10.56 %.
Frequently Asked Questions
Q1
What does the Z-Score Table Imply?
The z score table helps to know the percentage of values below (to the left) a z-score in a standard normal distribution.
Q2
What are the Types of Z Score Table?
There are two z-score tables which are:
- Positive Z Score Table: It means that the observed value is above the mean of total values.
- Negative Z Score Table: It means that the observed value is below the mean of total values.
Q3
What is Z Score and How is it calculated?
A z score is simply defined as the number of standard deviation from the mean. The z-score can be calculated by subtracting mean by test value and dividing it by standard value.
So, z = (x −μ)/ σ
Where x is the test value, μ is the mean and σ is the standard value.
FAQs
z score = (x – μ)/ σ
The area under the normal distribution curve shows probability. To calculate the area under the curve, we calculate the z-score and construct a z-score table that describes the percentage of area to the left of the z-score to give a defined comparison.
What is the difference between Z table and z-score table? ›
A z-table shows the percentage or probability of values that fall below a given z-score in a standard normal distribution. A z-score shows how many standard deviations a certain value is from the mean in a distribution.
What is the Z table for the normal distribution? ›
A standard normal table (also called the unit normal table or z-score table) is a mathematical table for the values of ϕ, indicating the values of the cumulative distribution function of the normal distribution.
What is the Z 1.96 from a table? ›
From the table, z = 1.96. Therefore 95% of the area under the standard normal distribution lies between z = -1.96 and z = 1.96.
What is the formula for the z-score of a sample distribution? ›
The mechanics of finding a probability associated with a range of sample means usually proceeds as follows. Convert a sample mean ¯X into a z-score: Z=¯X−μσ/√n Z = X ¯ − μ σ / n . Use technology to find a probability associated with a given range of z-scores.
What is the z-score for 95 confidence interval? ›
The Z value for 95% confidence is Z=1.96.
How to use z table to find critical value? ›
The z critical value can be calculated as follows:
- Find the alpha level.
- Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test subtract the alpha level from 0.5.
- Look up the area from the z distribution table to obtain the z critical value.
There are three variables to consider when calculating a z-score: the raw score (x), the population mean (μ), and the population standard deviation (σ). To get the z-score, subtract the population mean from the raw score and divide the result by the population standard deviation.
What is 1.645 in Z table? ›
A z-score of 1.645, for instance, is associated with. 95 , meaning that 95% of the values in the distribution fall below the value with that z-score (alternatively, we can say that there is a 95% chance a randomly selected value will be below the value with that z-score).
What is the Z table at 90%? ›
Step #5: Find the Z value for the selected confidence interval.
| Confidence Interval | Z |
|---|
| 80% | 1.282 |
| 85% | 1.440 |
| 90% | 1.645 |
| 95% | 1.960 |
3 more rowsUsing the symmetry property of the distribution, we find z(0.975) = –z(0.025) = –1.96. Therefore, the middle 0.95 of the normal distribution is bounded by –1.96 and 1.96.
What is the formula for calculating the z-score? ›
The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.
How do you find the z-score of a number in a data set? ›
How do you find the z-score with mean and standard deviation? If you know the mean and standard deviation, you can find the z-score using the formula z = (x - μ) / σ where x is your data point, μ is the mean, and σ is the standard deviation.
How do you find the critical value of a Z score table? ›
Z Critical Value
Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test subtract the alpha level from 0.5. Look up the area from the z distribution table to obtain the z critical value. For a left-tailed test, a negative sign needs to be added to the critical value at the end of the calculation.
