- Calculators
- ::
- Statistics and probability
- ::
- Z - score calculator
Z - score calculator
This calculator solves three types of problems related to z-scores. 1. Find the area (probability) to the left, right, or between two Z-scores. 2. Find the z-score if the cumulative probability level (p-value) is given. 3. Find the z-score based on the raw value, mean and standard deviation of a population.
About this calculator
In simple terms, the z-score measures how many standard deviations the observed value is above or below the population mean. Z scores were created so that we could easily compare some data to the average value. For example, if a person's height has a z-score of 0.15, it means he is somewhat higher than the average furthermore we can calculate that he is taller than 55% of the population.
How to calculate z - score?
Case 1: If we know the raw score (x), mean (μ) and standard deviation (σ), we calculate the z-score using the following formula.
Z = (x - μ)/σ
Example: Given x = 72, μ = 65, and σ = 5, the z-score would be
Z = (x - μ)/σ = (72-65)/5 = 1.4
Case 2:If we know the p-value, we can calculate the z-score using the standard normal table.
For example if p = 0.345 than, using the standard normal table we can find that the z-score is -0.399.
Case 3: If we are given a dataset, then we need to apply the following steps.
1. Calculate the mean μ using the formula μ = Σx/n,
2. Calculate the standard deviation using σ formula σ2 = Σ(x - μ)2 / (n-1) ,
3. Apply the same formula we used in Case 1: Z = (x - μ)/σ.
Example: The dataset of exam scores is 45, 51, 67 and 55. Find the z-score for x = 60.
Step1: Find the mean.
μ = (45+61+67+55)/4 = 57
Step2: Find the standard deviation.
σ2 = [(45-57)2+(61-57)2+(67-57)2+(55-57)2]/(4-1)
σ2 = [(144+16+100+4]/3
σ2 = 88
σ = 9.38
Step3: Calculate z
z = (60 - 57)/9.38 = 0.319
How to interpret z - score?
Basically, the z score shows how many standard deviations you are above or below the population mean. Approximately 68% of the data have a z-score of -1 to 1. This means that 68% of observations are less than one standard deviation away from the mean. If your z score on the exam is 1.5, you are far above the average. On the other side, a z-score of -0.25 indicates that you are slightly below average.