The Z-score Calculator is a statistical tool used to determine the standard deviation of a data point from the mean of a dataset.
The Z-score represents the number of standard deviations a data point is from the mean of the dataset. This information is crucial for identifying outliers, comparing data points, and understanding the distribution of data.
Z-score Chart
| Z-score | Probability |
|---|---|
| -3 | 0.13% |
| -2.5 | 0.62% |
| -2 | 2.28% |
| -1.5 | 6.68% |
| -1 | 15.87% |
| -0.5 | 30.85% |
| 0 | 50.00% |
| 0.5 | 69.15% |
| 1 | 84.13% |
| 1.5 | 93.32% |
| 2 | 97.72% |
| 2.5 | 99.38% |
| 3 | 99.87% |
This Z-score Tabular Chart shows that a Z-score of 0 represents the mean, a Z-score of 1 represents a data point that is 1 standard deviation above the mean, and a Z-score of -1 represents a data point that is 1 standard deviation below the mean.
Z-score Formula
The Z-score formula is:
Z = (x - μ) / σWhere:
- Z is the Z-score
- x is the data point
- μ (the Greek letter “mu”) is the mean of the dataset
- σ (the Greek letter “sigma”) is the standard deviation of the dataset
Let’s use the exam score example to check the formula:
- Exam score (x): 90
- Mean (μ): 80
- Standard deviation (σ): 5
Z = (90 - 80) / 5 = 2This means that the student’s score of 90 is 2 standard deviations above the mean of 80.
How is the z-score calculated?
To calculate the Z-score, you need to follow these steps:
- Find the mean (μ) of the dataset.
- Calculate the standard deviation (σ) of the dataset.
- Subtract the mean (μ) from the data point (x).
- Divide the result from step 3 by the standard deviation (σ).
Let’s go through an example:
Suppose you have a dataset of exam scores: 85, 92, 78, 90, 83.
Find the mean (μ):
- Add up all the scores: 85 + 92 + 78 + 90 + 83 = 428
- Divide the sum by the number of scores: 428 / 5 = 85.6
- The mean (μ) is 85.6.
Calculate the standard deviation (σ):
- Subtract the mean (85.6) from each score: -0.6, 6.4, -7.6, 4.4, -2.6
- Square each of these differences: 0.36, 40.96, 57.76, 19.36, 6.76
- Add up the squared differences: 125.2
- Divide the sum by the number of scores minus 1 (5 – 1 = 4): 125.2 / 4 = 31.3
- Take the square root of the result: √31.3 = 5.6
- The standard deviation (σ) is 5.6.
Subtract the mean (μ) from the data point (x):
- Let’s use the score of 90 as the data point (x).
- 90 – 85.6 = 4.4
Divide the result from step 3 by the standard deviation (σ):
- 4.4 / 5.6 = 0.786
The Z-score for a score of 90 in this dataset is 0.786, which means the score is 0.786 standard deviations above the mean.