Z-Score
Calculate z-score and normal distribution probabilities
How to use Z-Score
- 1Enter the original fraction that you want to standardize.
- 2Enter the average of the data set to the standard difference.
- 3Reads the z fraction and the numerical position it reflects.
What's Z score?
z = (x − μ) / σZ marks indicate how many standard differences a value is above or below the average. Positive is above average and negative is below average.
Z fractions allow you to compare the values of different distributions by a uniform scale and to identify off-group values.
The formula for Z fractions (standard fractions) is z = (x - μ) /. It answers the question:** How many standard deviations of this value from the average **. The value of this is that it allows the data of different scales to be compared - for example, 85 points for a student in mathematics (average 80, standard 5) and 90 points in language (average 85, standard 8), and Z = 1, language Z = 0.625, indicating that mathematics is relatively better, although the absolute language is higher.
Empirical law under normal distribution (68-95-99.7): About 68 per cent of the data fell within ±1 standard deviations, 95 per cent fell within ±2 and 99.7 per cent within ±3. This means that | > > 2 is relatively rare (about 5%) and | > > 3 is extremely rare (0.3%). That's why many quality control and abnormality tests use ±3 as a warning line -- observations beyond this range, probably not random fluctuations.
| Z Score | Percentage | Meaning | Common scenes |
|---|---|---|---|
| −2 | 2.3% | Far below average | Attention |
| −1 | 15.9% | Lower than average | Near the normal lower limit. |
| 0 | 50% | Equal to average | Medium |
| 1 | 84.1% | Higher than average | Good. |
| 2 | 97.7% | Much above average | Excellent. |
| 3 | 99.9% | Very few. | Very high |
Percentage of different Z fractions in standard normal distribution
Frequently asked questions
z score is 0. What does that mean?
indicates that the value is exactly equal to the average of the data set.
The higher the score, the better?
Not necessarily. It represents only the distance from the average, and whether it is “good” depends on the circumstances.
How to use the z fraction for the percentile?
The z fraction can be found in the standard normal distribution table, or the corresponding percentage can be obtained in combination with the cumulative distribution.
Z, is the score and the percentage one?
No, but they can be converted through a normal distribution. The Z score is "a few deviations from the average standard" and the percentage is "more than one percent" Under the standard normal distribution, Z = 1 corresponds to 84.1 percent (over 84%). ** This conversion takes place only when data are subject to normal distribution** - If the data distribution is clearly skewed (e.g., income distribution), the Z fraction-percentage correspondence will deviate from the index value.
Is the sample still working when it's small?
Be careful. Z fractions assume that the overall standard deviation is known and the sample size is large enough (usually n > 30). Sample volume hours, with an estimated instability of the standard deviation of the sample, should use the t-distribution instead of the normal distribution (t test). t The thicker end of the distribution implies a need for greater value in the same amount of statistics to achieve prominence. The smaller the sample, the greater the difference between the t distribution and the normal distribution - which is why it is more difficult to draw significant conclusions from small sample studies.