The standard deviation measures the dispersion of data in a distribution, while the Z-score indicates how many standard deviations a particular data point is from the mean of that distribution.
Let’s look at the difference between Z-score and standard deviations.
| Aspect | Standard Deviation | Z-Score |
|---|---|---|
| Definition | Measures the dispersion or spread of a data set. | Standardizes individual data points in terms of how many standard deviations they are from the mean. |
| Symbol | σ (sigma) | Z (Z-score) |
| Formula |
σ = √ [(Σ(xi – μ)²) / n] |
Z = X–μ/σ |
| Interpretation | Provides information about the spread of data points around the mean. | Indicates the relative position of a data point within a distribution. A positive Z-score means above the mean, and a negative Z-score means below the mean. |
| Unit of Measurement | Same unit as the data points being measured. | Dimensionless (no unit). |
where:
- N is the number of data points.
- Xi is each individual data point.
-
is the mean of the data set. - X is the individual data point.
- μ is the mean of the distribution.
- σ is the standard deviation of the distribution.
In summary, while the standard deviation gives a measure of how spread out the data is, the Z-score allows for a standardized comparison of individual data points in terms of their deviation from the mean.