# Z Score Calculator

## Calculator Use

The z-score is the number of standard deviations a data point is from the population mean. You can calculate a z-score for any raw data value on a normal distribution.

When you calculate a z-score you are converting a raw data value to a standardized score on a standardized normal distribution. The z-score allows you to compare data from different samples because z-scores are in terms of standard deviations.

A positive z-score means the data value is higher than average. A negative z-score means it's lower than average.

You can also determine the percentage of the population that lies above or below any z-score using a z-score table.

## Using the Z-Score Calculator

This calculator can find the z-score given:

- A raw data point, population mean and population standard deviation
- Sample mean, sample size, population mean and population standard deviation
- A sample that is used to calculate sample mean and sample size; population mean and population standard deviation

With the first method above, enter one or more data points separated by commas or spaces and the calculator will calculate the z-score for each data point provided from the same population.

With the last method above enter a sample set of values. Enter values separated by commas or spaces.

You can also copy and paste lines of data from spreadsheets or text documents. See all allowable formats below.

## Z-Score Formula

When calculating the z-score of a single data point x; the formula to calculate the z-score is the difference of the raw data score minus the population mean, divided by the population standard deviation.

\[ z = \dfrac{x - \mu}{\sigma} \]- \(z = \) standard score
- \(x = \) raw observed data point
- \(\mu = \) population mean
- \(\sigma = \) population standard deviation.

When calculating the z-score of a sample with known population standard deviation; the formula to calculate the z-score is the difference of the sample mean minus the population mean, divided by the **Standard Error of the Mean for a Population** which is the population standard deviation divided by the square root of the sample size.

- \(z = \) standard score
- \(\overline{x} = \) sample mean
- \(\mu = \) population mean
- \(\sigma = \) population standard deviation.
- \(n = \) sample size

Options

54

65

47

59

40

53

54,

65,

47,

59,

40,

53,

or

42, 54, 65, 47, 59, 40, 53

65 47

59 40

53

or

42 54 65 47 59 40 53

54 65,,, 47,,59,

40 53