A z-score can be defined as how many standard deviations away a value X from the mean. If a value is exactly an average, z-score will be 0. Values above the mean have positive z-scores whereas values below the mean have negative z-scores.
The Z score is computed as:
$$ Z = \frac{(X-\mu)}{\sigma} $$
Here μ is the Population mean and σ is the population standard deviation. X is called as raw score.Solution
$$ Z = \frac{(X-\mu)}{\sigma} $$
Assumptions of Z Score
- Normal Distribution: Z score assumes that the data is normally distributed.
- Independence: Z score assumes that the data points are independent of each other. If the data points are not independent, the Z-score may not accurately show the variation within the data.
- Sample Size: The sample size must be sufficiently large to provide a representative sample of the population.
- Continuous Data: The data must be measured on a continuous scale. If the data is measured on a categorical or ordinal scale, the Z-score may not be appropriate.
- Outliers: The Z-score assumes that the data does not contain outliers or extreme values.
Examples of Z Score
Example 1: Suppose a student scored 75 on a test with a mean of 65 and a standard deviation of 5. The Z-score for this student would be:
Z = (75 - 65) / 5 = 2
Example 2: Suppose the heights of a group of people are normally distributed with a mean of 68 inches and a standard deviation of 3 inches. A person who is 71 inches tall would have a Z-score of:
Z = (71 - 68) / 3 = 1
Example 3: Suppose a company's sales are normally distributed with a mean of $50,000 and a standard deviation of $10,000. If the company had sales of $75,000 in a given month, the Z-score for that month's sales would be:
Z = (75,000 - 50,000) / 10,000 = 2.5