Z score (or standardized) score measures how many standard deviations away from the mean the given data point lies. The importance of the Z score is that if a random variable follows normal distribution then the corresponding Z score follows the standard normal distribution.
The formula for the Z score is given as,,
Z = (X-µ)/σ where,
X is the given raw data value,
µ is the mean and σ is the standard deviation of the distribution.
How do you find Z score?
- Subtract the mean from the given data value.
- Divided the result by the standard deviation to get the Z score.
Example: In a class of 50 students, the average height is 1.7m with a standard deviation of 0.2m. If the height of Tom is 1.8m, express it as a Z score.
Solution: Given, X=1.8m
µ = 1.7m and σ = 0.2m
Then we apply the formula for the Z score as,
Z = (X-µ)/σ = 1.8-1.7/0.2 = 0.1/0.2 = 0.5 which is the required Z score
Use of Z score:
- We use the Z score of the sample mean as the test statistic in large sample testing for equality of means. (We use T scores for small samples).
- The Z score gives us a standardized value using which we can compare two data points coming from different populations.
- We can use the Z score to calculate probabilities for the normal distribution.
Z score for sample mean:
For a large sample, the sample mean follows the normal distribution and in this case the formula for Z score of sample mean is,
Z score for sample mean= (x̄-µ)/(σ/√n)
where, x̄ is the sample mean and n is the sample size.