Z score – Definition & Examples

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

• 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?

1. Subtract the mean from the given data value.
2. 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:

1. 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).
2. The Z score gives us a standardized value using which we can compare two data points coming from different populations.
3. We can use the Z score to calculate probabilities for the normal distribution.

Z score for the sample mean:

For a large sample, the sample mean follows the normal distribution and in this case, the formula for the Z score of the sample mean is,

Z score for sample mean= (x̄-µ)/(σ/√n)

where x̄ is the sample mean and n is the sample size.