Question

How do you convert a data value to a z-score?

What is a z score?

What is the empirical rule?

Answer 1

Solution:

Part i) How do you convert a data value to a z-score?

To convert a data value to a z-score, we use following formula:

where x is the data value, is the population mean of the given data and is the population standard deviation of the given data.

If these two population parameters are unknown, then we use their sample estimates sample mean and sample standard deviation s.

Suppose x value = 10 , and then

z score for x = 10 is :

Part ii) What is a z score?

z score follows the standard Normal distribution with mean = 0 and standard deviation = 1.

Thus z score is called as value of a standard Normal distribution.

It gives the relatioship between values in the data with mean and standard deviation of the random variable x.

For example : z = 0 is the value corresponding to mean of the random variable x.

z < 0 , means x values are below mean value.

z > 0, means x values are above mean value.

Part iii) What is the empirical rule?

For bell shaped distribution ( Normal distribution )

1) 68% of the data falls within 1 standard deviation from mean
that is :

2) 95% of the data falls within 2 standard deviation from mean
that is :

3) 99.7% of the data falls within 3 standard deviation from mean
that is :