Z Score Formula

What is Z Score in Statistics?

Statistics is a branch of Mathematics that deals with the collection, classification and representation of the data. Z score is a numerical value that indicates the position of the raw score with respect to the mean of the given data sample. In other words, Z score gives an idea of how far the data point is from the mean of the dataset. Z score is considered when the standard deviation units are used to measure the distance between the raw score and the mean of the dataset. The distribution of Z scores have zero as its mean and 1 as its standard deviation. 


Why is the Z Value Formula Important?

Z score can be statistically defined as the standard deviation of a raw score from its mean. It is one of the most important numerical measurements used in statistics. Z score is also most of the times regarded as the standard score. A Z score can be placed on the normal distribution curve. Z scores may be either positive or negative or null. Positive value of Z - score is the indication that the Z score is above the mean whereas the negative value indicates that the Z score is below the mean. To calculate Z score, we subtract the population mean from an individual raw score followed by the division of the difference obtained by the standard deviation of the population. This process involving the conversion is referred to as standardizing or normalizing.


The Z Score Formula

The concept of what is Z score in statistics can be better understood by the Z score formula. The Z value formula is given as:

z = (x - μ)/σ

In the above formula,

Z indicates the value of Z score

x is the raw score or the data point or the observation whose Z score is to be determined

μ represents the mean of the population

σ is the general representation of the Standard deviation


Z Statistic Formula Derivation

A series of steps explained in this section can be used to arrive at the standard formula to calculate Z score. 


Step 1:

The first and the foremost thing in calculating the Z score is the determination of the mean.Mean of the given data set is computed based on the data points or the raw scores or the observations and the total number of scores in the data set. The formula to calculate the mean of the given data is as follows. 

μ = \[\frac{\sum_{i}^{n}x_{i}}{N}\]

In the above formula, 

μ is the mean of the data set

xi is the raw score or the data point or the observation

N is the total number of scores or data points in the data set


Step 2:

The second important step to calculate Z score using the formula is the computation of the standard deviation. Standard deviation of the given population is evaluated on the basis of the mean of the population, the individual raw score and the total number of data points in the given data set. 

σ = \[\sqrt{\frac{\sum_{i}^{n}(x_{i} - \mu)^{2}}{N}}\]

In the above formula,

σ is the standard deviation

xi is the individual raw score or data point

μ is the mean of the given data set 

N is the total number of data points in the entire population


Step 3:

The value of mean and standard deviation thus obtained is substituted in the Z value formula to obtain the Z score. The summation of the difference between the mean and the individual data points is divided by the standard deviation to obtain the final value of Z score.

 z = \[\frac{x - \mu}{\sigma}\]


Z Score Formula Example Problems

  1. It was found that the mean of the marks scored by students of a class in Mathematics examination is 85 and its standard deviation was found to be 2. What will be the Z score of Sandesh, a student who had scored 79 marks in the Mathematics examination? What does the value indicate?

Solution:

Given data:

Mean μ = 85

Standard deviation σ = 2

Score of the individual student x = 79

The Z score formula is given as:

z = \[\frac{x - \mu}{\sigma}\]

z = \[\frac{79 - 85}{2}\]

z = -6/2

z = -3

Z - score is negative and hence it indicates that the individual score is lesser than the mean of the given data set. 


Fun Facts About Z Value Formula

  • The data point score and the mean score of a population are identical if the value obtained when we calculate Z score is zero.

  • The value is at a distance equal to one standard deviation from the mean when the value of Z score is 1. 

FAQ (Frequently Asked Questions)

1. What is Z Score in Statistics?

Z score is the standard deviation of the given data point or the raw score from its mean. Z score is calculated by a standard formula. The formula for calculating Z score formula is:

z = (x - μ)/σ

In this formula, 

Z indicates the numerical Z score value

x represents the raw data score or data points

Μ indicates the mean of the given data

σ is the representation of standard deviation of the population

2. How is Z Score Useful in Statistics?

Z score which is also known as standard score is very useful in statistical analysis because of the following reasons:

  • The Z score can be used to determine the probability of occurrence of a score within the normal distribution.

  • Two or more raw scores from different normal distributions can be analyzed and compared with the help of Z value formula. 

  • The comparison is done by standardizing the data points or raw scores into Z score values.