Everything about z-score in statistics

Standardized normal distribution or Z-score

It is a special type of normal distribution where the mean is zero(0) and the standard deviation is 1. The standard normal distribution is always centered at 0 and intervals increase by 1. The z score tells how many standard deviations a data value is above or below the mean.

For example, a Z-score 0 means it is on the mean. A Z-score of -2 means it is 2 standard deviations to the right or 2 standard deviations below the mean. -1.5 z-score tells that it is -1.5 standard deviation to the left or -1.5 standard deviation below the mean.
Z score is used to calculate how much area that specific z-score is associated with and you can find that exact area using the z-score table.

Formula of Z score:Z=(X-μ)/σ
Here,
σ=Standard deviation
μ=mean
X=Distribution value.

Here the distribution will be same, only the scale has changed. Suppose you have 10, 20, 30, 40, 50 in the distribution. Now if you apply standard normal distribution or z-score then 30 will convert into 0, 40 and 50 will be converted into 1 and 2. 10 and 20 will be converted into -1,-2.

To decide which Z score is better you have take two things in mind:
1. If the Z score is positive or above from the mean then that z score which is more far from mean or big z score value is better.

2. If the Z score is negative or below from mean then that z score which is less far from the mean or small z score value is better.

Example 1:If X is distributed with a mean of 100 and standard deviation
of 50, then the Z value for X=200 is ,
Z=(X-μ)/σ=(200-100)/50=2.0

What happens is here, the distribution will be the same but in the normal distribution you have value 200 and now after applying z score, that value become 2.0. Now you can say that X is 2 standard deviations left or above from the mean

Example 2:
Jack scored 90 in Physics. The class average is 85. The standard deviation is 10. Jenny scored 85. The class average is 70. The standard deviation is 8. Who scored better in their class?

For Jack:
X=90, μ=85, σ=10, Z=?
Z=(90-85)/10=0.5

For Jenny:
X=85, μ=70, σ=8,Z=?
Z=(85-70)/8=1.9

Jack and Jenny's z score is positive or above from mean. But Jenny Z's score is more or you can say more far from the mean so Jenny did better.