Standard Score

What is the Standard Score ?

Also known as a Z-score, Z-value, Normal Score or Standardized Variable.
A standard score indicates how many standard deviations a value is above or below the mean.
The distribution of standard scores is normal if and only if the distribution of the values is normal.
If the distribution of the values is normal, then the distribution of the standard scores is a standard normal distribution.
Standard scores correspond very closely to standard deviations.
Standard Deviations and standard scores provide us with a way of describing the position of an individual value in a larger distribution.

Using Standard Scores

It is usual to refer to standard scores as plus or minus.
A standard score of +1 indicates that the value is exactly 1 standard deviation above the mean.
A standard score of 3 (or +3) indicates that the value is 3 standard deviations above the mean.
Standard scores allow us to compare results from different distributions in a meaningful way.

Calculating Standard Scores

We can calculate a standard score using the following formula:
standard score = (x - mean) / (the standard deviation)
where x is the value you want to convert to a standard score.

Area Underneath

By using the areas underneath normal distributions we can calculate probabilities of different outcomes.
By converting normally distributed scores into standard scores we can ascertain the probabilites of obtaining specific ranges of scores.

If you are 1 standard deviation above the mean then you have a percentile rank of 84%.

Example - How did I compare

Lets imagine that you got 62% in a maths test and wanted to know how this compared with the rest of the class.
The teacher says the average score was 56% and that the standard deviation was 5.
We know the scores are normally distributed so 68% is within 1 standard deviation of the mean.
This means that 68% scored between 51% (56-5) and 61% (56+5)
This tells us that 68% of the class scored between 51% and 61%
This tells us that 15.5% of the class scored below 51% and 15.5% of the class scored over 61%
With a score of 62% we are therefore in the top 15.5%

Example - What is your exact position though ?

To find our exact position we can calculate the standard score and then convert this to a probability.
You can use the NORM.DIST to convert straight into a probability.
This tells us that (100 - 88.49) = 11.5% of the class got more than 62%.

Or you can convert into a standard score first using the STANDARDIZE function.
and then use the standard normal function NORM.S.DIST to convert this to a probability.

Example - Who is in the top 10%

What score would you need to be in the top 10%

Important

You can use standard normal tables to give the probabilities of a particular score occurring.