What is statistics? – Z score – Standard score – Part I

Statistical Definitions – Raw score
A raw score is any number of test performance, observation, result that statistical or mathematical operations has NOT acted upon it. That is, it is the original score before any statistical manipulation or transformation.

z transformation - z score - raw score - standard normal score

Z scores or z values or Standard scores or normal scores – Theoretical Definition
The Z score or z value or Standard Score or Normal Score is score/s of Standard Normal Distribution (pdf). As it was told, the z score distribution has an Arithmetic Mean that equals to zero (0) and a Standard Deviation that equals to one (1). Therefore, when a Raw score is transformed to Z score, automatically, its Arithmetic Mean is transformed to equal Zero (0) and its SD to equal one (1).

z score transformation

The Population Standard Deviation must be known and the underlying population or sample must follow a Normal Distribution. Then, Z scores can show:

i) How many Standard Deviations below or above the Population Arithmetic Mean a Raw score is.
ii) When Raw Scores that are coming from different samples BUT from the same underlying population are transformed into Z scores, can be compared in a standardized way. That is, these scores are transformed to something that includes the same statistical properties.

z transformation_z_scores_z_values_mean_0_sd_1

Statistical Formula to transform Raw scores into z scores
The formula for transforming, known also as “standardizing or normalizing”, a “X” Raw score into Z score, by knowing its Population Arithmetic Mean and its Population Standard Deviation is the following one:

z=\frac{X-\mu }{\sigma }

z transformation_z_scores_z_values_mean_0_sd_1_formula

i) If we know before hand the Population Mean, and the Population Standard Deviation, then the Z score can be transformed back into a Raw score, if we know its z score value:

X=(z*\sigma)+\mu

ii) We can also find what is the Standard Deviation of the Underlying Population by knowing beforehand the Population Mean \mu, the Raw Score: X as well its z value z by using the following formula:

\sigma=\frac{X-\mu}{z}

iii) Finally, if we know the sample mean and we would like to transform it to Z-score, then we must use this Statistical formula:

z=\frac{\bar{x}-\mu }{\sigma /\sqrt{N}}

Symbol Explanation
\bar{X} is the sample mean
\mu is the population mean
\sigma is the population Standard Deviation

population vs Sample_symbols

Statistical Example Ι
Let’s say you are in a school class which represents a whole population. You are interested in to learn about your z score in order to be able to compare your test performance with the performance of other classmates or with the Arithmetic Mean of the class (population mean). Let’s say that your performance in a Literature Test is X=15, the Mean of the class for this test is \mu=17 and the Standard Deviation of this class in this test is \sigma=2. Therefore:

X=15,
\mu=17 and
\sigma=2

By replacing these values into the formula, we get as a result:

z=\frac{15-17 }{2}=\frac{-2}{2}=-1

z_value_z_score_class_example_0

This result shows that:
i) Your performance in the Literature Test is 1 Standard Deviation below the Mean and that
ii) Your test performance was about 34.1% lower than the mean of the class in this test.

Statistical Example II
Let’s say that your classmate had a test performance in Literature of X=16 The population mean and Standard Deviation remain the same. We get the following result:

z=\frac{16-17 }{2}=\frac{-1}{2}=-0.5

i) Your test performance in Literature is 15 while your classmate has a test performance in Literature 16. The mean of the class in this test is 17. You have a Ζ value or Z score of -1 and your classmate -0.5. Therefore:
ii) Your test performance in literature is 1 Standard Deviation below the mean of the class in this test
iii) While the performance of your classmate in Literature test is half Standard Deviation below the mean (0.5).
iv) That is, you had a performance that was 34.1% and your classmate about 17% lower than the mean of the class, in this test.
v) In other words, these z scores differ by half Standard Deviation or about 17%.

z_value_z_score_class_example

Statistical Example ΙΙΙ
Let’s say that now you are in the Gymnastic class and your test performance is X_{1}=12, while your female classmate had a test performance of X_{2}=15. The mean of the class for this test is 15 and the Standard Deviation is 5:

X_{1}=12, X_{2}=15,
\mu=15 and
\sigma=5, therefore:

You have a z score of: z=\frac{12-15 }{5}=\frac{-3}{5}=-0.6
while your female classmate has a z score of: z=\frac{15-15 }{5}=\frac{0}{5}=0

i) Here, the difference between your raw score and the raw score of your classmate is three (3) points
ii) While the difference between your z score and the z score of your classmate is 0.6 Standard Deviations
iii) In other words, your performance was 20% lower than the mean of the class in this test, and
iv) The performance of your classmate was neither higher or lower than the mean of the class in this test (0%)
v) Therefore, one performance is lower / higher than the other performance about by 20%.

z_value_z_score_class_example_gymn

Table and z value comparisons
The below table shows what happens when the population mean / the mean of the class for a number of tests remains the same when the Standard deviation of the population changes each time.

Let’s say that each row represents a different test in the class. The first column shows your raw score in a test and the second column shows the raw score of your classmate in the same test. The final two columns show your Raw scores and your classmate’s Raw scores transformed into Z scores.

TestYour Score: X1Score of your classmate: X2Standard Deviation of the class (σ)Mean of the class (μ)Your z score (Ζ1)The z score for your classmate (Z2)Your performance
Classmate perf.
Difference of perf.
1st1215115-3049.9%-0%=49.9%
2nd1217315-10.6734.1%+20%=54.1%
3rd1220515-0.6120%+34.1%=54.1%
4th1201015-1.40.547.7%+17%=74.7%

First test results
i) Here, the difference between your raw score and the raw score of your classmate is three (3) points (12-15=-3)
ii) while the difference between your z score and the z score of your classmate is 3 Standard Deviations (-3-0=-3).
iii) In other words, your performance was 49.9% lower than the mean of the class in this test, and
iv) Where the performance of your classmate was neither higher or lower than the mean of the class in this test
v) Therefore, one performance is lower / higher than other performance about by 49.9%.

z_score_examle__I_Table

Second test results
i) In the second test, your raw score and the raw score of your classmate has a difference of 5 points (12-17=-5)
ii) while the corresponding z scores or z values differ by 1.67 Standard Deviations (-1-0.67=-1.67).
iii) In other words, your performance was about 34.1% lower than the mean of the class in this test,
iv) Where the performance of your classmate was about 20% higher than the mean of the class in this test,
v) Therefore, one performance is lower / higher than other performance about by 54.1%.

z_score_examle__II_Table

Third test results
i) In the 3rd test, the difference of your raw score and the raw score of your classmate was 8 points (12-20=-8)
ii) While the corresponding z scores or z values differed by 1.6 Standard Deviations, the same as before (-0.6-1=-1.6)!
iii) In other words, your performance was about 20% lower than the mean of the class in this test,
iv) Where the performance of your classmate was about 34.1% higher than the mean of the class in this test,
v) Therefore, one performance is lower / higher than other performance about by 54.1%.

z_score_examle__III_Table

Fourth test results
i) In the extreme case that your performance in a test was only 1 point (worst result) and the performance of your classmate in the same test was 20 (best result) and the Standard Deviation of the class in this test was 10 and the corresponding mean was 15, then you may see that your raw score and the raw score of your classmate differ by 19 points (1-20=-19), then
ii) The corresponding z scores differ by 1.9 Standard Deviations (-1.4-0.5=-1.9)
iii) In other words, your performance was about 47.7% lower than the mean of the class in this test,
iv) Where the performance of your classmate was about 17% higher than the mean of the class in this test,
v) Therefore, one performance is lower / higher than other performance about by 74.7%.

z_score_examle__Iv_Table

z_score_standard_score_team_finish