A score that has the same

meaning for all individuals

(regardless of age) and

indicates the degree to

which what is being

measured is normal or

abnormal. To understand

why this is needed,

consider the following

scenario. Imagine that the

same 50-item math test

was given two people,

ages 10 and 40.

The ten-year-old answered 20 items correct and the 40-year-old answered 25 items

correct. Who did better? We cannot say based on the raw (non-standardized) scores

(i.e., 20 and 25) because we do not know what is normal performance for each age

group. That is, a score of 20 out of 50 might be great for a 10 year old but terrible for a

40 year old. If so, the raw score (20) does not have the same meaning across individuals

in different age groups. This is where standard scores come in. To calculate a standard

score, one needs to know the average score and standard deviation (SD) of the

reference group (in this case, the age group). The standard deviation is a number that

reflects how much scores tend to vary away from the average score in the reference

group.

The average score (also known as the mean score) and SD are

known in advance. For the example above, assume the mean score

on the math test for ten-year-olds is 19 (SD = 5) and that the mean

score for forty-year-olds is 40 (SD = 7).

The formula for calculating a standard score is as follows: raw score

minus the mean score, divided by the SD. For the ten-year-old, the

calculation is as follows: 20-19, divided by 5, which equals 0.2. For

the forty-year old, the calculation is as follows: 25-40, divided by 7,

which equals -2.14. The resulting standard scores are known as Z

scores. The Z scores provide an indication of how far the score is

away from the normal (average) score of the reference group.

On a Z score scale, a score of 0 is normal and at the 50th percentile. A percentile reflects the percent of

people that one scored better than. So if a score is at the 50th percentile, this means that the score is

better than 50% of people who took the test. At the same time, the scores is also worse than 50% of

people who took the test. Thus, the 50th percentile is directly in the middle. A Z score of +1 is at the 84th

percentile, meaning it is better than 84% of people who took the test. A Z score of -1 is at the 16th

percentile, meaning it is better than only 16% of people who took the test. Thus, 34% of people fall within

one unit (known as a standard deviation) below the mean (i.e., 50-34=16) and 34% of people score one

standard deviation above the mean (50+34=84). To put it another way, 68% of people will obtain a

standard score that is within one standard deviation above or below the mean score.

So let’s return to our example. The z score for the 10-year-old is 0.2. This score is at the 58th percentile,

meaning it is better than 58% of other 10-year-olds who took the same math test. This is an average

score. The Z score for the 40-year-old is -2.14. This score is at the 2nd percentile, meaning it is only

better than 2% of other 40-year-olds who took the same math test. In other words, 98% of 40-year-olds

who originally took the math test performed better than this individual. So even though the 40-year old had

a higher raw score than the 10-year-old, the 10-year-old’s performance is actually better and the standard

score helps determine this.

Some people take Z scores and convert them to other types of standard scores such as scaled scores, T

scores, or deviation IQ scores. All of these scores can be used interchangeably. As mentioned above, the

50th percentile for a Z score is 0. The 50th percentile for a scaled score is 10, for a T score is 50, and for

a deviation IQ is 100. The standard deviation unit for a Z score is 1, for a scaled score is 3, for a T score

is 10, and for a deviation IQ is 15. To make things more confusing, deviation IQ scores are sometimes

referred to as “standard scores” but technically, all of the prior scores are standard scores. To see how

these scores are all interchangeable and what the associated percentiles are, please visit the

MedFriendly standard score to percentile conversion table.