Showing posts with label standard scores. Show all posts
Showing posts with label standard scores. Show all posts

Friday, January 14, 2022

z-scores or standard scores

 A z-score tells you the distance of the score from the arithmetic mean of a set of scores that are normally distributed.

The z-score represents standard deviation units thus, a z-score of 1 means it is one standard deviation above the mean of the set of scores. A z-score of minus one (-1) means the score is one standard deviation below the mean of the set of scores.

The z-scores are often plotted along the x-axis of  a normal distribution, which is sometimes called the bell curve.

Use lower case italics when reporting z-scores in APA style. The upper case Z is a different score.

You can calculate a z-score by subtracting a raw score from the mean and dividing by the standard deviation of the set of scores.

Example: A raw score on a test = 60. If the mean = 50 and the standard deviation = 10 then (60-50) = 10 and 10 divided by 10 = 1.0. The z score is 1.0, it is one standard deviation above the mean.

Most z- scores fall between -3.0 and +3.0 but it is possible to have scores beyond - 3or + 3.

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Normal curve or distribution


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Wednesday, October 25, 2017

How to Compare Test Scores

When counselors and psychologists report test scores, they often report one of the scores found in the table below. When several tests are used, it is helpful to know how the scores compare from one test to another.

A good place to begin is to locate the average score-- that's the row where z = 0. Then look at the broad middle range between z = -1 and z = 1. About 68% of people score between z = -1 and z = 1.

Intelligence Tests use Standard Scores abbreviated as SS. These scores take the place of the old IQ score. An average IQ is 100 -- about 68% of people score between 85 and 115.

Here's a table from Appendix B of Applied Statistics: Concepts for Counselors

Each row contains the equivalent score on a different scoring system. For example, a z-score of 1 equals a T score of 60, and a standard score of 115. The score is at the 84th percentile.


Table Notes

z-scores, M = 0, SD = 1
T-scores, M = 50, SD = 10
SS-Standard Scores, M = 100, SD = 15
PR- Percentile Rank

 Find more practical concepts in Applied Statistics: Concepts for Counselors

An inexpensive reference book for students and counselors

Learn more at the book's website  counselorstatistics

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Thursday, October 12, 2017

INTELLIGENCE TESTS - What Counselors & Psychologists Know

Intelligence tests (IQ tests) are in the news lately as people banter about terms from many decades ago. IQ tests are widely used because they measure the ability of people to solve various problems, predict academic achievement, and help with job placement in some settings. The tests also help neuropsychologists assess functioning in people with impairments due to head injuries and brain diseases.

During part of my childhood, I passed a facility where American IQ testing began. I saw people on swings and on the grounds of the Vineland Training school in Vineland NJ. It turns out that a little over 100 years ago, American psychologist, Henry Goddard, brought a test by French scientist, Alfred Binet, to the New Jersey Training School for Feeble-Minded Girls and Boys in Vineland, NJ. The test was modified and widely used in the U.S.

What tests are used today?

Today, a number of tests are available in the US and elsewhere. Popular American tests are the Wechsler Intelligence Scales and the Stanford-Binet Scale. The tests are regularly updated with new materials and tasks appropriate to people of different ages. Several other tests are also available such as the Kaufman Assessment Battery, Woodcock-Johnson Tests of Cognitive Ability, and the Differential Ability Scales.

A full scale test can last over an hour, so it is not surprising that a number of shorter tests are available. The shorter tests are considered "screening tests" because they include fewer subtests (or sets of tasks), to measure problem-solving skills. It is common to use a test of verbal ability such as vocabulary and a test of "nonverbal" ability such as tasks that require solving visually presented tasks.

Intelligence tests yield a variety of scores that recognize people have different abilities. This fits with common sense as we observe people with different abilities--strong verbal skills, incredible abilities to design complex structures, create various artistic works, and so forth. Still, many people want to know their IQ-- a short hand way of identifying an overall general ability. The overall score is controversial but remains in use.

In years past, the IQ (intelligence quotient) was measured as a ratio of chronological age to mental age. Mental age referred to a person's score on test tasks compared to others of the same age. As I posted previously, there are problems with age scores. Today, the scores on tests of intelligence compare people of similar ages to their age peers. For traditional reasons, the average IQ (or standard score) has an arithmetic average (M, mean) of 100 and a standard deviation (SD) of 15 points (read more about a few statistics).

It turns out that despite different test tasks and scales, people earn similar scores. An IQ score or standard score on one test is likely within a few points of the same score on a different test. As people age, the scores are more reliable-- that is stable. So, if an adult earns a score of 110 today, she would likely have a score within a few points of 110 in 2-years--unless something happened.

The stability of the scores make the tests useful when considering the effects of brain damage or disease. Of course, neuropsychologists use other tests as well (e.g., tests of memory, visual-spatial skills).

Read More


What is average intelligence?

That can be a trick question unless you clarify what you mean by intelligence. On tests that report standard scores, the average score is 100. Using the common standard deviation of 15 points, about 68% of our age peers will score beetween 85 and 115. Close to 95% of people the same age will have scores in the range of 70 to 130. As you can see, only a small percentage of people score above 130 or below 70.

The test scores compare people to others of the same age. The skill levels develop rapidly in young children. Several months can make a difference in average scores. In adults, scores vary in how they change for people in large age brackets. Some abilities decline more rapidly than others. For example, young adults tend to be faster than older adults when solving tasks requiring eye-hand coordination.

What are some problems with IQ tests?

Test scores do not capture the range of abilities of people who are differently abled. For example, those with severe visual impairments cannot see visual test tasks. And those with severe hearing impairments may not respond well to spoken instructions or auditory tasks. Clearly, it would be wrong to assume something about a person's intelligence using tests that are not designed for people who have limited vision, hearing, or some other similar condition.

Some clinicians fail to document vision, hearing, or other limitations. For example, many people show up for testing and leave their eyewear at home. A child may forget his glasses or hearing aids.

People with temporary limitations cannot take tests as well as they could at other times. If you cannot use your dominant hand due to an injury, you will have difficulty on tests that require using your hands.

People taking medication can respond differently when taking medicine that either helps or interferes with attention and concentration. Of course, illegal drugs can also affect the brain processes needed to remember instructions and solve problems.

People who are not fluent in the language of the test may have a difficult time depending on their language skills.

So called "nonverbal tests" measure different abilities than tests that include language so mistakes can be made when making judgments about general intelligence or ability.

Clinicians make mistakes in recording information, scoring, or writing reports.

Tests are not perfect measuring instruments. Even when administered to people under the best of circumstances, there is measurement error. Measurement error is usually more variable for children than for adults. Measurement error refers to a variation in scores from one administration to another.

What about labels?

I suppose we will have a hard time escaping labels. The words used for people getting high scores or low scores have changed over the years--too many to cover in this post. Insulting words about a person's intelligence were terms used many decades ago. Today, clinicians and organizations like schools use a variety of terms focused on helping high scoring students learn in more challenging environments. And students who score very low on several tests, are elligible for services designed to help them maximize their potential. Insurance programs use cutoff scores and other criteria when awarding benefits to people with severely impaired abilities.

Who administers IQ tests?

A variety of professionals are qualified to administer, score, and interpret IQ tests. They are most commonly used by School Psychologists in schools and private practices. But other psychologists who specialize in neuropsychology also use IQ tests as part of their assessment. Many school counselors also have the necessary skills. In some cases, a psychological technician will administer the tests but the interpretation is left to the clinian holding an advanced degree along with the appropriate license or certification.

Read more

Applied statistics: Concepts for counselors. 


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Thursday, September 21, 2017


What are age scores?

Age scores, also called age-equivalent scores, are supposed to help people understand how a person’s test score compares to other people of the same age. They are often provided to teachers and parents to show how children scored on achievement tests compared to their age peers. A common age-equivalent abbreviation is AE

Age scores are reported with a hyphen. The first number refers to age in years and the second number refers to the age in months. A score of 8-4 is supposed to mean a test performance typical of children age 8-years and 4-months.

The scores appear convenient and make a kind of common sense. An age score of 7-6 is supposed to mean that a child earned a test score similar to children age 7-years and 6-months. But there are problems with the scores.

What tests report age scores?

Age scores are commonly reported with results of achievement tests. They are sometimes reported with results of intelligence tests. Old intelligence tests reported a mental-age score (MA).

What’s the problem with age-equivalent scores?

The scores create an inaccurate impression of performance for children who are much younger or older than the age comparison group. An age score of 7-6 obtained by a 5-year-old does not account for all of the knowledge or ability that is typical of a child age 7-6. Similarly, a teen aged 13 may have different skills than represented by an age score of 7-6, which of course suggests a very low skill level for a 13-year-old.

The reliability of age scores for children whose actual age is much lower or higher than the reported age score varies from the reliability values for children close to the age-score. The reliability of scores is not a stable characteristic. Retesting can yield very different results on retesting when a very high or low score is obtained.

Age scores do not allow for an accurate comparison over time because the content of the tests and the abilities of children change dramatically as children age. Reading tests and reading abilities are very different for children age 7 and those age 14. Children learn very different math concepts at age 12 than do those at age 8. To say an 8-year old has an age score of 12-1 in math is hardly accurate.


Read more about age-equivalent and other scores in


Age score differences do not provide parents and teachers with an accurate picture of delays and advancement. A child who is one year behind peers in math at age 8 is further behind than is a child who is one year behind at age 15. The gains children make in reading, math, and other skills are greater in the early years of life than in later years.

Age units are inaccurate compared to other scores. Age scores report differences in months but child development is uneven—especially when it comes to mental abilities like reading comprehension, spelling, and visual memory. Comparisons do not make a lot of sense for children of the same chronological age who earn different age scores. Two children having the same chronological age of 8-years and 2-months but different reading comprehension test age scores of 6-10 and 9-3 have different skill levels. But concluding they are 2-years and 5-months apart on reading comprehension is not reasonable because only a small sample of skills are assessed on tests. Even worse might be the perception that the score difference is somehow fixed. Differences this large will almost always change over time.

There are problems with the samples of children at different ages when the age scores are very different from the chronological age. Consider children with an actual age of 8-3. If one obtains a test age score of 5-6 and another of 11-3. A lot of 8-year-olds can take a test designed for 8-year-olds but how many 5-year-olds or 11-year-olds take the same test? The problem is having an accurate sample group for comparison purposes.

What scores are better than age-equivalent scores?

Several scores are better than age-equivalent scores. Most tests report standard scores (SS) and national percentile rank (NPR) scores on children’s achievement tests. These scores compare children to others of the same age group.

What about grade scores?

Grade scores have the same problems as do age scores. A grade score is reported as a grade number with a decimal and a second number referring to the month of a school year. A grade score of 5.6 means the sixth month of grade 5. 

You can see my glossary of test and statistical terms at this website:

Sutton, G. W. (2017). Applied statistics: Concepts for counselors. Springfield, MO: Sunflower. Amazon  Paperback ISBN-10: 1521783926, ISBN-13: 978-1521783924

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Post Hoc Tests and Data Analyses

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