Showing posts with label Mean. Show all posts
Showing posts with label Mean. Show all posts

Thursday, January 20, 2022

variance and standard deviation

Variance is a measure of the dispersion of values in a distribution of values.

 In psychology and behavioral science statistics, the variance is typically a reference to the extent to which numerical values vary around the arithmetic mean of a data set. Theoretically, the values vary around a population mean but in most cases, researchers work with samples.

In statistics, write sigma squared for the population variance σ2

Write final form sigma squared for the sample variance ς2

In reports, write VAR for variance.

How it works

If we have a set of different numerical values such as scores on a test we can calculate a mean, which is the average of all the scores divided by the number of scores.

The difference of one score from the mean is a deviation score. X is a score and the Greek letter mu μ is the symbol for the population mean.

In a sample, which is what we normally have in psychology, we subtract a score X from the sample mean M. Thus, X - M = the deviation score. 

If a person earns a 7 on a test where the mean is 10 then their score is 3 points below the mean. The deviation score is -3 (minus three).

To find an average deviation for all the scores on the test, we must subtract each score from the mean. We end up with a set of positive and negative values.

We want to find an average but if we add the positive and negative values we will end up with zero. To solve the problem, 

square each deviation score

add them together to get the sum of the squares (SS)

 then divide by the number of scores (n) to get the variance.

The variance = the SS divided by n

Standard Deviation (SD)

The standard deviation is the average unit by which scores are distant from the mean.

Find the standard deviation by taking the square route of the variance.

In reports, write SD for standard deviation.

Examples in tests

A common mean for IQ tests is 100 and a common standard deviation is 15 points.

In personality tests, a common standard score is a T-score. The mean is T = 50 and SD = 10.

Key concepts in this post

deviation score


sum of squares

standard deviation


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Sunday, March 28, 2021

Skewed Distributions


Skewed Distributions*

Skewed distributions have one tail that is longer than the other tail compared to the "normal" distribution, which is perfectly symmetrical. Skew affects the location of the central values of the mean and median.

Positive Skew

Below is an image of positive skew, which is also called right skew. Skew is named for the "tail." If you had statistics, you may have heard a professor say, "the tail tells the tale." The tail is the extended part of the distribution close to the horizontal axis.

The large "hump" area to the left represents the location of most data. In behavioural science, the high part often refers to the location of most of the scores. Thus, in positively skewed distributions, most of the participants earned low scores and few obtained high scores as you can see by the low level of the curve, or the tail, to the right.

Negative Skew

As you might expect, negatively skewed distributions have the long tail on the left thus, they are also called left-skewed distributions. A negatively skewed distribution of test scores illustrates an easy test--just what students want. Teachers used to talk about grading on a curve. You can see that such grading could be good or bad for students depending on what curve the teacher uses.

Skewed distributions are nonnormal by definition. 

Recall that in the normal curve, the mean, median, and mode are all at the same point in the middle of the distribution. The value of skew in a normal distribution is zero. 

In skewed distributions, the mode is at the high point and it represents the most frequent value or test score. The mean is pulled in the direction of the long tail and the median falls between the mode and the mean.

Common test questions ask what happens to the mean in skewed distributions. Keep in mind that the mean is "pulled" toward the tail. The mean is an average and, as such, it is most susceptible to extreme scores.

Skew and Data Analysis

Most statisticians accept small deviations from normality when analysing data using procedures designed for a normal distribution like the Pearson r, t tests, and the parametric ANOVAs

The question of acceptable ranges of skew will yield different answers from different sources. A range of +1.5 to -1.5 is a common rule of thumb. An important consideration is the "true" nature of the measured variable. Scientists may argue for flexibility in analysing data from a sample if the variable is known to be normally distributed in the population.

Skewed data can be adjusted and should be adjusted before using parametric tests. One method of adjustment is to convert all scores to logarithms and perform the data analysis on these transformed values.

If the data are too skewed and it is inappropriate to transform the data, then analysts should use nonparametric statistical methods.


In statistics, the concept of moments is taken from physics. Moments refer to central values. The first moment is found by calculating the value of the mean. The first moment is zero.

The second moment is seen in the calculation of variance, which uses squared values.

The third moment is found by calculating skew and the fourth moment results in the calculation for kurtosis.

Learn more about behavioural statistics in Applied Statistics Concepts for Counselors on AMAZON   or   GOOGLE

Learn More about analyzing data  in Creating Surveys on AMAZON or GOOGLE

Please check out my website

   and see my books on   AMAZON       or  GOOGLE STORE

Also, consider connecting with me on    FACEBOOK   Geoff W. Sutton    

   TWITTER  @Geoff.W.Sutton    

You can read many published articles at no charge:

  Academia   Geoff W Sutton     ResearchGate   Geoffrey W Sutton 

*Photo credit- From Bing images labeled "Free to share and use."

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. 


You can also read more about the assessment of"thinking" in Creating Surveys.

Books by Geoffrey W. Sutton

Sunday, October 1, 2017

Take a brief Counseling Test Quiz 101

Can you answer these questions that every counselor ought to know?

Choose the BEST available answer.

I'll post the answers below.

1. If the correlation between a test of intelligence and a test of achievement is usually between .88 and .92, how well can you use the intelligence test results to predict achievement test results?

A. Very well
B. Moderately well
C. Not well at all
D. None of the above

2. A personality test score was high on a scale of Extraversion. The validity of the Extraversion scale was reported as .52 to .57 compared to similar tests. How much confidence should the person have that their score is "valid?"

A. A high degree
B. A moderate degree
C. A low degree
D. None of the above

3. School counselors administered a questionnaire to 1,000 students. They calculated results for answers about four school improvements rated on a scale of 1 to 5. Most of the scores were in the range of 18 to 20. The counselors reported a mean rating of 4.6 for each of the 4 items. Based on these data, what should they have reported?

A. The mean is fine-- an average is all that is needed.
B. They should report the Mean and Standard Deviation.
C. They should report the reliability with the mean.
D. They should report the median and range.

4. An agency director asks a counselor to determine if there was evidence of improvement in well-being for clients in one of three treatment groups. Assuming a normal distribution of the data, which of the following statistical procedures could provide the best answer?

A. An independent samples t test
B. A one-way analysis of variance
C. A two-way analysis of variance
D. A chi-square test


1. A. Other things being equal, the correlation between the two tests is strong thus, most of the time the intelligence test score will be a good predictor of the achievement test score. See Chapter 12 in Applied Statistics: Concepts for Counselors.

2. C. We do not know much about the validation of the Extraversion scale ; however, we know the validity values in the .50s are low so the best answer, given the limited data, is C. Validity coefficients range from 0.0 to 1.0. Important note: Validity is a product of the interpretation of data based on scores. Although it is common to refer to a test's validity, tests really do not have validity. Instead, there is a history of validity statistics and interpretations associated with validity. See chapter 20 in Applied Statistics: Concepts for Counselors.

3. D. The data appeared skewed given that 4 items on a 5-point scale would yield a maximum of 20. So, based on the limited data, the median would be the most typical value. When reporting the mean, counselors ought to report the standard deviation, but in this case, the median appears to be the best value. See Chapters 7-10 of Applied Statistics: Concepts for Counselors.

4. A one-way analysis of variance can be used to analyze data from two or more groups. If the overall value is statistically significant, t tests or other post hoc tests can be used to compare pairs of means. See Chapters 15-17 of Applied Statistics: Concepts for Counselors.

APPLIED STATISTICS: CONCEPTS FOR COUNSELORS is available as an eBook or paperback from AMAZON.

Book website

"If you need to review basic statistics and don’t know where to begin, this book is perfect! It makes difficult concepts easy to understand. I would recommend it for my undergraduate students who haven’t had Statistics in a while and need a refresher, or for graduate students facing their first graduate level research class!"
...Heather L. Kelly, Psy.D., Professor of Psychology, Evangel University
Springfield, Missouri, USA

You may also find this book relevant.

Sunday, September 3, 2017

Reporting Mean or Median

Who would think that a simple statistic like a mean or a median would make a difference?

In large samples involving thousands of people, and when data are normally distributed (close to the shape of a bell curve), the mean and median will be nearly the same. In fact, in a theoretical distribution called the normal curve, the mean, median, and  mode are in the middle.

But, many samples are not normal distributions. Instead, the often contain extreme scores called outliers or a lot of scores bunched up at high or low levels (skewed). Sadly, even people that understand statistics, continue to report the mean as if they are not thinking about their samples.

Suppose you work for a company where the top person earns $300,000 but most folks earn $30,000 to $60,000. Well that $300,000 is gonna skew results and the mean will look much higher than the median.

I ran some fictitious data on a sample of 10 people. Nine earn between $30 and $60K and one earns $300K. The Mean = $67K (standard deviation = 82.58), but the Median is only $38.5K and the Range = $270K.

Now those results are fictitious and it is a small sample so it magnifies the differences. But you know some folks are earning over $1,000,000.00 in some companies and lots of folks aren't earning anywhere near that amount.

So who cares? Well salaries make a lot of difference if you are arguing for a raise, considering a change of jobs, voting on budgets in not-for-profit organizations, and more. How motivating is it to give a donation to a company that helps the poor where the CEO pulls down nearly a million bucks a year and you get by on $65K-- or less?

But there's more. Teacher evaluations are usually skewed -- most students give high ratings-- so the median and range are more appropriate than the mean.

[ Read more about statistics in
Creating Surveys on AMAZON]

Real estate prices can be out-of-whack if you look at the mean price in a city where a few multimillion dollar homes pull the mean to a high level compared to the median price.

I see research papers where the scientists report the average age of people in surveys is 19 and they tell you thir sample was from a university. No problem with age 19 but when they report a Mean of 19 and a standard deviation of 5, there is a problem! If you understand standard deviations, you will know why they probably did not have a lot of 14-year olds in their university!

You can see that knowledgeable folks can play games with a simple statistic.

If you forgot about the meaning of some terms, here's a link to a free glossary.

A simple example

Counselors, teachers, and parents - think about test scores and how they are reported.  Test scores for students at school may be distorted by a few very high scoring or very low scoring students.

"Averages" can be deceiving.

Read more about basic statistics in APPLIED STATISTICS: CONCEPTS FOR COUNSELORS at



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