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.

Moments

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

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

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





ANSWERS



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  
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"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.

CREATING SURVEYS

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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