Based on the summary statistics you have provided I doubt whether your data set could have a skewness of 0.000 as this would be totally symmetric and your mean and median are very different. I don't think there are alternative ways of calculating skewness that would account for this large difference so perhaps you are reading the wrong figure from the software.

Enter the data into a single variable. Select Analyze - Descriptive Statistics - Descriptives, select the variable you have created and select skewness under the options.

The fact that Skewness is 0.000 is irrelevant. Take the following values 2, 3, 4, 5, 6, and run a procedure for Mean, Mode, Standard deviation and Skewness. Value of Skewness is 0.000, but when i calculate manually it is 1.26. My question is why is manually calculated Skewness different from SPSS calculation?

Boban, can you show us how you are doing the manual calculation of skewness, say for your example: 2,3,4,5,6 ? I mean the calculation or formula that yields you 1.26?

Mean is 4, Mode is 2 and Standard deviation is 1.58. So i calculate by this formula:

(Mean-Mode)/Standard deviation

(4-2)/1.58 = 1.26

Is it my way of calculating wrong or is it something else? I used SPSS and some other sites for calculating Skewness, and they all provide same value, but different from my "manual" value.

Hi Boban, thanks for your reply. The manual formula should be subtracting median instead of mode and also there should be a factor there. Look at this link, it will help: 

http://growingknowing.com/GKStatsBookSkewness.php

Hello again Ana.

It was useful, but i still get no match with value from SPSS software. It is most likely that SPSS program using some other formula to determine the value of Skewness. But i'm surprised to see that different programs are using different formulas.

If you (or somebody else) can find the exact formula for Skewness in SPSS, please post it here.

Best regards.

Mr. Sampath,

Thank you for joining the discusion. Let me answer your first question. I have misplaced the datadata, and i can't remember the exact values. I'm trying to find the data, but no luck so far. If i can find the data, i will post it. But i will repeat again, it's not the problem for getting value of Skewness 0.000 in SPSS software. The problem is that i have got different value when i calculate manually.

With formula provided by Ms. Donaldson and you, i'm getting the same value like in SPSS software. But another problem emerged. When i use another dataset: 5, 6, 7, 8, 10 i have got value of Skewness in SPSS 0.592, but manually i have got 0.321. In this dataset the Mean is 7.20, Median is 7 and Standard deviation is 1.92.

Well done Boban, S Sampath, Peter and all that contributed to this fruitful discussion. We can now say that Boban, you are not doing anything wrong. Notice that what happens is simply that SPSS (and Excel) use a different formula to calculate skewness from the one that is given in the textbooks (and in the website I passed on). See that in the website it says: "Caution - common mistakes?  You should be aware that computer applications like Minitab or Excel use a different formula for calculating skewness than most books show for manual calculations. Check with your teacher to see which calculation is acceptable.".  The skewness of your dataset 5, 6, 7, 8, 10 is around 0.32 when we use the usual formula but if you use the command SKEW in Excel you get 0.592. Of course it would be interesting to check what is the formula that Excel and SPSS (and I think Minitab) use and see what are the assumptions or conditions to use that formula or the one that appears in most textbooks. However, for the purpose of judging the degree of skewness, given the treshold values, the two results are not too distinct.

However, for the purpose of judging the degree of skewness, given the treshold values, the two results are not too distinct in this particular case.

Hi S. Sampath, I very much like your proposal to use the quartile-based skewness in this case. It is interesting to notice that all the approaches fall in the zone of very-small/essentially-none skewness. Best regards, Ana

Ana,

I agree with your observation. I think that it is the best to consulting with teachers or mentors about what is the most fitting formula. Your link was very helpful, and you highlighted very interesting and important thing from it.

As for Mr. Sampath observation, it is very good thinking, but i would still choose that distribution is asymetric, because of value 10 who will move the curve to the left. Because we don't have equally larger and smaller values around the Mean, we can not say that our distribution is symmetrical, or that Skewness = 0. 

Mr. Sampath,

I've provided the values for parameters (Mean=6; Mode=2; Standard deviation=3.16) from my question in description box. 

Hi Boban and S Sampath,

Thank you both for the food for thought and fruitful enjoyable discussion.

S. Sampath's calculation of the quartile-based skewness is correct, assuming the choice of 1st and 3rd quartiles is justified in our dataset. I agree with Boban wanting to see that the choice of mehtod reflects even a tiny departure of symmetry in the distribution.  

Let me try to explain my viewpoint. 

In the website http://quartilesdecilespercentiles.blogspot.com/  one can see that there are different methods to choose the second and third quartiles of a distribution. In method 1, when finding them, one doesn’t include the median or min or max into the two halves. In method 2, if median is a datum (like in our dataset), one includes it in both halves. To get the zero for the quartile-based skewness (as S. Sampath calculation) I think one would have to use method 1, and in that case the first quartile is 6 and the third quartile is 8, and the quartile based skewness exactly zero. This is not critical for the dataset we were discussing (5, 6, 7, 8 and 10) since 10 is very close to the rest.

A problem would come up if instead of 10 being the maximum so close to the rest, say, the dataset were: 5, 6, 7, 8  and 100  (or 5, 6, 7, 8  and 1000). Choosing 1st and 3rd quartiles with method 1 would not reflect the high skewness and would give 0 all the time.  

The method of choosing the first and third quartiles is therefore critical here. For the quartile-based skewness to be sensitive to departures from symmetry, it should reflect the gap between 8 and the maximum, so in that case (quartile based approach) I would choose method 2 for the choice of quartiles. With method 2, the first quartile would be 6 and the third quartile would be 9, and the quartiles-based skewness coefficient would be 0.14.

Notice that choosing 1st and 3rd quartiles with method 2,  if maximum was 20 the quartile-based skewness would be 0.65 and if the maximum were 100 the quartile based skewness would be 0.94.

This illuminating discussion with you two makes me think that quartile-based skewness has to be used very carefully and assume some knowledge of the actual distribution one is investigating.

I welcome your comments.

Best regards, Ana

Recent study has shown that the computation of skewness using SPSS is different from the value of Eview 7. We have modified method which is what is used in SPSS but the usual formula of computation of skewness can be found in Eview. So, if you compute the skewness using Eview, you will arrive at the same answer.  SPSS, Minitab, Excel function produce Skewness and kurtosis using modified approach but Eview produces the values using the commonly used expression for skewess and kurtosis.

I found that calculating skewness manually will not give accurate numbers, but can give estimation of skewness. however, statistical program has no error in doing this. Therefore, just do it using SPSS

Note that there are different formulas in use. For an overview see

Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183-189.

See also the ResearchGate thread at https://www.researchgate.net/post/Different_result_of_skewness_and_kurtosis-any_thoughts

Please, I hope that you can check your data , the median of your data is 6 not 2 and there is no mode in your data , and you can use spss for accurate answer for skewness and kurtosis 

Ms. Donaldson,

Your observation is very useful and always interesting.  Notice that, like in common mistakes with skewness calculation, "there is no universal agreement on choosing the quartile values" so i agree with your conclusion that quartile-based skewness had to be used very carefully. The question is whether SPSS is using some of these methods for calculating the value of SKewness?

Mr. Abidemi,

Thank you for your answer and participation in this discusion. You have provided very useful information.

Mr. Hassan,

Thank you for your answer and thank you for noticing and reporting my oversight. I have corrected my mistake so my data are now accurate. I must say that your observation on Mode value is very interesting, and, partially, i share the same opinion as you, because all values are repeating only one time, so the problem is what is correct Mode value. Nevertheless, the data that i have provided are extracted from SPSS software, so SPSS showed me that my Mode value in this case is the smallest value (in this case, 2). Now, if we say that in this dataset Mode don't exist, shouldn't SPSS software provided empty field instead? 

As for Mr. Ahmad, and Mr. Hassan advice on using SPSS to calculate value of Skewness, i accept your advices with respect. I will use SPSS for my calculations of Skewness value like i always do, but, as Mr. Enzmann said, there are different formulas in use for several statistical programs.

What i want to know is what is the formula that SPSS using, and why is that formula different from manually performed formula?

Or is it some kind of correcting in SPSS formula?

Thank you for your answers so far and thank you in advance for your upcoming answers. You have helped me greatly to expand my knowledge about this problem.

If you consult the article by Joanes & Gill (1998) you will see that SPSS is using formula G1, where the skewness is the sum of z^3 divided by n/((n-1)*(n-2)); z being the z-values of the values for which you want to calculate skewness. Note that here the z-values are calculated by z = (x - mean(x))/s(x) where x are the raw values and s is the estimated population standard deviation s = sqrt(sum(x - mean(x))^2/(n-1)). Some call this skewness measure bias corrected because the formula uses the *estimated* standard deviation of the population.

Excel, for example, uses Joanes & Gill's (1998) formula g1. The difference is that instead of the *estimated* standard deviation of the population, the *sample* standard deviation is used (s_sample = sqrt(sum(x - mean(x))^2/n) when calculating the z-values, thus the divisor in the sd-formula is n instead of n-1. Consequently, the g1 formula for the kurtosis is sum(z^3)/n  (much simpler and more easy to understand).

Formulas used by SPSS can be found in the manual "IBM SPSS Statistics 20 Algorithms" (the skewness formula is on p. 350), see for example ftp://public.dhe.ibm.com/software/analytics/spss/documentation/statistics/20.0/en/client/Manuals/IBM_SPSS_Statistics_Algorithms.pdf

As a side note: If you delve more deeply into the issue of "unbiased estimates" you will find that the n-1 formula of the variance produces an unbiased estimate of the population variance, but nevertheless the square root of this variance is still *not* unbiased, see http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation

ftp://public.dhe.ibm.com/software/analytics/spss/documentation/statistics/20.0/en/client/Manuals/IBM_SPSS_Statistics_Algorithms.pdf

http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation

For better understanding of the variation, read Jarqe-Bera and Adjusted Jarque-Bera normality tests

Will do, Mr. Abidemi. Thank you for your help.

try our paper to calculate skewness

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