it is related to the nature of the distribution but the amplitude is 0 to 3

Hello Dhritikesh,

The amount of departure from population values (0, 3) which can be considered "acceptable" due to sampling error when estimating skew or kurtosis depends on:

1. Sample size (N); and

2. Type I risk level (alpha level).

For N, smaller values mean that larger departures can be observed and the discrepancy still not be declared as statistically significant (all else equal);

For smaller alpha level, larger departures can be observed and the discrepancy still not be declared as statistically significant.

SE(skew) is estimated strictly from a function of N:: SE = sqrt([6*N*(N-1)] / [(N-2)*(N+1)*(N+3)])

SE(kurtosis) is also estimated as a function of N, albeit more complicated.

Good luck with your work.

When analyzing a random sample drawn from a normal distribution, it is expected that the values of skewness (Beta-1) and kurtosis (Beta-2) may deviate from their theoretical values of 0 and 3, respectively. However, there are certain ranges within which the sample values of skewness and kurtosis can fall for the parent population to be accepted as a normal distribution. These ranges can indicate the similarity between the sample and the theoretical normal distribution.

Here are some general guidelines for the acceptable ranges of skewness and kurtosis:

It's important to note that these ranges are general guidelines, and the acceptance criteria for skewness and kurtosis can vary based on the specific context, field of study, and sample size. Additionally, the assessment of normality should not solely rely on skewness and kurtosis but should also consider graphical methods (e.g., histograms, Q-Q plots) and formal statistical tests (e.g., Shapiro-Wilk test, Anderson-Darling test) for normality.

It's recommended to consult domain-specific guidelines or consult with a statistician or expert in your field for more specific ranges or criteria for accepting the normality of a sample based on skewness and kurtosis.

Sincere thanks to my dear

Hairi Ismaili , David Morse , Ma'Mon Abu Hammad and Samy Elhadi Oussadou

for providing answers to my question.

The answers, as I have found, contain lot of basic information.

However, I am still not clear about the acceptable ranges of Beta -1 and Beta -2 calculated from a random sample.

For example suppose, the calculated value of Beta-1 based on a random sample of size 20 is found to be 0.96.

Can we conclude that the parent population of the sample is normally distributed ?

Hello again Dritikesh,

Using the formula for SE(skew), given above in my post of 29 June 2023, we have:

Observed skew: 0.96

Sample size: 20

Approximate SE = 0.5121

Approximate z = 0.96 / 0.5121 = 1.9137

Non-directional prob > z = 0.0557

In other words, your sample scenario would be considered not to be statistically significant by most folks.

Good luck with your work.

@ David Morse ,

Sincere thanks to my dear respected David Morse for providing two answers.

For more clarification, I do request to give a conclusion of the following problem:

Suppose, a random sample of size 20, drawn from a population, is:

18 , 16 , 21 , 17 , 20 , 22 , 16 , 17 , 21 , 20 ,

16 , 19 , 22 , 24 , 14 , 19 , 23 , 16 , 20 , 15

Can the probability distribution of the parent population of the sample can be regarded as normal ?

Skewness (Beta-1) and Kurtosis (Beta-2) are measures used to assess the departure from normality in a distribution. While the theoretical values for skewness and kurtosis of a normal distribution are indeed 0 and 3 respectively, in real-world scenarios, samples drawn from a normal distribution may exhibit slight deviations from these theoretical values due to sampling variability.

For a sample to be considered as drawn from a normal population, certain guidelines are often used regarding the ranges of skewness and kurtosis values. However, it's important to note that there's no strict cutoff, and different statisticians might use slightly different criteria. Here are some common guidelines:

Keep in mind that these are just rough guidelines, and the context of the analysis, as well as the size of the sample, should also be considered. Additionally, it's worth noting that statistical tests for normality, such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test, can provide more formal assessments of normality. These tests consider both skewness and kurtosis, along with other aspects of the distribution.

Here's some additional information on interpreting skewness and kurtosis in the context of assessing normality:

When interpreting skewness and kurtosis values, it's essential to consider the overall shape of the distribution and the context of the analysis. Additionally, it's worth noting that while skewness and kurtosis provide useful information about the shape of a distribution, they are just two aspects of assessing normality. It's often recommended to use multiple methods, including graphical methods and formal statistical tests, to comprehensively assess the normality of a distribution.