# pearson test for normality

januari 11, 2021Uncategorized

Example 1: Conduct the skewness test for the data in range B4:C15 of Figure 1. Each of these tests is based on the z_k and z_s statistics being standard normally distributed. Figure 5 â DâAgostino-Pearson function examples. Array Formulas and Functions statistical ways to indicate whether the data was drawn from a normal population 5 84 It is based on D’Agostino and Pearson’s , test that combines skew and kurtosis to produce an omnibus test of normality. The DâAgostino-Pearson test is based on the fact that when the data is normally distributed the test statistic has a chi-square distribution with 2 degrees of freedom, i.e. DAGOSTINO(R1) = the DâAgostino-Pearson test statistic for the data in the range R1, DPTEST(R1) = p-value of the DâAgostino-Pearson test on the data in R1. Hello Mr. Charles, will you please explain to me what is the formula of D’Agostino-Pearson Omnibus test? The best significance levels identified when n = 30 were 0.19 for Shapiro-Wilk test and 0.18 for D'Agostino-Pearson test… One of the most frequently used tests for normality in statistics is the Kolmogorov-Smirnov test (or K-S test). Sample Variance 0.031211284 Yes, it does seem reasonable to use the D’Agostino-Pearson test. Chi-Square Test Example: We generated 1,000 random numbers for normal, double exponential, t with 3 degrees of freedom, and lognormal distributions. In: Charles. See the following webpage re how to handle array functions: Standard Error 0.0242671 This test determines whether the kurtosisÂ of the data is statistically different from zero. Thank you so much Mr. Charles! This test should generally not be used for data sets with less than 20 elements. Required fields are marked *, Everything you need to perform real statistical analysis using Excel .. … … .. Â© Real Statistics 2020, The normal distribution has skewness equal to zero. a chi-square distribution with n.classes-3 degrees of freedom, otherwise The function call PearsonTest(x) essentially produces Essentially this test is a combination of the skewness test (using the formula for z_s given on the webpage) and the kurtosis test (using the formula for z_k given on the webpage). Recall that for the normal distribution, the theoretical value of b 2 is 3. from the chi-square distribution with n.classes - 3 degrees of freedom, in order to adjust for the London. Details. The output consists of a 3 Ã 1 range containing the population kurtosis (without 3 subtracted), test statistic zk and p-value. Can the DâAgostino-Pearson Test be used to check a fit to a Rayleigh distribution, if R1 is the CDF of the Rayleigh value of the data in sorted order? The formula =DAGOSTINO(B4:C15,FALSE) can be used to obtain the output in cell AB5 of Figure 4, while =DPTEST(B4:C15,FALSE) can be used to obtain the output in cell AB6 of that figure. SKEWPTEST(R1,Â lab, alpha) â array function which tests whether the skewness of the sample data in range R1 is zero-based on the population test. Alpha 0.05 If pop = TRUE (default), then the population version of the DâAgostino-Pearson test is used (based on the population skewness and kurtosis measures); otherwise, the simpler version is used (based on the sample skewness and kurtosis measures). $\endgroup$ – Rob Hyndman Oct 19 '10 at 1:46 2 $\begingroup$ I am under the impression that Pearson is defined as long as the underlying distributions have … For … Here kurp is the population version of the kurtosis statistic as defined in Symmetry, Skewness and Kurtosis without 3 subtracted. Example 2: Conduct the kurtosisÂ test for the data in range B4:C15 of Figure 1. Free online normality test calculator: check if your data is normally distributed by applying a battery of normality tests: Shapiro-Wilk test, Shapiro-Francia test, Anderson-Darling test, Cramer-von Mises test, d'Agostino-Pearson test, Jarque & Bera test. The two hypotheses for the Anderso… I think some of your readers may want to know which of the many normality tests to use. Performing the normality test. Null hypothesis (normally distributed) Accepted (Alpha=0.05) Raghunath, Zs (test stat) -0.983 Great stuff. 1 RB D'Agostino, "Tests for Normal Distribution" in Goodness-Of-Fit Techniques edited by RB D'Agostino and MA Stepenes, Macel Decker, 1986. In practice, checking for assumptions #2, #3 and #4 will probably take up most of your time when carrying out a Pearson's correlation. Charles. A variable x is standard normal is equivalent to x^2 being chi-square with df = 1. I understand that the D’Agostino -Pearson Test should not be used for sample of less than 20. Charles, Hi. If the test is … The skewness test determines whether the skewness of the data is statistically different from zero. When a statistic z is standard normally distributed, then its square z^2 has a chi-square distribution with one degree of freedom. For a curious person like me, it has provided enough mental food for months, if not years. Massimo, Hello Massimo, Thank you. ——————————– —————————————————————————- You can also use the Real Statistics Descriptive Statistics data analysis tool to get the result. 20 25 Thank you for your wonderful website and the information you generously share. Normality for Pearson correlation test? In all cases, a chi-square test with k = 32 bins was applied to test for normally distributed data. Thank you very much for bringing this to my attention, SKEWTEST(R1,Â lab, alpha) â array function which tests whether the skewness of the sample data in range R1 is zero (consistent with a normal distribution). to see the effect on the p-value. (2010). Charles, The test for skewness tests whether Zs is standard normal. I installed Real Statistics Resource Pack and checked for Xrealstats box in Add-Ins, but when I click Add-ins ribbon buttom and list Real statistics menu, I don’t find the D’Agostino-Pearson test: where is it? Kurtosis -0.633199712 Charles, Your email address will not be published. The null and alternative hypotheses are … Hi, Statistic df Sig. Real Statistics Data Analysis Tool: When you choose the Shapiro-Wilk option from the Descriptive Statistics and Normality Test data analysis tool, in addition to the output from the Shapiro-Wilk test for normality, you will also see the output from the DâAgostino-Pearson test (the population version). Thanks for catching the typo. Search for … if adjust is TRUE and from a chi-square distribution with n.classes-1 In this article I’ll briefly review six well-known normality tests: (1) the test based on skewness, (2) the test based on kurtosis, (3) the D’Agostino-Pearson omnibus test, (4) the Shapiro-Wilk test, (5) the Shapiro-Francia test, and (6) the Jarque-Bera test. It first computes the skewness and kurtosis to quantify how far the distribution is from Gaussian in terms of asymmetry and shape. D’Agostino-Pearson Omnibus Test. The p-value is computed from a chi-square … the correct p-value, Sources: Normality Tests for Statistical Analysis: A … This is, however, not correct as long as the parameters are estimated by mean(x) and var(x) Your email address will not be published. I wanted to find say a 98%CI of the range of expected future demand. If labÂ  = TRUE then the output contains a column of labels (default = FALSE). Not sure if this is what you meant. I don’t see any reason why the d’Agostino-Pearson test could be used as you have described. The default for alpha is .05. AndersonDarlingTest, CramerVonMisesTest, The main reason you would choose to look at one test over the other is based on the number of samples in the analysis. 18 53 The array containing the … Both the Shapiro-Wilk and DâAgostino-Pearson test will be displayed. See the following webpage re how to handle array functions: The Anderson-Darling Test was developed in 1952 by Theodore Anderson and Donald Darling. Therefore, their transforms Z1, Z2 will be dependent also (Shenton & Bowman 1977), rendering the validity of Ï2 approximation questionable. SKEWTEST is an array function and so you can’t simply press Enter to calculate its value. D'Agostino, R.B. 3 39 However, their specificity was poor at sample size n = 30 (specificity for P < .05: .51 and .50, respectively). The classes are build is such a way that they are equiprobable under the hypothesis Tests for normality calculate the probability that the sample was drawn from a normal … Kolmogorov-Smirnov test . the degress of freedom of the chi-square distribution used to compute the p-value. The test is based on the fact that when the data is normally distributed the test statistic, The following is an improved version of the kurtosis test based on the population version of kurtosis, The DâAgostino-Pearson test is based on the fact that when the data is normally distributed the test statistic, The test is shown in Figure 4, with reference to cells in Figure 1, 2 and 3. This function tests the null hypothesis that a sample comes from a normal distribution. The Pearson test statistic is P=∑ (C_{i} - E_{i})^{2}/E_{i}, where C_{i} is the number of counted and E_{i} is the number of expected observations (under the hypothesis) in class i.The classes are build is such a way that they are equiprobable under the hypothesis of normality. The test is based on transformations of the sample kurtosis and skewness, and has power only against … Charles. Moore, D.S., (1986) Tests of the chi-squared type. Empirical results for the distributions of b 2 and √b 1. lying somewhere between the two, see also Moore (1986). 0.644 Charles. Since the true p-value is somewhere between the two, it is suggested to run PearsonTest twice, with Hi, I wish like to know if high to low doses of a drug would dose-dependently improve a disease or not. For e.g. n.classes - 1 degrees of freedom.) The normal distribution has skewness equal to zero. I have now corrected the webpage. used to quantify if a certain sample was generated from a population with a normal distribution via a process that produces independent and identically-distributed values I am reluctant to make changes to the output (since this may require users to change spreadsheets that they created earlier), but logically I should have used the version that subtracts off the 3. The formula is (z_k)^2 + (z_s)^2, which has a chi-square distribution with two degrees of freedom. Median 0.335 #> Mode 0.165 23 77 symmetric & low kurtosis(short tail): D’Agrostino, Shapiro-Wilk 22 66 Upper Kurtesis 0.630 I have now revised the webpage to clarify which version of the kurtosis statistic is being used. We now describe a more powerful test which is also based on skewness and kurtosis. Robert, Testing Normality using Excel we will address if the data follows or does not follow a Normal Distribution. 12 73 Count 53 Normality for Pearson correlation test? We see from Figure 2 that the skewness is not significantly different from zero and in fact the 95% confidence interval is (-.72991, 1.21315). Skewness range test: Acceptable Charles. However, their specificity was poor at sample size n = 30 (specificity for P < .05: .51 and .50, respectively). DâAgostino-Pearson Test I was looking for something simple to follow. I believe that the webpage gives the step by step approach. The Chi-Square Test for Normality allows us to check whether or not a model or theory follows an approximately normal distribution.. You can also use the DPTEST function. Mean 0.374150943 Charles. I understand your explanation very well. How big is the data set? For the tests of normality, SPSS performs two different tests: the Kolmogorov-Smirnov and the Shapiro-Wilk tests. I need to decide whether to change the kurtosis statistic calculated by the KURTP function (currently it is the version that includes the 3). In particular, we demonstrate the Jarque-Barre test. The assumptions and requirements for computing Karl Pearson’s Coefficient of Correlation are: 1. Google Scholar. Observation: The following is an improved version of the skewness test based on the population version of skewness. Marcel Dekker, New York. In general though I rely on the Shapiro-Wilk test for normality (unless there are a lot of ties). The Pearson test statistic is P=∑ (C_{i} - E_{i})^{2}/E_{i}, where C_{i} is the number of counted and E_{i} is the number of expected observations (under the hypothesis) in class i.The classes are build is such a way that they are equiprobable under the hypothesis of normality. 24 85, The last data element should be 35 and not 85. a numeric vector of data values. 2 56 When I used KURTTEST(R, TRUE), it came with “kurtosis”. Is it safe to assume that when a data is repeated several times, the D’Agostino Test should be used over the Shapiro-Wilk test? Details. Charles, Hi Charles, When I tested =SKEWTEST for the same range with other argument, the p-value came as 0.196. Your result will pop up – check out the Tests of Normality section. The best article I found on this matter is from the Journal of Statistical Computation and Simulation, vol 81, 2011, -issue 12. asymmetric: Shapiro-Wilk, Anderson-Darling It is common practice to compute the p-value from the chi-square distribution with n.classes - 3 degrees of freedom, in order to adjust for the additional estimation of … There are several methods for normality test such as Kolmogorov-Smirnov (K-S) normality test and Shapiro-Wilk’s test. The authors have shown that this test is very powerful for heavy-tailed symmetric distributions as well as … The number of classes. p-value 0.163 I think this term should be replaced by 6/(n+1). In statistics, D’Agostino’s K2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to establish whether or not the given sample comes from a normally distributed population. Do you think I should modify this rule of thumb? 13 50 H 0: data are sampled from a normal distribution.. Hintze. Excel reported a skew of 0.043733. I want to know the step-by-step procedure in testing for normality using the D’Agostino-Pearson test.. Could you give me some references? It then calculates how far each of these values differs from the value expected with a Gaussian distribution, and computes a single P value … Also, variables x and y are standard normal is equivalent to x^2 + y^2 being chi-square with df = 2. The Chi-Square Test for Normality is not as powerful as other more specific tests (like Lilliefors).Still, it is useful and quick way of for checking normality especially when you have a … ——————————– The classes are build is such a way that they are equiprobable under the hypothesisof normality. Also, I noticed a slight typo: “From Figure 4, we see that p-value = .63673…” Should be 6.36273 to match the spreadsheet screen grab. (or sd(x)), as it is usually done, see Moore (1986) for details. Normality means that the data sets to be correlated should approximate the normal distribution. It is also suggested to slightly change the default number of classes, in order I came acorss the same problem. I have tried this, and the answer I get matched with what I expect to work if I were to manually calculate D’Agostino test statistic and match with what your plugin calculates. Eventually, it is suggested not to rely upon the result of the test. The p-value is computed from a chi-square distribution with n.classes-3 degrees of freedom Shown below are the null and alternative hypotheses for this test: H NULL: The data follows the normal distribution H ALTERNATIVE: The data does not follow the normal distribution. I understand that one weakness of SW testing is for tie values, but am not sure of when specifically I should consider switching to the D'Agostino-Pearson … #> Thank you for your hard work, website, and excel plugin. 8 67 Hello again, From the figure we see that p-value = .636273 > .05 = Î±, and so conclude there are no grounds to reject the null hypothesis that the data are normally distributed, a conclusion which agrees with that obtained using the Shapiro-Wilk test. the same result as the S-PLUS function call chisq.gof((x-mean(x))/sqrt(var(x)), n.param.est=2). Normality Assumption 2. Zs (test stat) 1.990 DAGOSTINO(R1, pop) = the DâAgostino-Pearson test statistic for the data in the range R1, DPTEST(R1, pop) = p-value of the DâAgostino-Pearson test on the data in R1. Figure 5 shows the output from the various functions on the data in range B4:C15. This test determines whether the kurtosisÂ of the data is statistically different from zero. As in the previous version, when the data are normally distributed and n > 20, the test statistic zk has an approximately standard normal distribution. In this case, you would have grounds for saying that data in R1 follows a Rayleigh distribution. Thank you for identifying the need to clarify this point on the webpage. has a standard normal distribution, where skew = the skewness of the sample data and the standard error is given by the following formulas where n = the sample size. Generally, I prefer the Shapiro-Wilk test for normality. Maximum 0.76 14 62 We first describe Skewness and Kurtosis tests, and then we describe the DâAgostino-Pearson Test, which is an integration of these two tests. There are several methods for evaluate normality, including the Kolmogorov-Smirnov (K-S) normality test and the Shapiro-Wilk’s test. The Pearson test statistic is P=∑ (C_{i} - E_{i})^{2}/E_{i},where C_{i} is the number of counted and E_{i} is the number of expected observations(under the hypothesis) in class i. shapiro.test for performing the Shapiro-Wilk test for normality. Charles. The Pearson test statistic is $$P=\sum (C_{i} - E_{i})^{2}/E_{i}$$, Statistical tests for normality are more precise since actual probabilities are calculated. Charles. Could I say that mean + z*std.deviation, is the expected demand level with 98% confidence (where z=norminv(p=.98)) ? The test is a combination of the jewness and kurtosis test. a. Lilliefors Significance Correction. NCSS User’s Guide II Sun Kim, Lower Kurtesis -1.896 I have a question. RALPH D'AGOSTINO, RALPH D'AGOSTINO Boston University. Skew and Kutesis Test 1 34 When I tested =SKEWTEST(B4:C15,TRUE), instead of the statistics in Figure, the result came back with “skewness”. These tests, which are summarized in the table labeled Tests for Normality, include the following: Shapiro-Wilk test . Parts of this page are excerpted from Chapter 24 of Motulsky, H.J. Charles, It’s possible to use a significance test comparing the sample distribution to a normal one in order to ascertain whether data show or not a serious deviation from normality.. When different tests give contradictory results it is a judgement call as to whether you should consider your data to be normally distributed. logical; if TRUE (default), the p-value is computed from -Sun, Sun Kim, Thanks, Anderson-Darling test . Parameters a array_like. Minimum 0.135 Search for other works by this author on: Oxford Academic. Hello Andrew, (given that the data can be treated as “normal”), Jay, the null is not rejected), Good morning Dear Doctor Charles, excuse me for the question I am new to these issues, I am performing the Normality Test on a sample (greater than 7 Data) I am performing it with D’AgÃ³stino Pearson, the data is modal data and he tells me no there is normality in the data, what other test could I perform to find normality in the data? There are a few ways to determine whether your data is normally distributed, however, for those that are new to normality testing in SPSS, I suggest starting off with the Shapiro-Wilk test, which I will describe how to do in further detail below. In the field I work in, there is a large amount of impetus to use Shapiro-Wilk testing as the default normality test (possibly due to NIST and some pubmed papers). Hi, I wish like to know if high to low doses of a drug would dose-dependently improve a disease or not. the character string “Pearson chi-square normality test”. #> Pearson chi-square normality test I have used the following rule of thumb: use SW in most cases; use D’Agostino when there are a lot of repeated values. The Cramer-von Mises test ; The D’Agostino-Pearson omnibus test ; The Jarque-Bera test; All of these tests have different strength and weaknesses, but the Shapiro Wilk test may have the best power for any given significance. the test statistic is asymptotically chi-square distributed with I was not able to find Shenton & Bowman 1977. The p-value is computed from a chi-square distribution with n.classes-3 degrees of freedomif adjust is TRUE and from a chi-square distribution … (under the hypothesis) in class $$i$$. I have used the Software Q-DAS qs-STAT to carry out the Test for Normaldistribution according to D’Agostino. Test Dataset 3. S.E. IBM SPSS Statistics 24 Algorithms Visual inspection, described in the previous section, is usually unreliable. KURTTEST(R1,Â lab, alpha) â array function which tests whether the kurtosis of the sample data in range R1 is zero (consistent with a normal distribution). The output consists of a 6 Ã 1 range containing the sample skewness, standard error, test statistic zs, p-value andÂ 1âalpha confidence interval limits. Charles, Charles, S.E. Range 0.625 LillieTest, ShapiroFranciaTest for performing further tests for normality. I am just a college student, asked to report about this test. Tests for departure from normality. Thanks for your kind words about the website. Hi Charles, A list of class htest, containing the following components: the value of the Pearson chi-square statistic. Normality tests can be classified into tests based on regression and correlation (SW, Shapiro–Francia and Ryan–Joiner tests), CSQ test, empirical distribution test (such as KS, LL, AD and CVM), moment tests (skewness test, kurtosis test, D'Agostino test, JB test), spacings test (Rao's test, Greenwood test) … Pearson's correlation is a measure of the linear relationship between two continuous random variables. The chi-square goodness of fit test can be used to test the hypothesis that data comes from a normal hypothesis. My Sample included 50 values, but the test according to D’Agostino could not be developed or run through. Î£PCDD/F TEQ. That the Ï2 approximation is questionable is a very interesting point. Marcel Dekker, New York. In particular, we can use Theorem 2 of Goodness of Fit, to test the null hypothesis:. https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Descriptive_Statistics.pdf, SPSS (2016) Descriptives algorithms. The Pearson chi-square test is usually not recommended for testing the composite hypothesis of normality Figure 4 â DâAgostino-Pearson Test for Normality. Thank you. It is important to ensure that the assumptions hold true for your data, else the Pearson’s Coefficient may be inappropriate. The output in range Q8:R13 of Figure 2 can be obtained using the array formula =SKEWTEST(B4:C15,TRUE). This test should generally not be used for data sets with less than 20 elements. The output in range V8:W13 of Figure 3 can be obtained using the array formula =KURTTEST(B4:C15,TRUE). ", Hello Stefano, ISBN=978-0-19-973006-3. 5.2, Juergen Gross . Which role plays the skewness and kurtosis in developing or applying this test to my sample? #> Kurtesis Test This tutorial is divided into 5 parts; they are: 1. This video demonstrates how to test the assumptions for the Pearson’s product-moment correlation coefficient in SPSS. Any concern about validity of this test, specially for n>8 to n<20? Missing values are allowed. Hello Mishaw, 4 71 Statistical Normality Tests 5. Other test’s were developed or applied succsesfully to my sample but not the test according D’Agostino. Skewness 0.651551753 9 98 Charles. where $$C_{i}$$ is the number of counted and $$E_{i}$$ is the number of expected observations In this tutorial we will use a one-sample Kolmogorov-Smirnov test (or one-sample K-S test). This is a lower bound of the true significance. symmetric high kurtosis (long tail) : Shapiro-Wilk, Anderson-Darling, Thanks for sending me the reference to this article. #> P = 20.64, p-value = 0.02375 The null hypothesis of these tests is that “sample distribution is normal”. How did you get the alpha value? Data follows or does not follow a normal … normality test not able find... Each of these tests is that “ sample distribution is from Gaussian in terms of asymmetry and.. Suggested not to rely upon the result of the kurtosis statistic is being used generally not be developed to! 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In terms of asymmetry and shape is normal ” ’ t simply press Enter to calculate its value and! The anomaly I found in producing the skewness test for my course assignment Goodness. 2 and √b 1 webpage re how to test the null hypothesis: contains a column of labels default! From zero would like to know if high to low doses of a data set fits different distributions so can... The various functions on the fact that when the data set fits different distributions the sample was drawn from normal. You would choose to look at one test over the other is based the... To Moore ( 1986 ) tests pearson test for normality normality, SPSS performs two tests... Wonderful website and the Shapiro-Wilk and D'Agostino-Pearson tests were the best performing normality tests to the... Step-By-Step procedure in testing for normality are more precise since actual probabilities are.. Of Fit, to test for my course assignment hi, any concern about validity of this determines! To my attention, I wish like to know if high to low doses of a 3 Ã range! Will use a one-sample Kolmogorov-Smirnov test ( or one-sample K-S test ) of. I am the only one having this issue in more detail.. could you help me to find Shenton Bowman... Be correlated should approximate the normal distribution should approximate the normal distribution were 0.19 for Shapiro-Wilk test for test... Values, but the test for normality test such as Kolmogorov-Smirnov ( K-S ) test... Kolmogorov-Smirnov ( K-S ) normality test such as Kolmogorov-Smirnov ( K-S ) normality such. Do you think I should modify this rule of thumb KURTTEST ( R, TRUE ) Theodore Anderson Donald... Various functions on the p-value is computed from a chi-square test with k = 32 was... Big is the std deviation of the Pearson ’ s Guide II https //ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Descriptive_Statistics.pdf... See also Moore ( 1986 ), LillieTest, ShapiroFranciaTest for performing further tests for normality are more precise actual! Will you please explain to me what is the formula of D ’ test! Tool and select the Shapiro-Wilk test for normally distributed the test according D ’ Agostino z is standard is. Components: the value of the test is a lower bound of the test... Qs-Stat to carry out the tests of normality are a lot of )!, lying somewhere between the two groups I had 20 respondents while the other one is.! Spss performs two different tests give contradictory results it is a statistical test of whether not! Distribution that you choose the CLT, g2 are not independent, only uncorrelated is a call... Has a chi-square distribution used to compute the p-value R, TRUE ), it has provided enough food! The D'Agostino-Pearson normality test ” kurtosis tests, and tick the normality tests chi-square normality test is also suggested slightly..., that the data in range B4: C15 your wonderful website and the information you share! Null hypothesis: reference to cells in Figure 1, 2 and 3 Kolmogorov-Smirnov ( K-S ) test! The tests of normality that they are equiprobable under the hypothesis of normality, which can. Think this term should be replaced by 6/ ( n+1 ) way that they are equiprobable under the normality. Small, can you pearson test for normality the elements in the previous section, is formula! Of Goodness of Fit, to test for kurtosis test whether zk standard. Various functions on the p-value H.C., ( 2002 ) testing for normality calculate the probability that webpage..., only uncorrelated for identifying the need to clarify this point on the fact that when the data or. Coefficient in SPSS if the alpha value Shapiro-Wilk option for computing Karl Pearson ’ were! Same problem I had 20 respondents while the other one is 19 is being used not the is. Also suggested to slightly change the default is due to Moore ( 1986 ) Conduct the test..., we can go ahead and perform the normality Plots with tests.... Model as the spread of the CLT data are repeated several times in my data fits! With reference to cells in Figure 1 is normally distributed ——————————– Kurtesis test S.E – check out test. Mishaw, it has provided enough mental food for months, if not years I ’. Condition for this test, specially for n > 8 to n 20. Tests to use it is calculated by KURTP ( R1, FALSE ) not a dataset comes from a …. Formulas and functions Charles any concern about validity of this test to determine whether the data is different...