If b = FALSE then it is assumed that n1 and n2 are sufficiently large so that the approximation described previously can be used. Indeed, the p-value is lower than our threshold of 0.05, so we reject the I agree that those followup questions are crossvalidated worthy. It seems straightforward, give it: (A) the data; (2) the distribution; and (3) the fit parameters. If method='exact', ks_2samp attempts to compute an exact p-value, that is, the probability under the null hypothesis of obtaining a test statistic value as extreme as the value computed from the data. exactly the same, some might say a two-sample Wilcoxon test is Is it possible to do this with Scipy (Python)? KS uses a max or sup norm. For example, $\mu_1 = 11/20 = 5.5$ and $\mu_2 = 12/20 = 6.0.$ Furthermore, the K-S test rejects the null hypothesis It provides a good explanation: https://en.m.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test. Where does this (supposedly) Gibson quote come from? The data is truncated at 0 and has a shape a bit like a chi-square dist. Learn more about Stack Overflow the company, and our products. The best answers are voted up and rise to the top, Not the answer you're looking for? Here, you simply fit a gamma distribution on some data, so of course, it's no surprise the test yielded a high p-value (i.e. So, heres my follow-up question. Example 1: Determine whether the two samples on the left side of Figure 1 come from the same distribution. Are you trying to show that the samples come from the same distribution? Is a PhD visitor considered as a visiting scholar? less: The null hypothesis is that F(x) >= G(x) for all x; the To learn more, see our tips on writing great answers. Further, just because two quantities are "statistically" different, it does not mean that they are "meaningfully" different. X value 1 2 3 4 5 6 Finally, note that if we use the table lookup, then we get KS2CRIT(8,7,.05) = .714 and KS2PROB(.357143,8,7) = 1 (i.e. [3] Scipy Api Reference. The KS test (as will all statistical tests) will find differences from the null hypothesis no matter how small as being "statistically significant" given a sufficiently large amount of data (recall that most of statistics was developed during a time when data was scare, so a lot of tests seem silly when you are dealing with massive amounts of To test the goodness of these fits, I test the with scipy's ks-2samp test. Any suggestions as to what tool we could do this with? According to this, if I took the lowest p_value, then I would conclude my data came from a gamma distribution even though they are all negative values? Learn more about Stack Overflow the company, and our products. For example, ks_2samp(df.loc[df.y==0,"p"], df.loc[df.y==1,"p"]) It returns KS score 0.6033 and p-value less than 0.01 which means we can reject the null hypothesis and concluding distribution of events and non . * specifically for its level to be correct, you need this assumption when the null hypothesis is true. x1 tend to be less than those in x2. It seems straightforward, give it: (A) the data; (2) the distribution; and (3) the fit parameters. against the null hypothesis. For instance it looks like the orange distribution has more observations between 0.3 and 0.4 than the green distribution. Uncategorized . Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. scipy.stats.ks_2samp. How to interpret the ks_2samp with alternative ='less' or alternative ='greater' Ask Question Asked 4 years, 6 months ago Modified 4 years, 6 months ago Viewed 150 times 1 I have two sets of data: A = df ['Users_A'].values B = df ['Users_B'].values I am using this scipy function: @whuber good point. To build the ks_norm(sample)function that evaluates the KS 1-sample test for normality, we first need to calculate the KS statistic comparing the CDF of the sample with the CDF of the normal distribution (with mean = 0 and variance = 1). famous for their good power, but with $n=1000$ observations from each sample, Is it possible to do this with Scipy (Python)? Why does using KS2TEST give me a different D-stat value than using =MAX(difference column) for the test statistic? Finite abelian groups with fewer automorphisms than a subgroup. You can find tables online for the conversion of the D statistic into a p-value if you are interested in the procedure. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. The calculations dont assume that m and n are equal. My only concern is about CASE 1, where the p-value is 0.94, and I do not know if it is a problem or not. KS2TEST(R1, R2, lab, alpha, b, iter0, iter) is an array function that outputs a column vector with the values D-stat, p-value, D-crit, n1, n2 from the two-sample KS test for the samples in ranges R1 and R2, where alpha is the significance level (default = .05) and b, iter0, and iter are as in KSINV. empirical CDFs (ECDFs) of the samples. two-sided: The null hypothesis is that the two distributions are where KINV is defined in Kolmogorov Distribution. You can have two different distributions that are equal with respect to some measure of the distribution (e.g. Am I interpreting this incorrectly? When I compare their histograms, they look like they are coming from the same distribution. When you say it's truncated at 0, can you elaborate? Is it correct to use "the" before "materials used in making buildings are"? 2. What do you recommend the best way to determine which distribution best describes the data? dosage acide sulfurique + soude; ptition assemble nationale edf rev2023.3.3.43278. How can I make a dictionary (dict) from separate lists of keys and values? Finally, we can use the following array function to perform the test. Why are trials on "Law & Order" in the New York Supreme Court? finds that the median of x2 to be larger than the median of x1, On the medium one there is enough overlap to confuse the classifier. If method='asymp', the asymptotic Kolmogorov-Smirnov distribution is The medium one (center) has a bit of an overlap, but most of the examples could be correctly classified. Making statements based on opinion; back them up with references or personal experience. I'm trying to evaluate/test how well my data fits a particular distribution. So i've got two question: Why is the P-value and KS-statistic the same? KolmogorovSmirnov test: p-value and ks-test statistic decrease as sample size increases, Finding the difference between a normally distributed random number and randn with an offset using Kolmogorov-Smirnov test and Chi-square test, Kolmogorov-Smirnov test returning a p-value of 1, Kolmogorov-Smirnov p-value and alpha value in python, Kolmogorov-Smirnov Test in Python weird result and interpretation. Why is this the case? The only problem is my results don't make any sense? 11 Jun 2022. A place where magic is studied and practiced? Is it a bug? Imagine you have two sets of readings from a sensor, and you want to know if they come from the same kind of machine. I am believing that the Normal probabilities so calculated are good approximation to the Poisson distribution. Two-sample Kolmogorov-Smirnov test with errors on data points, Interpreting scipy.stats: ks_2samp and mannwhitneyu give conflicting results, Wasserstein distance and Kolmogorov-Smirnov statistic as measures of effect size, Kolmogorov-Smirnov p-value and alpha value in python, Kolmogorov-Smirnov Test in Python weird result and interpretation. See Notes for a description of the available By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. And how to interpret these values? To do that I use the statistical function ks_2samp from scipy.stats. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. It only takes a minute to sign up. When txt = TRUE, then the output takes the form < .01, < .005, > .2 or > .1. What can a lawyer do if the client wants him to be acquitted of everything despite serious evidence? Sure, table for converting D stat to p-value: @CrossValidatedTrading: Your link to the D-stat-to-p-value table is now 404. If method='auto', an exact p-value computation is attempted if both its population shown for reference. The significance level of p value is usually set at 0.05. Using K-S test statistic, D max can I test the comparability of the above two sets of probabilities? In the figure I showed I've got 1043 entries, roughly between $-300$ and $300$. is about 1e-16. The medium classifier has a greater gap between the class CDFs, so the KS statistic is also greater. Not the answer you're looking for? Already have an account? The values in columns B and C are the frequencies of the values in column A. I tried to implement in Python the two-samples test you explained here [1] Adeodato, P. J. L., Melo, S. M. On the equivalence between Kolmogorov-Smirnov and ROC curve metrics for binary classification. I was not aware of the W-M-W test. So I conclude they are different but they clearly aren't? There is a benefit for this approach: the ROC AUC score goes from 0.5 to 1.0, while KS statistics range from 0.0 to 1.0. alternative is that F(x) > G(x) for at least one x. CASE 1: statistic=0.06956521739130435, pvalue=0.9451291140844246; CASE 2: statistic=0.07692307692307693, pvalue=0.9999007347628557; CASE 3: statistic=0.060240963855421686, pvalue=0.9984401671284038. Is it possible to create a concave light? It only takes a minute to sign up. Can airtags be tracked from an iMac desktop, with no iPhone? Does Counterspell prevent from any further spells being cast on a given turn? Your question is really about when to use the independent samples t-test and when to use the Kolmogorov-Smirnov two sample test; the fact of their implementation in scipy is entirely beside the point in relation to that issue (I'd remove that bit). from the same distribution. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 43 (1958), 469-86. For this intent we have the so-called normality tests, such as Shapiro-Wilk, Anderson-Darling or the Kolmogorov-Smirnov test. [2] Scipy Api Reference. As such, the minimum probability it can return Somewhat similar, but not exactly the same. If you assume that the probabilities that you calculated are samples, then you can use the KS2 test. E.g. From the docs scipy.stats.ks_2samp This is a two-sided test for the null hypothesis that 2 independent samples are drawn from the same continuous distribution scipy.stats.ttest_ind This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values. The p-values are wrong if the parameters are estimated. Este tutorial muestra un ejemplo de cmo utilizar cada funcin en la prctica. When txt = FALSE (default), if the p-value is less than .01 (tails = 2) or .005 (tails = 1) then the p-value is given as 0 and if the p-value is greater than .2 (tails = 2) or .1 (tails = 1) then the p-value is given as 1. the empirical distribution function of data2 at Acidity of alcohols and basicity of amines. If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? To test this we can generate three datasets based on the medium one: In all three cases, the negative class will be unchanged with all the 500 examples. In this case, probably a paired t-test is appropriate, or if the normality assumption is not met, the Wilcoxon signed-ranks test could be used. Can you please clarify the following: in KS two sample example on Figure 1, Dcrit in G15 cell uses B/C14 cells, which are not n1/n2 (they are both = 10) but total numbers of men/women used in the data (80 and 62). is the magnitude of the minimum (most negative) difference between the I tried this out and got the same result (raw data vs freq table). Astronomy & Astrophysics (A&A) is an international journal which publishes papers on all aspects of astronomy and astrophysics Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. MathJax reference. It seems to assume that the bins will be equally spaced. This is explained on this webpage. Am I interpreting the test incorrectly? The p-value returned by the k-s test has the same interpretation as other p-values. Kolmogorov-Smirnov scipy_stats.ks_2samp Distribution Comparison, We've added a "Necessary cookies only" option to the cookie consent popup. The KOLMOGOROV-SMIRNOV TWO SAMPLE TEST command automatically saves the following parameters. So the null-hypothesis for the KT test is that the distributions are the same. It returns 2 values and I find difficulties how to interpret them. How do you get out of a corner when plotting yourself into a corner. Suppose, however, that the first sample were drawn from Do you think this is the best way? The closer this number is to 0 the more likely it is that the two samples were drawn from the same distribution. statistic value as extreme as the value computed from the data. To this histogram I make my two fits (and eventually plot them, but that would be too much code). In some instances, I've seen a proportional relationship, where the D-statistic increases with the p-value. Connect and share knowledge within a single location that is structured and easy to search. In the latter case, there shouldn't be a difference at all, since the sum of two normally distributed random variables is again normally distributed. Why is there a voltage on my HDMI and coaxial cables? The two-sided exact computation computes the complementary probability hypothesis that can be selected using the alternative parameter. Nevertheless, it can be a little hard on data some times. The classifier could not separate the bad example (right), though. The KS statistic for two samples is simply the highest distance between their two CDFs, so if we measure the distance between the positive and negative class distributions, we can have another metric to evaluate classifiers. @O.rka But, if you want my opinion, using this approach isn't entirely unreasonable. The medium one got a ROC AUC of 0.908 which sounds almost perfect, but the KS score was 0.678, which reflects better the fact that the classes are not almost perfectly separable. On a side note, are there other measures of distribution that shows if they are similar? @O.rka Honestly, I think you would be better off asking these sorts of questions about your approach to model generation and evalutation at. Copyright 2008-2023, The SciPy community. The same result can be achieved using the array formula. Both ROC and KS are robust to data unbalance. It is more a matter of preference, really, so stick with what makes you comfortable. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Why are trials on "Law & Order" in the New York Supreme Court? Are there tables of wastage rates for different fruit and veg? Finally, the formulas =SUM(N4:N10) and =SUM(O4:O10) are inserted in cells N11 and O11. It is important to standardize the samples before the test, or else a normal distribution with a different mean and/or variation (such as norm_c) will fail the test. Hi Charles, Making statements based on opinion; back them up with references or personal experience. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Thanks for contributing an answer to Cross Validated! Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. edit: par | Juil 2, 2022 | mitchell wesley carlson charged | justin strauss net worth | Juil 2, 2022 | mitchell wesley carlson charged | justin strauss net worth Sign in to comment For each galaxy cluster, I have a photometric catalogue. That seems like it would be the opposite: that two curves with a greater difference (larger D-statistic), would be more significantly different (low p-value) What if my KS test statistic is very small or close to 0 but p value is also very close to zero? Even if ROC AUC is the most widespread metric for class separation, it is always useful to know both. I want to test the "goodness" of my data and it's fit to different distributions but from the output of kstest, I don't know if I can do this? [I'm using R.]. We can do that by using the OvO and the OvR strategies. When the argument b = TRUE (default) then an approximate value is used which works better for small values of n1 and n2. Is a PhD visitor considered as a visiting scholar? Statistics for applications There are several questions about it and I was told to use either the scipy.stats.kstest or scipy.stats.ks_2samp. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. scipy.stats. OP, what do you mean your two distributions? alternative is that F(x) < G(x) for at least one x. Finally, the bad classifier got an AUC Score of 0.57, which is bad (for us data lovers that know 0.5 = worst case) but doesnt sound as bad as the KS score of 0.126. The test statistic $D$ of the K-S test is the maximum vertical distance between the Therefore, for each galaxy cluster, I have two distributions that I want to compare. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 1 st sample : 0.135 0.271 0.271 0.18 0.09 0.053 scipy.stats. Anderson-Darling or Von-Mises use weighted squared differences. The scipy.stats library has a ks_1samp function that does that for us, but for learning purposes I will build a test from scratch. Computes the Kolmogorov-Smirnov statistic on 2 samples. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The result of both tests are that the KS-statistic is $0.15$, and the P-value is $0.476635$. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. KS2PROB(x, n1, n2, tails, interp, txt) = an approximate p-value for the two sample KS test for the Dn1,n2value equal to xfor samples of size n1and n2, and tails = 1 (one tail) or 2 (two tails, default) based on a linear interpolation (if interp = FALSE) or harmonic interpolation (if interp = TRUE, default) of the values in the table of critical values, using iternumber of iterations (default = 40). thanks again for your help and explanations. We carry out the analysis on the right side of Figure 1. How to follow the signal when reading the schematic? It differs from the 1-sample test in three main aspects: We need to calculate the CDF for both distributions The KS distribution uses the parameter enthat involves the number of observations in both samples. We can calculate the distance between the two datasets as the maximum distance between their features. Is a collection of years plural or singular? I just performed a KS 2 sample test on my distributions, and I obtained the following results: How can I interpret these results? I already referred the posts here and here but they are different and doesn't answer my problem. All of them measure how likely a sample is to have come from a normal distribution, with a related p-value to support this measurement. In Python, scipy.stats.kstwo just provides the ISF; computed D-crit is slightly different from yours, but maybe its due to different implementations of K-S ISF. For example I have two data sets for which the p values are 0.95 and 0.04 for the ttest(tt_equal_var=True) and the ks test, respectively. Two-Sample Test, Arkiv fiur Matematik, 3, No. How to prove that the supernatural or paranormal doesn't exist? Use MathJax to format equations. Had a read over it and it seems indeed a better fit. Hypothesis Testing: Permutation Testing Justification, How to interpret results of two-sample, one-tailed t-test in Scipy, How do you get out of a corner when plotting yourself into a corner. distribution functions of the samples. For Example 1, the formula =KS2TEST(B4:C13,,TRUE) inserted in range F21:G25 generates the output shown in Figure 2. In this case, Topological invariance of rational Pontrjagin classes for non-compact spaces. Has 90% of ice around Antarctica disappeared in less than a decade? I wouldn't call that truncated at all. Help please! This isdone by using the Real Statistics array formula =SortUnique(J4:K11) in range M4:M10 and then inserting the formula =COUNTIF(J$4:J$11,$M4) in cell N4 and highlighting the range N4:O10 followed by Ctrl-R and Ctrl-D. If R2 is omitted (the default) then R1 is treated as a frequency table (e.g. Connect and share knowledge within a single location that is structured and easy to search. Suppose we wish to test the null hypothesis that two samples were drawn Please see explanations in the Notes below. A p_value of pvalue=0.55408436218441004 is saying that the normal and gamma sampling are from the same distirbutions? Excel does not allow me to write like you showed: =KSINV(A1, B1, C1). When I apply the ks_2samp from scipy to calculate the p-value, its really small = Ks_2sampResult(statistic=0.226, pvalue=8.66144540069212e-23). Staging Ground Beta 1 Recap, and Reviewers needed for Beta 2. numpy/scipy equivalent of R ecdf(x)(x) function? The best answers are voted up and rise to the top, Not the answer you're looking for? the median). If lab = TRUE then an extra column of labels is included in the output; thus the output is a 5 2 range instead of a 1 5 range if lab = FALSE (default). Note that the alternative hypotheses describe the CDFs of the So, CASE 1 refers to the first galaxy cluster, let's say, etc. distribution, sample sizes can be different. yea, I'm still not sure which questions are better suited for either platform sometimes. (this might be a programming question). What is the point of Thrower's Bandolier? from a couple of slightly different distributions and see if the K-S two-sample test A Medium publication sharing concepts, ideas and codes. How about the first statistic in the kstest output? Also, why are you using the two-sample KS test? Movie with vikings/warriors fighting an alien that looks like a wolf with tentacles, Calculating probabilities from d6 dice pool (Degenesis rules for botches and triggers). As stated on this webpage, the critical values are c()*SQRT((m+n)/(m*n)) A priori, I expect that the KS test returns me the following result: "ehi, the two distributions come from the same parent sample". Then we can calculate the p-value with KS distribution for n = len(sample) by using the Survival Function of the KS distribution scipy.stats.kstwo.sf[3]: The samples norm_a and norm_b come from a normal distribution and are really similar. 2. If the sample sizes are very nearly equal it's pretty robust to even quite unequal variances. The ks calculated by ks_calc_2samp is because of the searchsorted () function (students who are interested can simulate the data to see this function by themselves), the Nan value will be sorted to the maximum by default, thus changing the original cumulative distribution probability of the data, resulting in the calculated ks There is an error Using Scipy's stats.kstest module for goodness-of-fit testing. THis means that there is a significant difference between the two distributions being tested. How do I make function decorators and chain them together? For example, perhaps you only care about whether the median outcome for the two groups are different. The results were the following(done in python): KstestResult(statistic=0.7433862433862434, pvalue=4.976350050850248e-102). So let's look at largish datasets Asking for help, clarification, or responding to other answers. It is most suited to I am curious that you don't seem to have considered the (Wilcoxon-)Mann-Whitney test in your comparison (scipy.stats.mannwhitneyu), which many people would tend to regard as the natural "competitor" to the t-test for suitability to similar kinds of problems. ks_2samp interpretation. I am currently working on a binary classification problem with random forests, neural networks etc. The null hypothesis is H0: both samples come from a population with the same distribution. For each photometric catalogue, I performed a SED fitting considering two different laws. How can I proceed. So I dont think it can be your explanation in brackets. 99% critical value (alpha = 0.01) for the K-S two sample test statistic. can discern that the two samples aren't from the same distribution. Charles. Recovering from a blunder I made while emailing a professor. Para realizar una prueba de Kolmogorov-Smirnov en Python, podemos usar scipy.stats.kstest () para una prueba de una muestra o scipy.stats.ks_2samp () para una prueba de dos muestras. If you're interested in saying something about them being. the test was able to reject with P-value very near $0.$. Is there a reason for that? In Python, scipy.stats.kstwo (K-S distribution for two-samples) needs N parameter to be an integer, so the value N=(n*m)/(n+m) needs to be rounded and both D-crit (value of K-S distribution Inverse Survival Function at significance level alpha) and p-value (value of K-S distribution Survival Function at D-stat) are approximations. . That isn't to say that they don't look similar, they do have roughly the same shape but shifted and squeezed perhaps (its hard to tell with the overlay, and it could be me just looking for a pattern). The test only really lets you speak of your confidence that the distributions are different, not the same, since the test is designed to find alpha, the probability of Type I error. Asking for help, clarification, or responding to other answers. suppose x1 ~ F and x2 ~ G. If F(x) > G(x) for all x, the values in Check out the Wikipedia page for the k-s test. used to compute an approximate p-value. Two-sample Kolmogorov-Smirnov Test in Python Scipy, scipy kstest not consistent over different ranges. Value from data1 or data2 corresponding with the KS statistic; MathJax reference. We generally follow Hodges treatment of Drion/Gnedenko/Korolyuk [1]. scipy.stats.kstest. with n as the number of observations on Sample 1 and m as the number of observations in Sample 2. I really appreciate any help you can provide. It should be obvious these aren't very different. The approach is to create a frequency table (range M3:O11 of Figure 4) similar to that found in range A3:C14 of Figure 1, and then use the same approach as was used in Example 1. If you dont have this situation, then I would make the bin sizes equal. I am sure I dont output the same value twice, as the included code outputs the following: (hist_cm is the cumulative list of the histogram points, plotted in the upper frames). After some research, I am honestly a little confused about how to interpret the results. It is widely used in BFSI domain. Really, the test compares the empirical CDF (ECDF) vs the CDF of you candidate distribution (which again, you derived from fitting your data to that distribution), and the test statistic is the maximum difference. I think. that the two samples came from the same distribution. Charle. ks_2samp interpretation. K-S tests aren't exactly It only takes a minute to sign up. The statistic I am not sure what you mean by testing the comparability of the above two sets of probabilities. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 2nd sample: 0.106 0.217 0.276 0.217 0.106 0.078 There is clearly visible that the fit with two gaussians is better (as it should be), but this doesn't reflect in the KS-test. (If the distribution is heavy tailed, the t-test may have low power compared to other possible tests for a location-difference.). What is the right interpretation if they have very different results? Is there a proper earth ground point in this switch box? You may as well assume that p-value = 0, which is a significant result. But in order to calculate the KS statistic we first need to calculate the CDF of each sample. If KS2TEST doesnt bin the data, how does it work ? where c() = the inverse of the Kolmogorov distribution at , which can be calculated in Excel as. On the image above the blue line represents the CDF for Sample 1 (F1(x)), and the green line is the CDF for Sample 2 (F2(x)). GitHub Closed on Jul 29, 2016 whbdupree on Jul 29, 2016 use case is not covered original statistic is more intuitive new statistic is ad hoc, but might (needs Monte Carlo check) be more accurate with only a few ties It is distribution-free. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The following options are available (default is auto): auto : use exact for small size arrays, asymp for large, exact : use exact distribution of test statistic, asymp : use asymptotic distribution of test statistic. The values of c()are also the numerators of the last entries in the Kolmogorov-Smirnov Table. Learn more about Stack Overflow the company, and our products. Is it possible to rotate a window 90 degrees if it has the same length and width? More precisly said You reject the null hypothesis that the two samples were drawn from the same distribution if the p-value is less than your significance level. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. were drawn from the standard normal, we would expect the null hypothesis
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