# The True Points of Henna

Hey guys, in our big analysis post on regular and advanced henna and knighthood requirements we estimated points earned by using henna, so for a range of 1 to 10 we estimated 5.5, for 1 to 9 we estimated 5, etc. Following this work, we have continued collecting data and analyzing the results and here we share with you the true point returns, which are, unfortunately, a lot lower than we expected (which means that it will cost more or take more time to increase the stats).

We have prepared the analysis both for regular henna and advanced henna, which we share with you below starting with regular henna:

## Results of Tattoo Upgrade with Regular Henna

Here Cosmos collected the data for regular henna tattoo cleansing (115 cleanses) writing down each time how many points increase was achieved and comparing it with our expected return analysis and has prepared the below results chart from all the findings:

- Overall average received was
**4.2 versus the expected 5.0; this is 16% less**on top of the already difficult task - Consequently 11,600 regular henna was used compared to the expected 10,370
- 10 points increase was never received in 115 cleanses, but 1 point increase was received 16 times (13.9% or roughly 1 in 10)

Here are the increases received: 2, 6, 5, 6, 6, 9, 4, 2, 4, 5, 1, 4, 3, 7, 4, 3, 9, 6, 4, 3, 2, 6, 2, 9, 9, 8, 7, 9, 7, 7, 7, 4, 1, 2, 1, 4, 4, 1, 6, 2, 1, 1, 3, 3, 4, 7, 2, 6, 2, 5, 4, 4, 5, 6, 1, 3, 3, 4, 8, 6, 7, 2, 2, 8, 1, 3, 1, 1, 8, 3, 3, 4, 4, 1, 6, 6, 8, 1, 1, 3, 4, 7, 8, 6, 7, 2, 6, 2, 7, 4, 2, 3, 2, 5, 4, 2, 7, 5, 7, 2, 6, 3, 1, 2, 1, 3, 2, 5, 1, 6, 7, 3, 2, 4, 2.

## Results of Tattoo Upgrade with Advanced Henna

Advanced henna analysis was done by Jesse Kabam S67 – thanks! Here is a summary of the data for advanced henna:

There are a total of 199 cleanses. The data seems similar to regular henna results. For every range the determined mean is less than the expected. I calculated the 95% confidence interval. For each row that says FALSE there is only a 5% chance the true mean is within that interval. That suggests that **each row that says FALSE the stat obtained is not evenly distributed** over the range and not random.

Range |
Expected | Determined | Std.Dev. | N | 95%CI | Within Confidence Interval |

3000-3249 |
24.5 | 23.2 | 2.192201 | 30 | 0.784455 | FALSE |

3250-3499 |
24 | 23.4 | 1.703225 | 56 | 0.446094 | FALSE |

3500-3749 | 23.5 | 23.0 | 1.400577 | 55 | 0.370147 |
FALSE |

3750-3999 | 23 | 22.5 | 2.136863 | 46 | 0.617513 |
TRUE |

4000-4249 | 22.5 | 21.3 | 0.492366 | 12 | 0.278577 |
FALSE |

thanks Cosmos great analisis as always! i have to ask you please can you update the mount table reference with new mounts? thanks again for the hard work,

thanks buddy

yes i know we are behind in updating that reference, we must do it, need to find time

it is a good try to describe henna –

BUT your examples are statistically too

small to make good descriptions; so the numbers

you have derived are not meaningfull.

i assume you used (for adv henna) the normal-destribution (or students’ t) this gives you only a good explanation if your sample size is at least 100. the bigger the sample size the better the explanation.

your total sample is 199. but your study is about sample sizes of 56 and smaller… so your statistic is not significant… SRY

another problem is your description of a confidence-intervall: a CI has an upper and a lower bound and the boundries are right with prob 95%. and if you made a test if your calculated mean is in that intervall you have also not enough datas to ensure statistical significance.

so it is better to show your datas and make a diagramm of relative frequencies (eg. histogram) or other statistics, which don’t use assumptions of distributions or complicated statistics.

and another addition: maybe for this application, you should also proof a 90% CI (if you would have a bigger sample size in each category)…

Hi donqui, I asked Jesse to reply to your comment, thanks

Thank you for the comments! An increase of sample size will decrease the confidence interval by 1/square root of N. Regardless of N, the 95% confidence interval will be the 95% confidence interval – it is neither more or less significant with an increase (or decrease) of N. Also, the column labeled 95% CI is the one-sided range of the 95% confidence interval from the determined mean. The range of the 95% confidence interval is the mean +/- the one-sided range. For four of the five tattoo stat ranges for advanced henna the expected mean is not within the 95% confidence interval.

Of course confidence interval is not a measure of randomness; it is a measure of the confidence that the given range includes the true mean. However, considering this and the observation (for both regular henna and advanced henna) each determined mean has a negative bias from the expected mean strongly suggests the distribution is not evenly distributed and not random.

hey, yes, but it says nothing, if the expected mean is not element of the ci – you cannot reject that the expected one is the real mean. for this you would have to test many times and bigger samples.

what you did is like throwing a dice 6 times and there were no 6 and you say the dice is unfair…

the law of the large numbers say, the sample mean converges to the real mean the bigger the sample is.

so maybe a bigger samplesize and a hypothesis-testing would be more meaningfull.

that is the point i made.

one addition: if you use a normalization (you substract your mean and devide by your sample variance)- it does not hold for small sample sizes. if you roll a dice twice you can not derive a ci and interprete it. this holds only for bigger sample sizes (i remember my prof said 100 or larger). the ci is an approximation and does not work for too small samples.

and from wiki:

A 95% confidence interval does not mean that for a given realised interval calculated from sample data there is a 95% probability the population parameter lies within the interval, nor that there is a 95% probability that the interval covers the population parameter.[11] Once an experiment is done and an interval calculated, this interval either covers the parameter value or it does not, it is no longer a matter of probability. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval.

Out of curiosity, were you able to calculate the Degrees of Freedom (DoF)? And were you able to plot a histogram of the results?

I would also be really curious to see if there is some sort of persistence or ‘memory’ to the results, ie: does the next henna roll depend partly on the previous henna roll?

It would also be interesting to see if there would be a different average value if you used a different tattoo each time you did a henna upgrade, that might take a bit of work, but it might be an interesting experiment.

As far as the Confidence Interval (CI), Really, the DoF helps determine the ‘width’ of the CI, So fewer DoF simply indicates a wider range of values where we cannot reject the hypothesis.

In this case, the hypothesis is that the value 4.5 is the mean of the henna tattoo upgrade. Since it falls outside the the CI, we can reject the hypothesis. Mind you, that doesn’t absolutely verify that 4.5 isn’t the mean, it just says that 4.5 is very unlikely to be the mean given that distribution.

but the less datas you use the less meaningfull is that. think on my example, you throw a dice only 3 times or 6 times – you may have luck and throw 3 times a six, one times 5,4 and 3 – your ci would not include 3.5 the trueth mean of a fair dice. so you can construct many other examples, if you throw less enough and either have luck or do not have luck your result and ci will be biased.

but if you throw the dice 30, 50 or 100 times your ci would be a good measure.

nice statistical calculation…std dev & 95% CI…bring back memories T.T

thank you

How about we put it in layman’s terms = if you’re not a casher how long will it take to make KH? My estimate is slim to never…..LOL How about it Cosmos? BTW you do seen to put a lot of effort into this hope it doesn’t affect your job performance….just kidding well done C!

hi buddy, we have already answered that in very clear blue bold letters in our reference post: //wartune.dolygames.com/complete-henna-analysis-and-reaching-knighthood-requirements/

quoting:

“To reach knighthood with regular henna we estimate that you will need around 339,694 regular henna (28.3 months or ~2 years).”

However, as the issue is more complex one should read the full conclusion at the bottom of the post and ideally the complete article(s) / analysis we published for henna.

Don’t lose hope, but we definitely have to play smarter and take some actions (i will make another post today i think about specific action to take)

Cheers,

Cosmos

I think you may have revealed an interesting fact. Either the top value of the range is excessively rare, or they messed up the math and you can’t ever get the top value. I just tried about 50 regular henna refreshes, and didn’t get the top once.

And your chart shows the same, top values are one less than expected. It otherwise looks like a pretty even distribution, and the advanced henna chart is very close to the expected value if you simply cut off the top.

We need to survey people to find out if anyone ever gets the maximum return from a henna or advanced henna refresh. I wasn’t keeping track before, but I will from now on.

Thank you for sharing your opinion MrFancyPants. Yes, why not, maybe they made a mistake and 10 does not show up or maybe it is there only for “marketing” purpose, hopefully more people give feedback on this.

Also feel free to email me if you have data or anything, if it is made with love I will be happy to publish it 🙂

Cheers,

Cosmos

That’s a very good observation. The same holds true for advanced henna data, none of the cleanses obtained the maximum stat increase.

To update:

I submitted a ticket on this, and R2 support says that they have tested it, and the max value will occur, but very rarely. They said they forwarded it to the developers, but in any case, what we thought is basically correct – the max value is effectively lower than advertised 🙁

Holy math geeks batman! Great job guys!

After playing wartune for 4 years, I noticed that luck in this game could be related to the hour of the game (e.g. I always avoid to open important chests around guild battles). Not sure if it’s a rule, but quite sure luck isn’t constant!!

When did you collect these data?? All in once? Maybe it was a “bad timing” to do that and you got worse results than average?

Regards,

Giulio

we would be happy to incorporate any concepts into analysis if there is data / evidence to support the concepts and at this time there is no evidence that opening chests at different times of the day makes any difference; if you have such proof send it over and we will be happy to have a look and publish if possible

omg – that would be very hard to proof if it is a random result or dependent on different times – so you would need many chests on different times all documented… then calculate the average 10 times or more on each time and test the hypothesis that the average is independent on the time, if you can reject this, everithing is ok, if not you would have had much work for nothing…

i do not mind other proofs for that – and i would say it is very interisting but a proof is too hard but if the average is totally different to others it would be a good intuition…

maybe i am too much influenced by my studies… tell me if i should not be such critical 😉

in my opinion is not so hard: probably you can’t prove it with event chests, since not enough, but could be doable with some other kind of chests that you can save for more time (spire, marriage, tok, etc.)

I still don’t have any proof, but a real evidence was first realized by myself with astros: much more yellow/red with same gold in the early morning, than afternoon!

Hope could help someone or someone will help us with it 🙂

Best,

Giulio

cause henna upgrades are diff enough would be fair just to lower the effekt stead of having a “random” result, cause like always when it comes to “random” r2 cheats.

– some examples ? cloud adventures: rolling dices should be average of 3.5 (tested and got a 2.6 *surprise*)

– chests from farm (10,20,30) average should be 20, after doing tables from beginning till last month got to an average of 14,78 each chest wich is signifficant loss against the statistic average it should be)

again: how often have you thrown the dice to calc that average? keep care on small samples! i think for an average should 30-50 times should be enough… but if you want to prove if r2 cheats us with averages 100 times and a hypothesis-test is the easiest way i know to do that – and then it is significant with a probability of 90 or 95 percents.

The real question is whether the values are normally distributed. Given even the evidence of what appears to be skewness or something like it above, it seems that the values are not normally distributed but have a long right tail.

yes, could be (for example) xi-destributed…

sry i meant chi-square-distributet… it is late now.. sry

after gettin tons of free adv henna (yes its irony) was llokin for some info about henna and the chance to get the max reward when using. and surprise: 199 tries no 10 ? ohhh nooooooo, r2/kabam cheats again on this one. pure statistic will bring a minimum of 1 times a “10” in all these tries (sure a 1% chance to get something can also get 1 a success of 10 and another the result of “none”, but in average there should be all numbers.

and just to say it : theres fault in average calculation:

average is 5.5 not 5 what shows that theres even more cheating than calculated

(why 5.5 ? 1+2+3+4+5+6+7+8+9+10 = 55 so average is 55/10=5.5)