The thanks is all mine for you choosing to spend your precious time on my videos. I am very glad you find them helpful. But the true superstars are people like yourself making the effort to learn and improve. Thank you again and best of luck!
Yes you are correct. I worked in higher ed for many years which is why I am sympathetic with professors. There isn't enough time in a 3-credit class to really teach deep understanding of many subjects. eLearning is not a a replacement, but a good complement to classroom learning. Thanks for the subscription and share with classmates who are also having challenges. Take care!
It's been a year since I've started my e-learning of statistics. This is by far the best lectures I've come across. It's just amazing how every question I had about statistics has been addressed in these lectures. Thank you so much! I'm just gonna go ahead and say it. I love you.
Do you know why high school teachers around the world are not allowed to tell ALL their students that they are brilliant and smart? It is called ego massaging and it is manipulative and abusive. Anyone with a MEd who is posting videos for people he has NEVER met, should know that obvious fact. But apparently it works when uploading TH-cam videos because people decide that they now …. Love you?! Get help.
Hi Brandon - first of all, your videos are SUPERB. By far the most helpful statistics videos on TH-cam. So thank you for your generosity and help. Can I (and no doubt thousands of others) ask you to make a video all about "degrees of freedom"? It comes up all the time and the books don't make it clear at all. Why do we use it? How does it work? And so on
Adam Abdulla Hello! Thank you your comments and for watching. Degrees of freedom, the bane of statistics. In a nutshell, it is the number of observations minus the number of estimated parameters needed to compute a statistic from a sample. So for something like a t-test or variance, we use the sample mean which is a parameter estimate of the population mean. So df is N-1, with the 1 being the number of parameter estimates we used; in this case the sample mean.
Adam Abdulla Hi Adam. I have not done a video specifically on degrees of freedom. But based on popular demand I may have to! I am always afraid of getting too far "into the weeds" and confusing people more than they already may be.
***** I think that would be really appreciated by everyone, especially as you - if anybody - can make it comprehensible. The trouble is all the books say: "It's what we use when we're estimating a parameter" or they talk about how many values are "free to vary" (and give an example like when you you know a mean and two out of three scores and can therefore work out what the third score must be), but they don't really RELATE this to the case at hand. And just because we're using the statistic to estimate a parameter why does that mean we divide by N-1 (as opposed to N-2, for example, or as opposed to just multiplying the number by some constant to make it bigger)?
Hi Brandon: I am a 54 year old professional, and I've been using your videos to update my quantitative skills. Your videos are extremely helpful, and you have a way of making things very clear. Thanks.
OMG, I've been watching your vids for the last month. BE ENCOURAGED!!! All the fluff, distractions, and redundancy makes learning new concepts so much easier. The presentations flow and builds upon each other. Thank you so much. I am paying a college professor to teach in one week what I can learn in 1/2 hr to an hour.
your videos in tandem with the chatgpt's "sky" voice have given me an insanely better grasp on this critical concept just in time for my final. thank you so much
These are great video's. I am in an online class and a non-stat major. The various videos have been a tremendous help as I am unable to participate in class lectures. Your slow and detailed explanation helps a lot.
Teaching is an art.. not everyone can do it.. you're born with it!!! Im a big fan of your videos , have great respect for style of detailed explanations in them.. I owe my grades in this subject entirely to you.. Ill let you know once the grades are out.. Thanks a lot!!!
It will be ok! There is 100% confidence that if you do NOT breathe, stats will not go well! :) We all want you around for a long while so stay centered. All the best, B.
You are awesome, I wish you were teaching us this semester, we are all so confused! Thank you for shedding light on the why's of what we are doing as that seems to be the piece that is lacking and leading to a lot of confusion.
Thanks so much for posting these videos!!! I am at a UC and have found myself relying mostly on youtube videos to learn the stuff for my classes. I cant imagine what I would have done without such well thought out, organized, & helpful videos such as the ones you make!!! Thank you!!!
Wow! Your videos are amazing! Before I tought that statistics was something I would newer understand. You are giving me hope! Now I think with some time invested, it might be possible to grasp parts of this dense statistical djungle. Tank you for taking time explaining!
These videos got me through my biostatistics class! I would love to have outlines available to make notes and have the formulas handy. Thanks for all the great help!
Hi Brad, Highly appreciate your work. The explanation flow of concepts is very good and very easy to understand if someone has just started. It would be really helpful if you can share the slides, makes easier to review and understand these concepts.
Like so many others have done I'd like to thank you for your great instructional videos! Some helped me quite a bit, but others left me more confused. The text we're using in my Activity based online college stats class never mentions alpha, but you do it regularly. You go to t-scores, yet there is nothing in my text that uses it. It's all based on z-scores, PHat, etc., it's enough to drive me nuts at times. If I make it out of this class with any credit at all it'll be a miracle.
Hi Brandon, firstly thank you so much for making such amazing content, one suggestion for us students definitely understanding the concept is super important, but I would really like if you include more solvable examples from exam perspective
Most of my previous stats professors believe that "CI is a range of values that we are confident contains the population parameter. E.g. when we don’t know the value of the parameter, such as average score (µ), we can estimate it by constructing an interval centered on the observed statistic (e.g., average score from the sample)." Wow, this blew my mind!
@kklsh88 Thank you for your comment. I do regret that the video did not meet your needs. In this case I am not sure how I could both slow parts down and make the video shorter. However all feedback is welcome and I appreciate yours as well. Hopefully you were able to find a resource that met your needs better. All the best in your work and studies! - B
Hello! Thank you for your comment! I have videos with examples for both cases. Here are some key words/phrases..."degrees of freedom" and "sample standard deviation" denoted as "s" always point to the t-distribution. The z-distribution is only used when sigma, the population standard deviation, is known or given. Let me know if you have any more questions! All the best, B.
Hello! Kudos to you on all of your hard work. In this example, the sample size (n) is 10. Therefore the degrees of freedom is n-1. So 10-1 = 9. The row in the t-table is one below the sample size. Did I understand your question correctly? If not please let me know. All the best! - B
(Oh my god that was fast) .. Thanks!!! I just subscribed so I will definitely keep in touch after each class... by the way..... this is a fun class, my professor just teaches it too fast as if everyone took this before and or is good/great at math..
1) Yes. The short answer is that degrees of freedom come into play when sigma is unknown. Think of it as a "correction" to having to estimate sigma. We would then use the t-distribution. 2) Proportion and probability only have the same meaning under a certain context. Most of the time not.
Hi. Your videos have been of great assistance in my preparation for my exams. I however seem to miss something: In the show-rooming example you indicate that the mean of the sample is 3661.5 seconds. How did you calculate this? I am writing tomorrow and I am probably just tired now an missing the obvious. Regards from South Africa.
Hi Susan! It is just a "given" in the problem. :) Good luck on your exams and remember to rest, eat well, and trust your talents. I know you will do great! - B
Do you have question example and solution for this lesson so i can understand it more😁 i kinda confused when you explain the “sample” part that make this so different… great video thanks🙏
Great video, found myself taking taking lots of notes. however, I felt there was 5 mins extra missing. I would like to have seen an example of how to use the formula. I'm now left surfing TH-cam for someone who shows an example of usage.
Hi Karen! Thanks for watching. Were you able to watch Part 2 of this video? I do the mathy stuff and examples after explaining the basic concepts. Statistics 101: Confidence Intervals, Population Deviation Unknown - Part 2 Let me know if you still have any questions after watching videos. All the best, B.
OH yeah, one more thing, do you think that you could author a text? I can only begin to imagine what a difference it would make if you were explaining things in the text book! It would be the difference between pass and fail. Thank you Dr. Foltz!
Hey Brandon, Thanks for the video. Can you shed more idea about the intuition behind why the t-distribution curve is fatter in the tails and shorter in middle? I am not able to understand why is that happening?
Great video and a clearer understanding of the this subject... Question do you know of a website that can provide examples or... can you make a video with examples... so when I'm reading I'm able to tell which is known and unknown. I have a business statistics book and the examples seem to read the same. I'm trying to find key words to help separate the two (z and t score formula)... Thanks!!!
Hey Brandon - Thank you for the video, it was vary informative and basic which really helped me to understand the concepts. I however have a problem with the fact you give to much time to basic manipulation in calculations and also reading out the same values over and over when I can see the numbers on the screen. This makes your videos a bit longer. Overall, everything is well presented and tailored for the understanding of the viewer. Thank you.
I'd like to learn from all of your videos especially Statistics 101, can I know the sequence of these videos? which video do I need to watch first?? Hope you can shed light on my concern. Thanks!
thanks for the video, it was very informative. I have a question, though: what is the real importance of degree of freedom? I understand dof is governed by sample size and as sample size increases the standard error decreases and the sample mean moves closer to the actual population mean. But what is the motivation behind calling the figure (n-1) as 'degree of freedom'? How does it push t-values more toward Z-values? I am sorry if these r too many questions all at once.
Great video. A couple of questions. Is the degree of freedom something only used when sigma is unknown? Does proportion and probability share the same meaning?
Brandon, your videos are very helpful so I appreciate that thank you. if you could clarify, what if x barred is .43 sample standard deviation is .07 and the sample size is 500. I am confused on how to calculate when n is greater than 30, I have an assignment due tomorrow if you could get back to me asap i would appreciate it
Amazingly amazing video. Now, I don't know why I am paying tuition to someone who can't explain (such a reasonable) concept reasonably. Brandon for the President!!
you talk about previous video.... Maybe put a link that brings you to the previous video, like the one you have for part 2. which would be helpful as i am trying to figure out which one was previous
Hello Brandon, A big thanks for all these great video series, watching them during corona lockdown, best investment of my time. My doubt is, my t-test gives a significant p-value but the confidence interval does not contain my sample mean. What am I doing wrong here ?
Good day Sir, I have a question but don't really know what to ask for. Here is my problem. I have large seta of data that are numbers that I get from substracting the price of a stock compared to a previous value. So I have more than a hundred values for the CHANGE of the stock price. What I want to know is what mag it'd of change is the most common. What magnitude of change is the average? So if I knew the stick price I could statistically predict what the "jump in price" is going to be. (Of course not knowing if it will go up or down) Thank you very much.
So I take three samples, I get standard deviations of three samples say s1, s2 and s3. Which one do I use as "s" to estimate population standard deviation i.e. s/Sqrt(3)?
Hi Brandon - Is it possible to have a Confidence Interval of a p-value. If I am expected to make decisions based on this value surely we need to know how reproducible it is given that samples taken from unknown populations are likely to bounce around. Books I have read seem to conflate Confidence Intervals and p-values without addressing this. We have confidence Intervals for most statistics we work with why not p values. Regards
A recent article in the daily newspaper indicated that the mean selling price of the homes in capital city is 357 dollars. Can we conclude that the mean selling price in the capital city is not different from 357 doller .use the 0.05 significance level where n= 52 pleas solve the math
The thanks is all mine for you choosing to spend your precious time on my videos. I am very glad you find them helpful. But the true superstars are people like yourself making the effort to learn and improve. Thank you again and best of luck!
Yes you are correct. I worked in higher ed for many years which is why I am sympathetic with professors. There isn't enough time in a 3-credit class to really teach deep understanding of many subjects. eLearning is not a a replacement, but a good complement to classroom learning. Thanks for the subscription and share with classmates who are also having challenges. Take care!
It's been a year since I've started my e-learning of statistics. This is by far the best lectures I've come across. It's just amazing how every question I had about statistics has been addressed in these lectures. Thank you so much! I'm just gonna go ahead and say it. I love you.
Do you know why high school teachers around the world are not allowed to tell ALL their students that they are brilliant and smart? It is called ego massaging and it is manipulative and abusive. Anyone with a MEd who is posting videos for people he has NEVER met, should know that obvious fact. But apparently it works when uploading TH-cam videos because people decide that they now …. Love you?! Get help.
@@lawjef not interested
Oh thank you so much! But all the amazing-ness goes to you and all the other viewers for taking the time and making the effort to learn. All the best!
Hi Brandon - first of all, your videos are SUPERB. By far the most helpful statistics videos on TH-cam. So thank you for your generosity and help. Can I (and no doubt thousands of others) ask you to make a video all about "degrees of freedom"? It comes up all the time and the books don't make it clear at all. Why do we use it? How does it work? And so on
Adam Abdulla Hello! Thank you your comments and for watching. Degrees of freedom, the bane of statistics. In a nutshell, it is the number of observations minus the number of estimated parameters needed to compute a statistic from a sample. So for something like a t-test or variance, we use the sample mean which is a parameter estimate of the population mean. So df is N-1, with the 1 being the number of parameter estimates we used; in this case the sample mean.
Thanks Brandon. I must confess I'm still confused! Do you have a video on it?
Adam Abdulla Hi Adam. I have not done a video specifically on degrees of freedom. But based on popular demand I may have to! I am always afraid of getting too far "into the weeds" and confusing people more than they already may be.
***** I think that would be really appreciated by everyone, especially as you - if anybody - can make it comprehensible. The trouble is all the books say: "It's what we use when we're estimating a parameter" or they talk about how many values are "free to vary" (and give an example like when you you know a mean and two out of three scores and can therefore work out what the third score must be), but they don't really RELATE this to the case at hand. And just because we're using the statistic to estimate a parameter why does that mean we divide by N-1 (as opposed to N-2, for example, or as opposed to just multiplying the number by some constant to make it bigger)?
Hi Brandon: I am a 54 year old professional, and I've been using your videos to update my quantitative skills. Your videos are extremely helpful, and you have a way of making things very clear. Thanks.
OMG, I've been watching your vids for the last month. BE ENCOURAGED!!! All the fluff, distractions, and redundancy makes learning new concepts so much easier. The presentations flow and builds upon each other. Thank you so much. I am paying a college professor to teach in one week what I can learn in 1/2 hr to an hour.
Absolutely love the positive pep talk in the beginning. Thank you!
I was just beginning to teach this after 8 years of teaching leave; thank you so much. Found it very very useful
your videos in tandem with the chatgpt's "sky" voice have given me an insanely better grasp on this critical concept just in time for my final. thank you so much
These are great video's. I am in an online class and a non-stat major. The various videos have been a tremendous help as I am unable to participate in class lectures. Your slow and detailed explanation helps a lot.
Thank you Brandon. You are a true gentleman with so much of humility inspite of sharing such a wealth of knowledge..
Teaching is an art.. not everyone can do it.. you're born with it!!! Im a big fan of your videos , have great respect for style of detailed explanations in them.. I owe my grades in this subject entirely to you.. Ill let you know once the grades are out.. Thanks a lot!!!
I cannot express how much this video has helped me. Thank you so much. This is brilliant!
It will be ok! There is 100% confidence that if you do NOT breathe, stats will not go well! :) We all want you around for a long while so stay centered. All the best, B.
You are awesome, I wish you were teaching us this semester, we are all so confused! Thank you for shedding light on the why's of what we are doing as that seems to be the piece that is lacking and leading to a lot of confusion.
Thanks so much for posting these videos!!! I am at a UC and have found myself relying mostly on youtube videos to learn the stuff for my classes. I cant imagine what I would have done without such well thought out, organized, & helpful videos such as the ones you make!!! Thank you!!!
Wow! Your videos are amazing! Before I tought that statistics was something I would newer understand. You are giving me hope! Now I think with some time invested, it might be possible to grasp parts of this dense statistical djungle. Tank you for taking time explaining!
I am so grateful for your video's! There is no way I would have gotten through stats without you!
The positive intro is very comforting. Thank you, very thoughtful. Great teacher!
These videos got me through my biostatistics class! I would love to have outlines available to make notes and have the formulas handy. Thanks for all the great help!
By far the best video I've seen on the many parts to statistics. Keep them coming.
THANK YOU
Hi Brad, Highly appreciate your work. The explanation flow of concepts is very good and very easy to understand if someone has just started. It would be really helpful if you can share the slides, makes easier to review and understand these concepts.
Like so many others have done I'd like to thank you for your great instructional videos! Some helped me quite a bit, but others left me more confused. The text we're using in my Activity based online college stats class never mentions alpha, but you do it regularly. You go to t-scores, yet there is nothing in my text that uses it. It's all based on z-scores, PHat, etc., it's enough to drive me nuts at times. If I make it out of this class with any credit at all it'll be a miracle.
Very good Brandon, thank you for putting the time and effort into making these videos.
You are simply AMAZING
Thank you so much for uploading these!
THANK YOU THANK YOU THANK YOU THANK YOU!!!!!!!!!!!!!
you are an amazing teacher
Hi Brandon, firstly thank you so much for making such amazing content, one suggestion for us students definitely understanding the concept is super important, but I would really like if you include more solvable examples from exam perspective
Most of my previous stats professors believe that "CI is a range of values that we are confident contains the population parameter. E.g. when we don’t know the value of the parameter, such as average score (µ), we can estimate it by constructing an interval centered on the observed statistic (e.g., average score from the sample)." Wow, this blew my mind!
Brandon, God bless you. Your video has being a big help to me, and I appreciate you man.
@kklsh88 Thank you for your comment. I do regret that the video did not meet your needs. In this case I am not sure how I could both slow parts down and make the video shorter. However all feedback is welcome and I appreciate yours as well. Hopefully you were able to find a resource that met your needs better. All the best in your work and studies! - B
Thank you Brandon for making statistics so understandable ❤
This is so good. Cant believe its all free
P(length of confusion interval = 0) = 1. Great video! Please, make more! (Multivariable statistics.)
Hello! Thank you for your comment! I have videos with examples for both cases. Here are some key words/phrases..."degrees of freedom" and "sample standard deviation" denoted as "s" always point to the t-distribution. The z-distribution is only used when sigma, the population standard deviation, is known or given. Let me know if you have any more questions! All the best, B.
Hello! Kudos to you on all of your hard work. In this example, the sample size (n) is 10. Therefore the degrees of freedom is n-1. So 10-1 = 9. The row in the t-table is one below the sample size. Did I understand your question correctly? If not please let me know. All the best! - B
(Oh my god that was fast) .. Thanks!!! I just subscribed so I will definitely keep in touch after each class...
by the way..... this is a fun class, my professor just teaches it too fast as if everyone took this before and or is good/great at math..
Very helpful videos Brandon. Thank you for your good work.
1) Yes. The short answer is that degrees of freedom come into play when sigma is unknown. Think of it as a "correction" to having to estimate sigma. We would then use the t-distribution. 2) Proportion and probability only have the same meaning under a certain context. Most of the time not.
Thank you very much. Best explanation!
your videos have helped so much thanks
Hi. Your videos have been of great assistance in my preparation for my exams. I however seem to miss something: In the show-rooming example you indicate that the mean of the sample is 3661.5 seconds. How did you calculate this? I am writing tomorrow and I am probably just tired now an missing the obvious. Regards from South Africa.
Hi Susan! It is just a "given" in the problem. :) Good luck on your exams and remember to rest, eat well, and trust your talents. I know you will do great! - B
Again i have to send tons of thanks for the vids. They r awesome
I AM A STUDENT IN AKIM STATE UNIVERSITY COLLEGE AND IT HAS HELP DURING MY EXAMS, THANKS SO MUCH KEEP SIR
This dude is better at stopping my panic attacks than any family member.
You are a great teacher.
Awesome video! Thank you!
Do you have question example and solution for this lesson so i can understand it more😁 i kinda confused when you explain the “sample” part that make this so different… great video thanks🙏
You are amazing!~ please, keep up the good work!
Great video, found myself taking taking lots of notes. however, I felt there was 5 mins extra missing. I would like to have seen an example of how to use the formula. I'm now left surfing TH-cam for someone who shows an example of usage.
Hi Karen! Thanks for watching. Were you able to watch Part 2 of this video? I do the mathy stuff and examples after explaining the basic concepts.
Statistics 101: Confidence Intervals, Population Deviation Unknown - Part 2
Let me know if you still have any questions after watching videos. All the best, B.
Wow!! Because of you Statistics is interesting
OH yeah, one more thing, do you think that you could author a text? I can only begin to imagine what a difference it would make if you were explaining things in the text book! It would be the difference between pass and fail. Thank you Dr. Foltz!
U r a saviour❤
Hey Brandon,
Thanks for the video.
Can you shed more idea about the intuition behind why the t-distribution curve is fatter in the tails and shorter in middle? I am not able to understand why is that happening?
Great video and a clearer understanding of the this subject... Question do you know of a website that can provide examples or... can you make a video with examples... so when I'm reading I'm able to tell which is known and unknown. I have a business statistics book and the examples seem to read the same. I'm trying to find key words to help separate the two (z and t score formula)...
Thanks!!!
Hey Brandon - Thank you for the video, it was vary informative and basic which really helped me to understand the concepts. I however have a problem with the fact you give to much time to basic manipulation in calculations and also reading out the same values over and over when I can see the numbers on the screen. This makes your videos a bit longer. Overall, everything is well presented and tailored for the understanding of the viewer. Thank you.
Thank you so much. I can breathe again!
You are a life saver.
I'd like to learn from all of your videos especially Statistics 101, can I know the sequence of these videos? which video do I need to watch first?? Hope you can shed light on my concern. Thanks!
Hi these videos are great, but can provide a link to the previous videos please? Thanks!
Hello! Here is a link to all the playlists. Everything is in order by topic and sequence. th-cam.com/users/BCFoltzvideos?view=1&flow=grid
Thankyou !
thanks for the video, it was very informative. I have a question, though:
what is the real importance of degree of freedom?
I understand dof is governed by sample size and as sample size increases the standard error decreases and the sample mean moves closer to the actual population mean. But what is the motivation behind calling the figure (n-1) as 'degree of freedom'? How does it push t-values more toward Z-values?
I am sorry if these r too many questions all at once.
I have to learn this in 2 days for an assessed test.
Great video. A couple of questions. Is the degree of freedom something only used when sigma is unknown? Does proportion and probability share the same meaning?
The best explaination
Brandon, your videos are very helpful so I appreciate that thank you. if you could clarify, what if x barred is .43 sample standard deviation is .07 and the sample size is 500. I am confused on how to calculate when n is greater than 30, I have an assignment due tomorrow if you could get back to me asap i would appreciate it
very very good presentation
Amazingly amazing video. Now, I don't know why I am paying tuition to someone who can't explain (such a reasonable) concept reasonably. Brandon for the President!!
Thanks a lot. you are truly awesome
Much appreciated.
you are the BEST !
Thanks, another related question is following: is it possible to compute the exact p-value of an estimate given the confidence interval?
you talk about previous video.... Maybe put a link that brings you to the previous video, like the one you have for part 2.
which would be helpful as i am trying to figure out which one was previous
Hello Brandon,
A big thanks for all these great video series, watching them during corona lockdown, best investment of my time.
My doubt is, my t-test gives a significant p-value but the confidence interval does not contain my sample mean. What am I doing wrong here ?
I am sorry , please Ignore my question, I interpreted it all wrong. Value of null (not the sample mean) should not be in confidence interval .
I really love your encouragement and your warnings!
Awesome
you are awesome Brandon :)
Please make some videos on Factor Analysis
How do you create your diagrams of the Curve? What software do you use that will allow you to draw the curve and shade/fill it in without any data?
Thank you so much
I am watching this because I am struggling. Neither I nor anyone around me has faith in me to pass this final exam today
Amazing, Amazing, Amazing:) Thank u!:D
Good day Sir,
I have a question but don't really know what to ask for.
Here is my problem.
I have large seta of data that are numbers that I get from substracting the price of a stock compared to a previous value. So I have more than a hundred values for the CHANGE of the stock price.
What I want to know is what mag it'd of change is the most common. What magnitude of change is the average?
So if I knew the stick price I could statistically predict what the "jump in price" is going to be. (Of course not knowing if it will go up or down)
Thank you very much.
Hi Brandon i study higher diploma in statistic thanks for helping and i need your advice teacher
very helpful video
thanks for help.
So I take three samples, I get standard deviations of three samples say s1, s2 and s3. Which one do I use as "s" to estimate population standard deviation i.e. s/Sqrt(3)?
Hi Brandon - Is it possible to have a Confidence Interval of a p-value. If I am expected to make decisions based on this value surely we need to know how reproducible it is given that samples taken from unknown populations are likely to bounce around. Books I have read seem to conflate Confidence Intervals and p-values without addressing this. We have confidence Intervals for most statistics we work with why not p values. Regards
Thanks
Help me a lot thx ~
A recent article in the daily newspaper indicated that the mean selling price of the homes in capital city is 357 dollars. Can we conclude that the mean selling price in the capital city is not different from 357 doller .use the 0.05 significance level where n= 52 pleas solve the math
You've taught me everything. Everything. Adopt me. Feed meh.
why do we work on a distribution of the sample means instead of plotting all the data from the samples themselves?
Hi Brandon. I trying to estimate the standard deviation so as to get a sample size for my research. How do i do this?
Oh yeah, I keep getting confused, the chi-square looks so much like the t- distribution.
The video is too long and important parts you skip too fastly. Didnt get anything
I was just beginning to teach this after 8 years of teaching leave; thank you so much. Found it very very useful