in this video we're going to talk about quartiles deciles and percentiles so what are they exactly quartiles divides the data into four equal parts what I like to think about is quarters four quarters adds up to one dollar now let's draw a number line and we're going to go from zero to a hundred but let's divide the number line into four equal parts so this is the first part second third and the fourth part this is q1 this is Q2 and here we have Q3 or the third quartile on the left this is the zero percentile the first quartile is the 25th percentile the second quartile is the 50th percentile and the third quartile is the 75th percentile and this goes up to 100. percentiles divides the data into 100 equal parts think of percentage one hundred percent represents the total of everything Q2 is basically the median of the entire data q1 is the median of the lower half of the data and Q3 is the median of the upper half of the data now let's talk about decels and deciles divide the data into 10 equal parts think of a decimeter a decimeter is basically one-tenth of a meter which means that it takes 10 decimeters to equal one meter so when you hear the word Desi think of a TEF so a decile is basically one-tenth of the data so it takes 10 deciles to cover the entire data so let's break up this data into 10 equal parts so this is going to be D1 this is d0 D2 D3 is the third decile D4 the fourth decile now Q2 is the same as D5 and this is D6 D7 d8 D9 d10 so what you need to know for instance is D4 represents the 40th percentile in this example D5 is the 50th percentile which is the same as the second quartile so we can write the percentage values here so D1 is the test percentile D2 is the 20th percentile D3 is the 30th D4 is the 40th D6 is the 60th and so forth so D9 will be the 90th percentile so now you can visually see how quartiles deciles and percentiles divide the data into different equal parts now what is the meaning of a percentile have you thought about that the 70th percentile for example is a data point where seventy percent of the entire data is less than or equal to the data point now it can also mean that 30 of the data is greater than or equal to the data point so what do we mean let's use the visual example so this would be zero this would be 70 and this would be a hundred so let's say if you took the SAT exam and your score ranks among the 70th percentile so your score is right here now seventy percent of students had a score that is equal to or less than your score so basically you scored better than 70 of the students who took the SAT at that time however 30 percent of the students who took the SAT did equal to or better than you did on the exam and so that's what the 70 percentile tells you it tells you that 70 of the data is less than the 70th percentile with thirty percent of the data is greater than the 70th percentile or it could be equal to it as well so now you know the meaning behind a percentile now let's say if we have the numbers 2 3 5 7 8 10 11 13 15 16 and 19. how can we find the three quartiles in this list of numbers so what is the first second and third quartile the best thing to do is to find the second quartile first the second quartile is basically the median of the entire data set and so let's find a middle number we can eliminate the first two and then the last two until we get the number in the middle which in this case is 10. so I'm going to get rid of the 10. I'm going to separate the data into two equal parts I'm going to put the 10 right here so 10 is our Q2 value that is the median of the entire data set q1 is the median of the lower half of the data set so q1 is 5. as you could see 5 is the middle number of these five numbers now what about Q3 Q3 is the median of the upper half of the data set so the middle number is 15. so that is the third quartile now it turns out that there's another way in which we can calculate q1 Q2 and Q3 using percentiles keep in mind q1 is basically the value of the 25th percentile to find the location of a percentile it's equal to K divided by a hundred times n Plus 1. so n is basically the number of data items in your list or the number of numbers in the set K is basically the subscript so if you want to find the 25th percentile K is 25. so let's illustrate this with an example so let's find the location of the 25th percentile so it's going to be 25 divided by 100. and there's one two three four five six seven eight nine ten eleven numbers in the list so n is 11. so let's get the answer what is 25 over a hundred divided by 25 is 4. so 25 divided by 100 is 1 4. and 11 plus 1 is 12. so we have 12 divided by 4 which is 3. so what does this number tell us the 3 indicates that the first quartile is in the third position it's the third item from the left so this is the first item this is the second item this is the third item so the third number from the left is five which is our first quartile as we can see here and so that's how you're supposed to use the formula now let's try another example let's calculate the second quartile so the second quartile is basically the 50th percentile which we know to be 10. so let's determine the location of the 50th percentile so it's K divided by 100 or 50 divided by 100 times n Plus 1. so n is not going to change it's still 11 for this list so 50 divided by 100 that's basically 0.5 or we could say it's 1 over 2. 11 plus 1 is 12. half of 12 is 6. so the second quartile is basically the sixth item from the left so this is the fourth item this is the fifth item and we can see here this is the sixth item which is ten and so that is the second quartile if you wish to calculate the third quartile it's basically the 75th percentile which we know it to be 15. so using the same formula we're going to calculate the location of this the 75th percentile so it's 75 over a hundred times eleven plus one 75 over 100 if you divide both numbers by 25 you can reduce it to 3 over 4. now 12 divided by 4 is 3 times the 3 on top that gives us 9. so 3 4 of 12 is 9. so the knife data point in this case is going to be 15. which is the third quartile let's try another example so this time we have an even number of items in this list so let's determine the first second and the third quartile so let's split the data into two equal parts so here we have eight numbers on the left and eight numbers on the right so this time the median is going to be the average of 15 and 16. if we add those two numbers and then divide by 2 this will give us the midpoint or the average of those two numbers which is 15.5 so that is our second quartile now to find the first quartile we need to find the middle number of the numbers on the left now they're even so we're going to split it here so that we have 4 on the Left 4 on the right so the median is going to be the average of 9 and 11. 9 plus 11 is 20 divided by 2 that's 10. so 10 is the first quartile now let's split the data on the right into two equal parts and so the average of 19 and 20 is going to be 19.5 19.5 is right in the middle of 19 and 20. so that's our third quartz hour now let's see if we can calculate that using the same process that we did before so let's calculate the location of the 25th percentile so it's K divided by 100 in which case k is 25 and there's 16 numbers in this list we have four four four and four so n is 16. so this is going to be 16 plus 1. so it's 1 4 of 17. now 17 divided by 4 that is not going to give us a whole number in fact we're going to get a decimal number so what is the 4.25 value well that's hard to tell when you get a situation like this you need to round up and down you need to look at the fourth value and the fifth value so this is the first second third fourth fifth value the fourth value is 9 and the fifth value is 11. so if you average four and five it will give you ten I mean if you average 9 and 11. it will give you 10. and so this is the location well the location of the 25th percentile is this number but the value of the 25th percentile is 10 which is q1 so that's how you could use this formula to help get you this answer now let's use the same process to calculate Q3 so Q3 is the 75th percentile which has a value of 19.5 so let's determine the location of the 75th percentile so at 75 divided by 100 times 16 plus 1. so it's three-fourths of 17. 17 times 3 divided by 4 is 12.75 so this is 6 7 8 9 10 11 12 and 13. so we can see that 12.75 is between these two numbers so it's going to make sense to average them so when you get a number like this you need to identify the 12th value which is 19 in this case as we see here and the 13th value which is 20. and so the 75th percentile is going to be the average of the 12 and the 13th value so we're going to take 19 and add it to 20 and then divide by 2. so we get 39 divided by 2 which is 19.5 here's another example using this list of numbers let's say if we want to calculate the value of the six decile how can we do so the sixth decile is basically the 60th percentile and so first we need to calculate the location of the 60th percentile so it's going to be 60 divided by 100. times 16 plus 1. so it's basically 0.6 times 17. which is 10.2 so the 60th percentile is going to be the average of the TEF number and the 11th number because 10.2 is between the 10th and the 11th number and then we're going to divide it by 2. so the 10th value we could see is 16. and the 11th value is 17. the midpoint between 16 and 17 is 16.5 so this is the 60th percentile based on the data that we have here that is of course if we follow the same process as we've been doing however I'm inclined to think that the 60th percentile is based on the 10th value in the list well for one thing 10 is very close to 10.2 and a 10th value in the list is 16. and here's why I'm inclined to think about intuitively the second quartile is the 50th percentile the third quartile is the 75th percentile so it's clear to see that our answer should be somewhere in this region so 16.5 is not a bad estimate however if we split this in the middle the midpoint between 50 and 75 if you add them up and divide by two that should give you 62.5 and 60 is to the left of 62.5 which puts it more closer to 16 then to 17. now granted the calculations will not be perfect because n is small but as you increase your n value as you increase the number of numbers in a list these calculations become more accurate so keep that in mind now there's something else that we can do with this list of numbers so far we talked about finding the value of a percentile for instance the value of the 50th percentile is 15.5 but what if we're given the value how can we find the percentile that corresponds to that value so let's use let's use 12 for example now we know that 12 is between the 25th percentile and the 50th percentile would you say this is the 30th percentile the 40th percentile the 35th percentile what would you say well if we're going to Ballpark in the average between 25 and 50 is 37.5 so it makes sense that 12 is between the 25th percentile and a 37.5 percentile but let's see if we can do some math to get the answer to find it you could use this formula it's X plus 0.5 y divided by n times 100. so what is X in this formula X is the number of numbers that is less than 12. so all these numbers are less than 12. so there's five numbers less than 12. that's X is 5. why is the number of times this number appears there's only one twelve so Y is going to be one if there were two twelfths y would be 2. now n is the number of numbers in the list we know we have a total of 16 numbers we got four in each quadrant and then we're going to multiply everything by 100. so this is basically 5.5 divided by 16 times 100. 5.5 divided by 16 is 0.34375 so times 100 this is 34.375 now let's round it to the nearest whole number so therefore we could say that the 34th percentile is equal to 12. or we could say that 12 is the 34th percentile and that's that is how we can find it and it makes sense because 34 is between 25 and 37.5 so that's a good approximation let's try another example what about the number 16 the ninth data point in the list what is the percentile that corresponds to that number now we know it's going to be between 50 and 62.5 but feel free to pause the video and calculate the answer there's no need for me to retype that formula so P of what subscript number is equal to the first 16 that we see well first let's determine X X is the number of items that is less than this 16 value so notice that we have eight numbers to the left of this number so the x value is eight now Y is the number of times 16 appears because 16 is the number that we're focused on notice that there's two sixteens in this case it's going to be two n is still 16 and then times 100. so 0.5 times 2 is 1 plus 8 that's 9. so this is going to be 9 over 16. times 100. and so this is going to be 56.25 and so we're going to round this to 56. so 16 is the 56 percentile or we could say that the 56 percentile is equal to 16. now Does this answer make sense well let's see 16 is between 50 and 62.5 and 56 is between 50 and 62.5 so saying that the first 16 is the 56 percentile is reasonable it makes sense based on the data that we see here now there's one more thing we need to talk about and that is the ability to make a cumulative relative frequency table and then to use it in order to calculate percentile values so let's work on this example let's create a cumulative relative frequency table and then let's calculate the fourth the seventh the third and the sixth decile in that order so to make a cumulative relative frequency table we need four columns the first one will contain the value the second will be the frequency the third will be the relative frequency and the fourth column will contain the cumulative and relative frequency values so the lowest value in this list is 2. now how many times does 2 appear one two times so the frequency is 2. now the next highest value is a 3. so we have one two three four threes in this list now the next value in the list is 4. so we have one two three fours in the list next up we don't have a five but the next number is a six and so there's only two sixes that I've counted now we do have a seven only one of them the next number is a nine and there's one two three four five nines and I'm gonna need some extra space here and then we have a 10 it turns out we have two tenths and the last number is a twelve but there's only one twelve now our next step is to take the sum of the frequency column so two plus four is six plus three is nine plus two is eleven and then 12 17 19 20. so this tells us that we have a total of 20 items in our list the next step is to calculate the relative frequency to do this take the frequency and divide it by the total frequency or the total number of items 2 divided by 20 is 0.10 4 divided by 20 and that's going to be 0.20 3 divided by 20 is point fifteen 2 divided by 20 is 0.10 1 divided by 20 is 0.05 divided by 20 that's 0.25 and 2 over 20 is 0.10 1 over 20 is 0.05 . now if you add all of these numbers you should get 1.00 which I'll mark in red now let's calculate the cumulative relative frequency so first we're going to start with this number 0.10 and then we're going to add 0.10 and the next relative frequency which is 0.20 . so this is going to give us 0.30 and then add 0.30 Plus 0.15 so that's going to be 0.45 and then add 0.45 plus 0.10 so that's 0.55 and then add in those two this is going to be 0.60 0.60 plus 0.25 that's 0.85 and 0.85 plus 0.10 that's 0.95 and then the last one is going to be one when you add 0.95 and 0.05 so that's how you can calculate the cumulative relative frequency now let's use it to calculate the deciles that we have listed here so let's start with the fourth decile D4 the fourth decile we know to be the 40th percentile the 40th percentile corresponds to a cumulative relative frequency of 0.4 notice that 0.4 is somewhere between 0.3 and 0.45 so here is a question for you well the 40th percentile be equal to three or will it be equal to four or will it be equal to the average of three and four which one of these three options is correct now you need to understand that the 30th percentile it ends with the last three out of the four threes that we have here and it begins with the first four out of the three fours that we have here so if you want to find the 30th percentile exactly it's really the average of three and four however if you want to find a percentile that is between two cumulative relative frequency values it's going to be the higher of the two numbers four is between a percentile of 30 and 45. the value 6 is between a percentile of 0.45 and 0.55 or 45 and 55. so the the knife value or the value of 9 is between the 60th percentile and the 85th percentile but if you want to find let's say the 85th percentile it is the average of 9 and 10. and I'm going to explain why visually so you can see it so because the 40th percentile is between these two numbers it's going to be four the 30th percentile ends with three but it begins with four so the 40th percentile has to equal four now what about the seventh decile or the 70th percentile what is the value that corresponds to that so the 70th percentile has a cumulative relative frequency of 0.7 which is between 0.60 and 0.85 now keep in mind the 60th percentile ends with 7 and begins with nine so the 70th percentile has to be one of the five nines that we have here so it's going to equal not now what about the third decimal so that's the 30th percentile notice that we do have an exact value here a cumulative relative frequency of 0.30 so 0.3 or the 30th percentile it ends with the last three and it begins with the first four so in this case it's going to be the average of those two numbers so the 30th percentile will be 3.5 now what about the sixth decile or the 60th percentile notice that we have that exact number in our chart or in our table so thus it's going to be the average of these two the 60th percentile ends with the last seven and starts with the first five so the average of seven and nine I meant to say it ends with the last seven but starts with the first nine so that's the average of those two numbers is going to be eight now let's confirm these answers let's see if we can make sense of it so we said that the fourth decile is four the seventh decimal is nine the Third decile is 3.5 and the sixth decile is eight now let's go ahead and put this list in increase in order you could also use the cumulative relative frequency table that we had to do this as well so if you go back to it we said that we had two twos and we had four threes and we had two sixes we had a seven and we have five nines and then we have two tenths and we have a 112. it looks like I'm missing something I'm missing the force so let's go back there were three fours and then we had two sixes one seven and then five nines and then there were two tenths and a 12. so notice that I'm grouping everything in pairs because I have 20 numbers and I want to divide it into 10 equal parts so you could see what the deciles are so this is going to be the first decile this is the second third fourth fifth six seven eight nine ten and we could say this is zero so let's start with the fourth DEC sound which we said was four so here's D4 notice to the left is the four and to the right it's a four so the fourth decile or the 40th percentile is four so because they were the same we didn't really have to average any numbers now let's talk about the seventh decile so here's the seventh decile to the left we have a nine to the right we had a nine so as we can see it's one of the five nines that we have in our list so this answer is accurate now the Third decile is between two different numbers as we can see here it ends with the last three and it begins with the first four so that's we average three and four and get 3.5 now the situation is the same for the sixth decile as we could see it ends with the last seven and begins with the first line so it's the average of seven and nine giving us eight so hopefully that makes sense and now you have a better understanding of quartiles deciles and percentiles and you know how to find them given a list of numbers thanks for watching