How To Find The Weighted Mean and Weighted Average In Statistics

In this video we will focus on solving problems that ask us to

find

the

weighted

mean

or the

weighted

average

, so let's start with this problem, so we want to calculate the weighted

mean

of the sample of numbers shown below. Well first calculate the mean of those numbers, the sample mean is basically the sum of all the values ​​of x divided by the number of values ​​we have in a list, so it will be 16 plus 20 plus 12 plus the next 2 16 and then plus 10 and so on and I'm running out of space here and we have a total of 10 numbers so n is 10. now let's get the sum of those 10 numbers so the sum I got is 170 divided by 10 numbers, which gives us a sample mean of 17.

Now there is another way we can get this answer so that we can also calculate the mean or more specifically the weighted mean if we take the sum of all the weights multiplied by the sample, It could be the weight it could be in frequency form or it could be in percentage form and then divided by the sum of the weights, so looking at the first number we are going to start with the lowest number which is 10. 10 has a frequency of 1 because there is only one ten in the list, so its weight is one, so it will be one multiplied by ten, the next number is twelve, it also has a weight or frequency of one, now the next number 16, there is four, so it has a weight or frequency of four, so it will be plus 4 times sixteen, then we have three twenty, so it will be plus three times twenty and then we only have twenty-four.

Now we have to take the sum of the weights, so it's one plus one, which is two more. four is six plus nine I mean plus three is nine plus one is ten so the sum of the weights or frequencies is 10 in this example, so 1 times 10 is 10 plus 12 plus 4 times 16 plus 3 times 20 plus 1 times 24 gives with the same sum of 170 dividing that by 10 we will get the same answer of 17. This is how you can use this formula to calculate the weighted

average

. Now let's look at another example in the class of 20, eight students averaged a score of 86, seven students averaged 74, and 5 students averaged a test score of 98.

What is the average test score for the entire class? class? We are going to use the same formula to calculate the weighted average, so let it be the sum of all the weights multiplied by each data point in our sample divided by the sum of all the weights, so first we have a weight of eight students and an average score of 86. Then we have seven students, so that's the frequency or the weighting. and the average for that is 74 and then we have five students with an average test score of 98. now we need to divide that by the sum of the weights, so 8 plus 7 is 15 plus 5 gives us the total of 20. so Let's do this step by step, first let's multiply 8 by 86, which will be 688, then we have 7 by 74, which is 518 and then 5 by 98, which is 490.

So, taking the sum of those three numbers 688, 518 and 490 , we have a the sum of 1696 divided by 20. gives us a weighted average of 84.8, so that is the average of all the test scores for the entire class. Now let's move on to the next problem number three in a certain university, 20 of the students have an average weight. 140 pounds, 35 percent of the students have an average weight of 160 pounds, 30 percent of the students have an average weight of 175 pounds, and 15 of the students have an average weight of 195 pounds, so it's a lot information, but according to your data, what is the average weight of all the students in this high school, so, according to the last two problems above, you know how to do this one, feel free to pause the video and try it if you want, so We will follow the same process as us.

We are going to multiply the weight by each data in the sample or in this case the average and then divide by the sum of all the weights then we could say x would be the average weight in pounds w would be the percentage that corresponds to the average weight, so for the first let's write this as w1 x1 plus w2 x2 plus w3 x3 and so on and then the sum of the weights will be w1 plus w2 plus w3 plus w4 if that makes it easier, for the first part we have 20 percent of students with a weight average of 140 pounds 20 will be the weight x1 will be 140 pounds so we can write w1 as a percentage 20 percent and then the total would be one hundred percent or we can write it as decimal point twenty where the total will be one, I prefer to use the decimal version percentage, so w1 will be 0.20 .now w3 will be 30 or 0.30 and the extreme is 175.

Then we have w4, which is point 15 times x4, that's 195. So if we add point 20 plus 0.35, that's 0.55, so 4 and then plus 0.3, that's 0.85 plus point fifteen, that will give us one or one hundred percent, so the sum of the four weights is one, so now let's connect the numbers and get the answer for this problem, so the weighted average for this example problem will be 165.75 pounds and that's it for this problem. This is how you can calculate the average weight of all students at this high school. Now let's move on to the next example problem number. Four, using the information shown below, calculate John and Kelly's final semester grade, so we will use the same formula to do this, it will be the sum of the product of the weight and the data point of our sample x1, for which is going to be w1 x1 plus w2 x2 and since we have five things we're dealing with, we're going to go up to w5x5 and then we're going to divide that by the sum of all the weights, since the weights are given as percentage values, the sum will be one hundred percent or one as a decimal number, so let's start by calculating the weighted average for John, so w1 will be 15 percent, so it's 0.15 and the homework score for John is 92, so that goes. to be x1, next we have w2, so that will be the weighted percentage for the test, which is 10 times the test score of 74. and then we will just repeat the process, so for the lab it will be point 20 and the score for that is 83. then we have the tests that are weighted 25 percent or 0.25 times the score of 76 and then the final exam is weighted 30 percent times the score of 88. so now let's divide by the sum of all the weights point fifteen plus point ten that's point twenty-five plus point twenty that's point four five plus point twenty-five that's 0.7 plus 0.3 that's one so you don't really need to put in the one since it doesn't work to change your answer so this turns out to be a final exam score or rather a final semester grade of 83.2 for John.

Now let's do the same with Kelly. Feel free to pause the video and try it yourself so the homework percentage is still 0.15 but Kelly's homework score is 100. so that's different and then for the test it will be 10 or 0.10 your test score is 82 and then it will be plus 0.20 times your lab score of 95 plus 0.25 times your test score of 70 and then plus 0.30 times your final exam score of 76 and we don't need to divide it by one , so we can leave it like that, so all we have to do is just plug these numbers into the calculator and see what the final result is to get the final answer. is 82.5, so in this example, John has a higher final semester grade than Kelly.

Now let's talk about why this is so if he notices that his test score and his final test score are significantly higher than Kelly's and these two categories have a higher weight than the other three. categories Even though Kelly did better on her homework lab and exam scores than John, the total weight of those three categories is 45 compared to the weight of the tests and final exam categories, which is 55 percent, because he did better in the areas that had more weight. or that had more weight, his score was slightly higher than Kelly's score and that's it for this problem.

Now you know how to calculate your final grade for the semester if you know the weighted percentage values ​​for each of these number five categories using the information shown below. the student's gpa or grade point average, so we'll use the same formula we've been using w1 x1 plus w2 x2 in this case up to w5 x5 divided by the sum of the weights we now need to determine what x is and what the weight is We want to calculate the average grade or the average grade, not the average hours, so we want the average of the points, so the points will be x, the weight will be the hours, w1 will correspond to chemistry.

Those will be the hours of chemistry, which are three x one, will be the chemistry score, which is three, so every time you have a b that corresponds to three and a corresponds to four points, a c corresponds to two and the course credit hours are just depends on you know what type of course you're trying for a lab, it's a one credit hour course in college, at least the college I attended, a typical chemistry course, might have three credits, some might even have four, but let's move on. and finish this problem so that w2 corresponds to physics, that is the hour credit for physics and the score for that x2 the score was 2. now for w3 it will be 1 and the score for that is 4. and then for w4 calculation has four credit hours you got an a in class so that's four points and then w5 is three credit hours for English multiplied by three points for a b now we have to add up all the credit hours so it will be 3 plus 3 plus 1 plus four plus three, so it's w one through w five, so now we can do the math three times three is nine three times two is six four times one is four four times four is sixteen three times three is nine and then we have three plus three, which is six seven eleven fourteen, so we have a total of fourteen credit hours, so 9 plus 6 plus 4 plus 16 plus 9 will be 44 divided by 14.

That gives this student an average grade point average of 3.14, which is close to a b, this is how you can calculate a student's GPA if you know the credit hours of the courses they have and if you know the letter grade for each of those courses number six Rachel mixes five gallons of twenty percent antifreeze solution with ten gallons of fifty percent antifreeze solution to form a new solution with a different antifreeze concentration. Will the new solution be closer to 20 percent or 50 percent concentration? what would i say? So let's say we have solution 1 and solution 2. so solution one has a volume of five gallons and solution two has a volume of ten gallons now the concentration of solution 1 is 20 the concentration of solution 2 is 50 we want to calculate the concentration of the combined solution then based on that, what is w and what is x x is what we want to

find

the average, so this is x one and that is x two, the weight in this case would be the volume which would be w2, well, What just happened there, sometimes this computer can malfunction. has problems, but this is what we have for this problem so far.

The combined solution or let's call it solution three will have a total volume of 15 gallons because once we mix a five gallon solution with a 10 gallon solution, we are going to get a 15 gallon solution, what we need to calculate is the average or concentration weighted of that solution. Now, if I were to average 20 and 50, assuming they have equal weights, the midpoint of 20 and 50 is 35. Now, because the weights are not equal, we know that the answer will not be 35, but is the answer more Close to 20 percent or is it closer to 50? What would I say? Take a minute and think about what the average concentration, that is, the new concentration of antifreeze, will be. should be between 20 and 35 or 35 and 50.

Now notice that solution two has more weight than solution one, therefore the combined solution should have a concentration between 35 and 50. So I am going to write that it should be between 20 and 35 or 35 and 50. 35 and 50 because solution 2 has more weight than solution 1. Therefore, its concentration will be closer to that of the solution, so now let's calculate the weighted average or the new concentration of the antifreeze solution, so it will be the sum of the multiplied weights. in this case the concentrations divided by the sum of the weights then it will be w1 x1 plus w2 x2 divided by w1 plus w2 so w1 is 5 gallons and x1 the concentration is 20 this time I will leave it in percentage i I am not going to convert it to point 20 because I want my final answer to be in percentage.

Now w2 is 10 gallons and x2 is 50. Now we are not taking the sum of the percentages that we want to average the percentages that we need to take. the sum of these two numbers w1 plus w2 is 5 plus 10. so we have 5 times 20, which is 100 and then plus 10 times 50, that will be 500 divided by 15. so that is 600 divided by 15 and that will give us a concentration weighted average of 40, so this is the answer, by the way, for those of you who like to focus on units, this is in gallons, so it is a percentage, meaning gallons percent, when you add these two ,will be in gallons, so the unit of gallons will be canceled leaving behind the unit percentage now, as we could see, our answer is within this range, it is between 35 and 50, so solution 3 has a concentration closer to solution two because solution two has more weight than solution one and that's how weighted averages work.

That's all for this video. Thanks again for watching.