Statistics on ACT Math Strategies for Mean, Medium, Mode

Statistics on ACT Math Strategies for Mean, Medium, Mode SAT / ACT Prep Online Guides and Tips Statistics questions on the ACT are often simpler than the statistics questions you have seen in class. Most all of the statistics questions on the ACT boil down to finding or manipulating means, medians, and modes of a set of numbers. If you are already familiar with these terms, you will have a good head-start on these types of problems. But even if you aren't familiar with these terms, most of ACT stats questions require that you understand and apply just a few key concepts (all of which we will go through in this guide). This will be your complete guide to ACT means, medians, and modes- what they mean, how you'll see them on the test, and how to solve even the most complicated of ACT statistics questions. What is a Mean, Median, or Mode? Before we look at how to solve these kinds of problems, let's define our terms: A mean is the statistical average of a group of numbers. In order to find the mean, we must add up the sum of the numbers in our set and then divide that sum by the amount of numbers in the set. (Note: on the ACT, the question will almost always use the word "average" instead of "mean.") What is the average speed of six runners if their race times were, in seconds: 85, 67, 88, 75, 91, and 80? To find the average (mean), we must find the sum of all the numbers and then divide that number by the total amount, which in this case is 6. $(85 + 67 + 88 + 75 + 91 + 80)/6$ $486/6$ $81$ The mean (average) race time is 81 seconds. The median is the number directly in the middle of a set of numbers, after they have been arranged in numerical order. (Note: the number will be halfway into the set, but is NOT necessarily the mid-value between the largest and smallest number.) For example, take a set of numbers {14, 15, 23, 37, 213}, the median would be 23, as it is in the middle of the set. This is true, despite the fact that 23 is NOT halfway between 14 and 213. If your set has an even amount of numbers, then you must take the mean (average) of both the numbers in the middle. Find the median value of the set of numbers {10, 2, 34, 47, 17, 8}. First, arrange the numbers in order from least to greatest. 2, 8, 10, 17, 34, 47 We have an even number of terms in our set, so we must take the average of the two middle terms. $(10 + 17)/2$ $27/2$ $13.5$ Our median is 13.5 The mode is the number or numbers in a set that repeat(s) most frequently. In the set of numbers {4, 6, 6, 4, 3, 6, 12}, our mode is 6. Even though the number 4 occurred twice, the number 6 occurred three times and is thus our most frequently appearing number. If each number in your set occurs only once, there is no mode. In the set of numbers {3, 11, 7, 23, 19}, there is no mode, since no number repeats. If multiple numbers in a set repeat the same number of times, your set will have more than one mode. In the set {4, 11, 11, 11, 13, 21, 23, 23, 23, 43, 43, 43}, we have three modes- 11, 23, and 43. All three numbers occur exactly three times and no other numbers occur more frequently, which means that we have multiple modes. The more you get used to statistics questions, the more quickly you'll be able to spot your answers. Typical Mean, Median, and Mode Questions Mean, median, and mode questions are fairly simple once you get the hang of how they work. Because these types of questions will appear 1 to 2 times on the test, you will see them in a variety of different forms. But always keep in mind that, no matter how unusual they look, mean, median, and mode questions will always break down to the concepts we outlined above in their definitions. For mean questions, there will be two types- weighted and unweighted averages. Unweighted averages are by far the most common, but you'll need to know how to tackle both. Unweighted Average Unweighted average questions are solved exactly how we found our means above. We simply find the sum of our set and divide this number by the amount of numbers in the set. The monthly fees for single rooms at 5 colleges are $\$ 370$, $\$ 310$, $\$ 340$ 380$, and $\$ 310$, respectively. What is the mean of these monthly fees? F. $\$ 310$G. $\$ 340$H. $\$ 342$J. $\$ 350$K. $\$ 380$ We must find the sum of our terms and divide by the amount of terms (in this case 5). $(370 + 310 + 380 + 340 + 310)/5$ $1710/5$ $342$ We have found our mean. Our final answer is H, 342. Weighted Average A weighted average, on the other hand, puts more emphasis on (gives more "weight" to) some numbers more than others. When this is the case, you must multiply each number in the set by its weight and then add their sums and divide as normal. Let us look at this process in action: In Karen's math class, the final class grade is determined by a combination of quizzes, homework, and test scores. Quizzes make up 30% of the final grade, homework accounts for 25% of the final grade, and test scores account for 45% of the final grade. Each assignment/test has a potential score of 100 points. Karen received a 92 and an 83 on her two quizzes, scores of 100 on her three homework assignments, and test scores of 78, 89, and 98. What is Karen's final grade in the class? First, we must find the average of each type of assignment as normal and then multiply that average by the weight allotted to the assignment. So, to find the number of total points she earns from her quizzes, we would say: $(92 + 83)/2$ $175/2$ $87.5$ She earned an average of 87.5 on her quizzes, but now we must multiply it by the percentage allotted to the quiz scores in terms of her overall grade (the weight). $(87.5)(0.3)$ $26.25$ Her quiz score will contribute 26.25 points towards her overall score. Now let us do the same for her homework. $(100 + 100 + 100)/3$ $300/3$ $100$ The homework is weighted as 25% of the grade, so we must multiply the average by its weight. $(100)(0.25)$ $25$ And again for her test scores. $(78 + 89 + 98)/3$ $265/3$ $88.33$ And again, we must multiply this average by the allotted weight. $(88.33)(0.45)$ $39.75$ Now, simply add them all together to find her final score. $26.25 + 25 + 39.75$ $91$ Karen's final grade in the class will be a 91. Now that we've seen our different types of mean questions, let's look at the other types of statistics questions on the ACT. Most all the statistics questions you'll see on the ACT will be on means/averages, but a few will involve medians. These are generally straightforward, so long as you understand how to find your median. What is the median of the following 7 scores? 42, 67, 33, 79, 33, 79, 21 A. 42B. 52C. 54.5D. 56E. 79 First, let us, as always, put our numbers in ascending order. 21, 33, 33, 42, 67, 79, 89 Since we have a set of 7 numbers, there is a number exactly in the middle of our set. Now that we've put them in order, we can see that the middle number is 42. Our final answer is A, 42. And lastly, mode questions very rarely show up on the ACT. You should still know what a "mode" means in case you do see a mode question on the test, but odds are you'll only be asked to find means and/or medians. Though the questions may appear different, just remember that they are all variations on the same few concepts. How to Solve Mean, Median, and Mode Questions Since you will see these questions multiple time on any given test, it can be easy to rush through them and/or underestimate them. But as you go through your test, remember to keep these ACT math tips in mind: #1: Always (always!) pay attention to exactly what the question is asking You will be asked to find means/averages the majority of the time, so it can become second nature to immediately start finding a mean when you come across an m-word in a math problem. It may seem obvious right now, but the pulse of a ticking clock and the adrenaline in your veins during the test-taking process can make it so that you misread the words in a math question, and try to find the mean instead of the median (or even vice versa). The test makers know how easy it is for people to make these kinds of errors and will provide bait answers to tempt anyone who makes a mistake. Always double-check that you are answering precisely the right question before you start solving the problem (and especially before bubbling in your answer!). #2: Write It Out Take the time to rearrange your set of numbers in order when dealing with medians and modes, and make sure you write out your equations when dealing with means. It can be tempting to solve problems like these in your head, but a single misplaced digit will give you a wrong answer. In order to avoid losing points to careless errors, always take a moment to write out your problem. It will not take as long as you think it will to reorganize your values and it will almost always lead you (quickly) to the right answer. #3: Use PIA/PIN When Necessary If you find yourself stuck on a problem and have some extra time to spare, don't hesitate to use your fallback strategies of plugging in answers or plugging in numbers where applicable. Always keep in mind that it will often take you a little longer to solve a problem using these techniques, but doing so will almost always lead you to the right answer. Practice and technique are required to master any skill, be it statistics questions or silly walks. Test Your Knowledge And now, let's put your knowledge of statistics to the test against real ACT math problems. 1. Tom has taken 5 of the 8 equally weighted tests in his U.S. History class this semester, and he has an average score of exactly 78.0 points. How many points does he need to earn on the 6th test to bring his average score up to exactly 80.0 points? A. 90B. 88C. 82D. 80E. 79 2. 3. What is the difference between the mean and the median of the set {3, 8, 10, 15}? A. 0B. 1C. 4D. 9E. 12 4. To increase the mean of 4 numbers by 2, how much would the sum of the 4 numbers have to increase? F. 2G. 4H. 6J. 8K. 16 Answers: A, B, A, J Answer Explanations: 1. In order to find out how much we need to increase our sum, we first need to find our original sum. Let us represent the original sum with the variable $x$ and use our mean equation to find it. $x/5 = 78$ $x = 390$ Let us use this original sum for our new mean equation with the set of 6 terms. We will represent the missing value with $y$ and set our equation to the needed 80 points. $(390 + y)/6 = 80$ $390 + y = 480$ $y = 90$ We have found the amount necessary to increase our sum in order to get an average of 90 with 6 terms. Our final answer is A, 90. 2. We are told that there are 43 soccer games, so we must find the percentage of each match and multiply this figure by the number of goals per match. For instance, there are 4 matches in which there were 0 goals. Which would give us: $(0)(4/43)$ $(0)(0.093)$ $0$ Now, we need to do the same for all the matches and add them together. $0 + (10/43)(1) + (5/43)(2) + (9/43)(3) + (7/43)(4) + (5/43)(5) + (1/43)(6) + (2/43)(7)$ $0.2325 + 0.2326 + 0.6279 + 0.6512 + 0.5814 + 0.1395 + 0.3256$ $2.79$ Finally, we need to round this number to the nearest 0.1, as we were told to. $2.8$ Our final answer is B, 2.8. 3. The numbers in our set are already in numerical order, so we do not need to rearrange them. Let us find our median: We have two numbers in the middle of our set, because there are an even amount of numbers in our set. This means we must take the average of the two middle numbers. $(8 + 10)/2$ $18/2$ $9$ Now let us also find our mean: $(3 + 8 + 10 + 15)/4$ $36/4$ $9$ We can see that the mean and the median are equal, so the difference between the two is 0. Our final answer is A, 0. 4. We have two different ways to solve this question- using algebra and using PIN. Let's look at both methods. Method 1: Algebra Let us represent both the sum and the mean by the variables $x$ and $y$, respectively in our mean equation. $x/4 = y$ $x = 4y$ Now, let's look at how this changes when we add 2 to to our mean. $x/4 = y + 2$ $x = 4(y + 2)$ $x = 4y + 8$ We can see that we need to add 8 to our previous mean of $4y$. Our final answer is J, 8. Method 2: PIN We could also use plugging in numbers in this case. So let us pick four numbers and find their mean. Let's just say our four numbers are: 3, 4, 6, and 10. (Why those numbers? Why not!) (3 + 4 + 7 + 10)/4 $24/4$ $6$ Now, we want to increase our mean by 2, which would make it: $6 + 2 = 8$ Which means that now we have: $(24 + x)/4 = 8$ $24 + x = 32$ $x = 8$ We can see that we need to add 8 to our sum in order to increase our mean by 2. Our final answer is again J, 8. (Or boy or other gender). Either way, go you! The raptors are proud. The Take Aways Once you know your way around the techniques of finding your means, medians, and modes, you will be able to tackle most any ACT question on the topic. All ACT statistics questions are simply variations on the same theme, so knowing your foundations is essential. As we saw above, there are often multiple ways to solve these types of problems, so don't hesitate to use PIA or PIN if you have the time to spare and if you feel uncomfortable with the algebra. Otherwise, always make absolutely sure you are answering the proper question and don't take for granted that these questions are simple (a careless error will still lose you precious points!). What's Next? You've tackled all there is to know about ACT stats questions and now you're hungry for more ACT math guides...right? Right! Well, lucky for you, we've got guides on all the ACT math topics you could ask for. Need to brush up on your formulas? How about your trigonometry? 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