book-1 Statistics (Mean,median,mode, etc)

Concept Range 1. Set x = {a,b,c} where a<b<c if (a+b)/2 = 3x-13 and (b+c)/2 = 3x+11 then find the range. 2. Set A: {x, x, x, y, y, y, 3x+y, x-y } If the median of set A is 10 and 0 < x < y, what is the range of set A? 3. Set S is comprised of six distinct positive integers less than 10. Which of the following must be true? I: The median is an integer II: The median is less than the average III: The range of digits in Set S is less than 8 (A) I only(B) I & II(C) II & III(D) III only(E) None of the above. 4. A list is comprised of five positive integers: 4, 4, x, 7, y. What is the range of the possible values of the medians? (A) 2(B) 3(C) 6(D) 7(E) Cannot be determined by information provided.

Highest value - the lowest value 1. Range c-a = 48 2. x-y, x,x,x,y,y,y,3x+y, median (x+y)/2 = 10, x+y = 20 range = 3x+y - (x-y) = 3x+y-x+y = 2x+2y = 2(x+y), 2(20) = 40 3. Ans E 4. 3 Hint: Figure out the smallest median possible and then the largest median possible.

Concept Weighted Average 1. Exam A, 20 students with average 90 Exam B, 30 students with average 82 Exam C, 50 students with average 66 What is the Weighted Average of all three exams? 2. Set A consists of 30 numbers, the average of which is 60. Set B consists of 29 numbers, the average of which is 70. Column A Column B The average of Set A and B 65 3. Set S and Set T both contain x elements. The average of Set S is 40. If the average of Set S and Set T combined is 50, which of the following must be true? I. The average of Set T is 60. II. The range of Set T is greater than that of Set S. III. x is an even number (A) I only (B) I & II (C) II only (D) I & III (E) All of the above 4. Set A contains 120 terms with an average of 8.2. Set B contains 240 terms with an average of 10.6. If Set A and B are combined what is the resulting average? (A) 8.4 (B) 8.8 (C) 9.0 (D) 9.8 (E) 10.1

Weighted Average: Case 1: If the population A and B are equal then weighted average is just the simple average. Case 2: If population A < B then weighted average will be closer to population B. Example Population A has 2 people of 20 each. Population B has 4 people of age 40. Therefore: (2*20) + (4*40) = 40 + 160 = 200. Next 200/6 = 33. 1. Method 1: 20.90 + 30.82 + 50.66 = 7560, Next 7560/100 = 75.6 Method 2: 20.90/100 + 30.82/100 + 50.66/100 = 18+24.5 + 33 = 75.6 2. B 3. A, ie I only 4. Method 1: 120->8.2 120->10.6 120->10.6 29.4 /3 = 9.8 Method 2 by approximate: Add first and second group we get 240 with average 9.4. Now 240 -> 9.4 120 -> 10.6 Therefore it must be between 9.4 and 10.00, since only option is 9.8 Ans (D) 9.8

Percentiles Percentiles is used for larger populations and are more precise than Boxplots. -If score is p%, it means the score is larger than p%. -Percentiles are from 0% to 99%, there is no 100% 1) Sasha took a nationwide standardized test that is graded on a scale from 20 to 60. Sasha got one of the best scores recorded on that this test. Column A Column B Sasha's score the percentile of Sasha's score 2) Alice took nationwide standardize test that is graded on a scale from 0 to 100. Alice scored the highest score recorded on this test. Column A Column B Alice's score the percentile of Alice's score 3) A large distribution of score is normally distributed Column A: score that's one standard deviation above the mean Column B: score that has the 80th percentile 4) It goes like the random variable x is normally distributed and values 650 and 850 are 60th and 90th percentiles of distribution. quantity a: the value at the 75th percentile of distribution of x quantity b: 750 Ans to 4 Follows: 4) We know 650 is the 60th percentile — it is just above the center hump of the Bell Curve (center hump = mean = median = mode = 50th percentile). 850 is the 90th percentile, way out on the arm of the Bell Curve. The height of the Bell Curve declines precipitously as we walk from X = 650 to X = 850. Suppose we walk halfway, out to X = 750 — the question is: of the slice of Bell Curve between 650 and 850, between the 60th percentile and the 90th percentile, is more than half before or after X = 750. Well, the curve is declining precipitously in this region, so the height of the curve is *much higher* before X = 750 than after X = 750. Another way to say it is: the Bell Curve is densest toward its center. Again, considering the slice between 650 and 850, more than half will be toward the center, to the left of X = 750. Therefore, the halfway percentile, the 75th percentile, has to be to the left of X = 750, in other words, has to have an X-value that's less than 750. Answer = A. The question is a deep *visualization* question.

(1) B; (2) D; (3) A 4. moved to left for better formatting.

Boxplots (Concept) is used for graphics representation. Min, Q1, Q2, Q3, Max Q2 is median Range = Max - Min Q1 = Median of Min and Q2 Q3 = Median of Q2 and Max 1Q = Min to Q1 2Q = Q1 to Q2 3Q = Q2 to Q3 4Q = Q4 to Max 1. Given a set = {2,4,7,9,4,5,9,4,9,2,11,2,3,4,3,4} Find all of the above. in ascending order: {2,2,2,3,3,4,4,4,4,4,5,7,9,9,9,11}

1. Min = 2 Max = 11 Median = 4 Q1 = 3, 25% percentile Q2 = 4, 50% percentile Q3 = 8, 75% percentile ---Following is not needed and not sure how it will work with odd number of set. 1Q = 2,2,2,3 2Q = 3,4,4,4 3Q = 4,4,5,7 4Q = 9,9,9,11

Concepts Mean (Average) - Properties of evenly spaced sets: a. Arithmetic mean and the median are the same b. Mean and the median is the average of the first and the last element. c. The sum is the mean multiplied by the number of items. 1. Find the missing number, given the average and the remaining numbers? Example find 'x' if the average of the numbers is 10 and the remaining numbers are 8, 7 , 9 , 8. Note: If the number are equally spaced then mean and the median is the same. If the number in the set are even then the mean and the median is the average of the two center numbers. 2. The mean of twenty-five consecutive positive integers numbers is what percent of the total? (A) 4% (B) 5% (C) 20% (D) 25% (E) Cannot be determined by the information provided. 3. On 4 sales, Matt received commissions of 300,40,x and 140. Without the x, his average commission would be 50 lower. What is x? 4. Matt gets a $1000 commission on a big sale. This commission alone raises his average commission by 150. If matt's new average commission is 400, how many sales has Matt made? 5. If X>0 and the range of 1,2,x,5,x^2 is 7, then what is the approximate average of the set? 6. Among the set {1,2,3,4,7,7,10,10,11,14,19,19,23,24,25,26} What is the ratio of the largest item in Q2 to the average value in the Q4? 7 Given X>2 Quantity A: The median of x-4,x+1,4x Quantity B: The mean of x-4,x+1, 4x 8. Quantity A: The number of multiples of 7 between 50 and 100 inclusive. Quantity B: The number of multiples of 9 between 30 and 90 inclusive. 9. If A is comprised of the following terms: 3x, 3x-4,3x-8,3x-12,3x-16,3x-20 Quantity A: The sum of all the terms in set A Quantity B 18x - 70 10. Column A: Sum of the multiples of 4 less than 100 Column B: Sum of the multiples of 5 less than 100 11. Taxi drives 113 miles at 50 mi/h and returns via the same route at 60 m/h A: Tax's average speed for the entire trip? B: 55m/h

1. 8+7+9+8 = 32, (32 + x) / 5 = 10, 32 + x = 50, x = 18 2. In the problem above we have 25 numbers (it doesn't really matter that they are consecutive). The mean will be 1/25 of the total, or (A) 4%. Method 2: 1+2+.....+25 = (25.26) / 2 = 325, x.325/100 = 13, therefore x = 4. Method 3: 13.100/13.25 = 4 . Ans is (A). 3. x=360 MH, book 5, page 95 4. 5 (MH, book 5, page 95) 5. Approximate 3 6. 20:49 (can't have ratio in decimal) 7. B (compare 1 vs x-1) 8. C 9. A Hint: the set is of numbers equally spaced. Use First and last element to calculate the mean and then multiply with the number of items in the set. 10. A Hint : evenly spaced and median. 11.Ans B Hint you have lot more 50's than 60's. Therefore it is close to 50. Another simple example: Distance 100 miles, traveling at 100m/h and returning at 1m/h. We get 200 miles in 101 hours and our average is less than 2m/h, much closer to the 1m/h.

Concepts Median (Middle) 1. A list is comprised of five positive integers: 4, 4, x, 7, y. What is the range of the possible values of the medians? (A) 2(B) 3(C) 6(D) 7(E) Cannot be determined by information provided. 2. The average of five positive integers is less than 20. What is the smallest possible median of this set? (A) 19(B) 10(C) 4(D) 3(E) 1 3. Set S is comprised of 37 integers. Quantity A: The median of set S Quantity B: The mean of the lowest and the highest term

1. Ans: 3 It doesn't matter if x or y take on a large or small value. The two possible medians are 7 and 4, therefore range is 3 2. Ans: 1 Solution: Since smallest given is one we can start with that. {1,1,1,1,1}, note that it doesn't say nowhere that you can't repeat a number. 3. Ans D Solution: Pick a set of all 1, this will make A and B equal. Now pick a set of 36 1's and the last number is 100. Now the answer is B. Therefore D

Standard Deviation (Concept) The Standard Deviation is a measure of how spread out numbers are. Mean, SD and Variance. Normal distribution: 1SD = 68% 2SD=95% 3SD = 99% 1. Calculate the mean, SD and Variance of the following: 5,3,7, 2 and 8 2) Consider the following sets: L = {3, 4, 5, 5, 6, 7} M = {2, 2, 2, 8, 8, 8} N = {15, 15, 15, 15, 15, 15} Rank those three sets from least standard deviation to greatest standard deviation. (A) L, M, N (B) M, L, N (C) M, N, L (D) N, L, M (E) N, M, L 3. Set A={1,2,3,4,5}, B = (440,442,443,444,445} A)Which set has the greatest SD? B)If each data point is increased by 7 does the SD change? C) if each point in the set is multiplied by 7, does the SD change? 4) Which set has the greatest SD? A={ 3,4,5,6,7} B={3,3,5,7,7} 5) View the Image A) -1 is at what percentile? B) 2 is at what percentile? C) only .1 or .2% is outside 3SD.

1. Mean = 25/5 = 5 Variance square the distance of each point from the mean and take average. Therefore Variance: 0^2 + 2^2 + (-2)^2 + 3^2 + (-3)^2 = 26, i.e. v = 26/5 = 5.2. SD = root (5.2) = 2.28. If normal distribution then 68% of the value are within 1SD etc and so on 2. L-> mean is 5 range is 4 M-> mean is 5 and range is 6 N -> mean is 15 and range is 0 Therefore the order is NLM, which is (D) 3. A) set B B) NOClick to zoom C) SD will change by a factor of 7 4. Note that we get the same range. Method 1: eliminate the common elements and we are left with 4,6 and 3,7. Set B has the greatest range, therefore it has higher SD. Method 2: Add up the difference: A= 2+1+1+2 = 6/5 B= 2+2+2+2 = 8/5 Therefore set B has higher SD 5) From attached image A) 50 -34 = 16% B) 50+34+13.6 = 97.6%