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## Statistics and Probability

### Basic Statistics - Median, Mean, Mode, Range

#### Q.01

'From a box containing 5 red balls, 3 white balls, and 2 blue balls, a trial of drawing one ball and observing its color before replacing it is repeated 5 times. Let X be the number of times a red ball or a white ball is drawn. Find the expected value E(X) and variance V(X) of X.'

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'What is the name for the average value of a variable x in a population called?'

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'Mean: A method of calculating the average of a group of numbers.'

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'How to represent population mean and population standard deviation?'

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"The following table shows the number of sales of women's clothing by size in two stores in a week. Find the mode for each size."

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'Basic Example 143 Frequency Distribution Table and Measures of Central Tendency\nThe table on the right shows the frequency distribution table of the number of lunch box sales in a shop over 30 days.\n1. The mode of the data is \\\square\.\n2. Using class values from the table, the mean of the data is \\\square\.\n3. When calculating the mean without using class values, the data\n\n| Class (units) | Frequency |\n|----------------|----------|\n| 100 and above, less than 120 | 3 |\n| 120 and above, less than 140 | 5 |\n| 140 and above, less than 160 | 11 |\n| 160 and above, less than 180 | 8 |\n| 180 and above, less than 200 | 3 |\n| Total | 30 |\n\nthe average falls into the range of more than \\\qquad\ units and less than \\\square\ units.'

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'The table on the right shows the data for variables x and y. (1) Calculate the interquartile range for the data of variables x and y respectively. Also, determine which data has a higher degree of variability. Compare based on the obtained interquartile ranges.'

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'Explain the measures of central tendency. Specifically, discuss the mean, mode, and median.'

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'Calculate the average of the following data: 2, 9, 6, -9, 1'

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'Explain quartile range and outliers, and provide examples.'

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'For example, when the data of variable x is 1, 2, 4, 4, 8, the mean is x = (1+2+4+4+8)/5 = 19/5 = 3.8, the variance is sx^2 = (1^2+2^2+4^2+4^2+8^2)/5-3.8^2 = 5.76, and the standard deviation is sx = sqrt(5.76) = 2.4. Let the variable y be the result of adding 3 to variable x, i.e., y = x + 3. The mean of y after the transformation y = x + 3 is y = ((1+3)+(2+3)+(4+3)+(4+3)+(8+3))/5 = (19+3*5)/5 = x + 3 = 6.8.'

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'The table on the right shows the frequency distribution of the minimum daily temperatures measured over 30 days in a certain city.'

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'The following data represents the commuting time of 8 high school students: 49, 52, 44, 50, 41, 43, 40, 49 (in minutes).'

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'Find the median of the following data: 8, 14, 22, 48, 97'

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'Translate the given question into multiple languages.'

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'When the data is arranged in ascending order, it becomes 5, 9, 10, 11, 12, 14, 15, 17, 20. The second quartile is Q₂ = 12, the first quartile is Q₁ = (9 + 10) / 2 = 9.5, and the third quartile is Q₃ = (15 + 17) / 2 = 16'

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'A class of 40 students took an exam, with 39 students taking it and 1 student absent. The average score of the 39 students who took the exam was 60, with a variance of 20. Later, the absent student took the same exam and scored 60. How will the average and variance of the 40 students, including this score, change compared to the values before recalculation? Choose one answer from the following options for each blank: (1) Remains the same (2) Increases (3) Decreases'

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'Please calculate the range, quartiles, interquartile range, first quartile range, third quartile range, and quartile deviation of the given dataset.'

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'Find the median of the following data: 11, 20, 20, 38, 39, 50, 51'

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'What is a representative value, and please explain its types.'

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'The table on the right shows the number of defective products x and y per hour for 5 hours each from two molding machines X and Y that can mold a certain product. (Unit is in pieces)\n\egin{tabular}{l||c|c|c|c|c}\n\\hline\ x \ & 5 & 4 & 8 & 12 & 6 \\\n\\hline\ y \ & 6 & 9 & 8 & 5 & 7 \\\n\\hline\n\\end{tabular}\n(1) Find the mean, variance, and standard deviation of x and y data. Round to the nearest second decimal place.\n(2) Compare the degree of dispersion from the mean of the data x and y based on their standard deviations.'

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Join our Discord#### Q.27

'The following data records the height growth of 6 first-year high school students over the course of a year: 20, 28, 19, 24, 21, 26 (in mm).'

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'When the data is arranged in order of magnitude, 46, 48, 49, 50, 50, 51, 51, 52, 52, 53, 54, 54, 55, 57, 58, the minimum value, first quartile, median, third quartile, and maximum value are 46, 50, 52, 54, 58'

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Join our Discord#### Q.31

"Calculate the interquartile range for both Player A and Player B's 15-year home run data, and determine which player has more dispersion in their data. Compare based on the obtained interquartile ranges."

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Join our Discord#### Q.33

'When the data is arranged in ascending order, the values are 20, 33, 40, 59, 60, 62, 64, 91, and since the total number of data points is 8, the median is the average of the 4th and 5th values.'

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'The following data represents the snowfall over a period of 10 days in cities A and B. City A: 3,10,8,25,7,2,12,35,5,18 (cm). City B: 5,20,16,34,10,3,12,52,6,23 (cm). (1) Calculate the first quartile, second quartile, and third quartile of the data from city A. (2) Determine the interquartile range and quartile deviation of the data from city A. (3) Which city, A or B, has a greater degree of variability in the data? Use the interquartile range to make a judgement.'

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'The table on the right is a frequency distribution table of the average daily temperatures in city A measured over a month. Calculate the mode for this data.'

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'The following data is the result of investigating how long it took 10 high school students to solve a problem: 10, 7, 9, 8, 9, 10, 12, 11, 13, 11 minutes. (1) Find the mean of this data. (2) There are errors in some of this data, the correct data is that out of 2 persons spending 9 minutes, one spent 10 minutes, 7 minutes, 12 minutes were actually 10 minutes, and 8 minutes. When these errors are corrected, will the mean, median, and variance of this data increase, decrease, or remain the same as before?'

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'Data Analysis (2) Problem related to data dispersion and quartiles: When the data is arranged in order of magnitude, find the following.'

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'The table on the right shows the frequency distribution table of the average daily temperature in city A measured over one month. Calculate the mode of this data.'

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"Let's consider the median. From the box plot of Test A, the median is 60 points. Also, since there are 30 data points, the median is the average of the 15th and 16th scores from the lowest. Explain why 60 points is the average of the 15th and 16th scores from the lowest."

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'The following data is about 8 high school students who challenged a maze at an amusement park, and it shows how many minutes it took them to successfully navigate through the maze.'

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'Explain the interquartile range of data. Describe how it is less affected when there are outliers in the data.'

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'Calculate the correlation coefficient for basic 154'

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"In a data set containing 85 observations. Which of the following statements about quartiles are true for any data set? (0) The mean lies between the first quartile and third quartile. (1) The interquartile range is larger than the standard deviation. (2) The number of observations smaller than the median is 49. (3) Removing one observation equal to the maximum does not change the first quartile. (4) When all observations smaller than the first quartile and larger than the third quartile are removed, the remaining number of observations is 51. (5) When all observations smaller than the first quartile and larger than the third quartile are removed, the range of the remaining data is equal to the original data's interquartile range."

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'The following data (1) represents the sleep times in minutes of 7 students on a Sunday.'

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'The following data examines how long it took 8 high school students to escape a maze in an amusement park: 7, 16, 11, 8, 12, 15, 10, 9 minutes. Find: (1) the average of this data. (2) There are some recording errors in this data, with 16 minutes being recorded as 15 minutes, 11 minutes as 8 minutes, and 9 minutes as 13 minutes. When these errors are corrected, will the average, median, and variance of the data increase, decrease, or remain the same? Provide the answer.'

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'List all the mathematical terms related to quartiles.'

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"The scores of a 50-point kanji 'reading' and 'writing' test for 10 students in a class are denoted as variables x and y, respectively. The figure on the right is a scatter plot of variables x and y. The data for variable x for the 10 students are as follows (in points): 13, 17, 20, 23, 28, 34, 36, 40, 44, 45. (1) Find the mean and median of the variable x data. (2) It is found that the variable y value of 13 points for the student with a variable x value of 40 points is incorrect, it is corrected to the correct value of 32 points. Find the median of the variable y data before and after correction. (3) In (2), let the correlation coefficient of x and y before correction be r1, and after correction be r2. Choose the correct pair of values (r1, r2) from the following (1) to (4). (1) (0.82, 0.98) (2) (0.98, 0.82) (3) (-0.98, -0.82)"

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'The histogram on the right shows the results of a survey of 25 students at a high school regarding the number of days they used the school bus in the past week.'

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'Explain the definitions of the representative values of data, and provide examples of when each is applicable. Mean, mode, median.'

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Join our Discord#### Q.51

'Calculate the average of the following data:\n6,8,22,18,2,6,11,0,17,7,2,14,8,11,4,8'

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'The following data represents the amount of snowfall in A city and B city over a period of 10 days:'

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'A population consists of two groups, A and B. When the data was aggregated, the number, average, and variance for each group were as shown in the table on the right. In this case, the overall mean of the population is _____, and the variance is _____.'

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Join our Discord#### Q.56

'The following data (1) represents the Sunday sleep duration of 7 students: (1): 410, 360, 440, 420, 390, 450, 400 (minutes). Compute the median of data (1). If the sleep durations of the next 3 students are added to data (1) to create data (2) as 420, 360, 430 (minutes), find the median of data (2).'

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'Find the quartiles of the following data: 5 cm, 9 cm, 18 cm'

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'The table on the right shows the frequency distribution table of test scores for 8 students. Assume that all scores are integers.'

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'The following data lists the monthly maximum temperatures of a city for a year.'

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'Given a data of size n with values x1, x2, ..., xn, the sum of these values divided by n is called the average of the data, denoted by x̅. For example: Find the average of the data 2, 3, 5, 6.'

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Join our Discord#### Q.63

'The following data is the results of a 20-point test for 10 students: 6, 5, 20, 11, 9, 8, 15, 12, 7, 17 (scores).'

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'Practice The following data is a list of monthly minimum temperatures for a city in a certain year.'

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Join our Discord#### Q.66

'179 range is 20 hours, interquartile range is 10 hours'

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'Problem regarding data dispersion and quartile range'

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'The following data are the results of a 10-point test for 5 students in Group A and 6 students in Group B. Group A: 5, 7, 8, 4, 9 Group B: 7, 10, 9, 4, 8, 6 (scores are in points)\n(1) Find the average of Group A and Group B data, rounding to 2 decimal places.\n(2) Find the average of the combined data of 11 students from Group A and Group B.\n(3) Find the median of the data for Group A and Group B respectively.'

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'Calculate the interquartile range for each dataset, and compare the degree of data spread using the interquartile range.'

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'Given a variable with data, having a mean of 50 and a standard deviation of 15. When modifying the data by increasing each of the 185 data points by 1.2 times and subtracting 5, determine the new mean and standard deviation.'

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Join our Discord#### Q.72

'Chapter 5 Data Analysis\n20 Data organization, Measures of central tendency\nProblem:\nFind the mean, median, and mode of the following dataset:\nDataset: 2, 4, 4, 5, 7, 9'

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'The following data shows the monthly minimum temperatures of a city for a certain year: (-12, -9, -3, 3, 10, 17, 20, 19, 15, 7, 1, -8) (in °C)'

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'For the given data of variable x, answer the following questions. 186, 514, 584, 598, 521, 605, 612, 577 (1) By setting y=x-570, find the mean value of the data of variable x. (2) By setting u=(x-570)/7, find the variance of the data of variable x.'

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Join our Discord#### Q.75

"Next, let's consider the impact on the mean and variance by adding one more data point. In CHECK 4-A, the number of test takers is only 39, so the effect of adding one more data point is significant, requiring a recalculation of the mean and variance. However, if the number of test takers is sufficiently large, then the impact of adding one more data point is very small, so recalculating the mean and variance would hardly result in any change."

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'The following data is about the commuting time of 8 high school students: 49, 52, 44, 50, 41, 43, 40, 49 (in minutes).'

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'The point representing the fifth smallest value in the vertical axis data is the point inside the dashed lines of (3) in Figure 2, and the value of the horizontal axis data of this point is less than 200,000 tons. According to the table in the problem statement, the average value of the horizontal axis (Tarra) data is 210,975 tons, which is greater than 200,000 tons. Therefore, it is correct.'

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"The following table shows the sales of women's clothing by size in two stores for one week. Find the mode for each set of data."

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'This is the amount of PET bottles waste collected at the garbage collection site in November. (1) Calculate the median and mean values. (2) It is known that one of the six numbers above is incorrect. According to the correct numbers, the median and mean values are 2.84 tons and 3.02 tons, respectively. Identify the incorrect number and find the correct number.'

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'(1) The interquartile range of data for x is 5, for y is 3; x has a greater degree of data dispersion (2) 1 item'

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'In general, the degree of data dispersion cannot be determined from the median or mode.'

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"Which graph is most appropriate for comparing an individual's results in multiple fitness tests with the average?"

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'For data A where 98 observations are 0 and the remaining 1 observation is 99, answer the following questions:\n1. What is the mean of data A?\n2. What is the interquartile range of data A?\n3. What is the standard deviation of data A?'

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'The table on the right shows the frequency distribution table of the daily minimum temperatures measured over 30 days in a certain city.'

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'The median is one of the possible values that represents the middle value of the data.'

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'The given data is the scores of 10 students in an English test at a school. Here, the value of a is a non-negative integer.'

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'The points within the dashed lines in Figure 2 (1) have become the minimum values for both the horizontal and vertical axes of the data. This means that the lowest catch was recorded in that year, which is correct.'

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"Which page should I refer to when I don't understand standard sample questions?"

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'Explain how to find the maximum and minimum values.'

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'71 (1) A: 53 books, B: 51 books; The dispersion of player A is larger. (2) A: Q1=38 books, Q2=44 books, Q3=48 books B: Q1=22 books, Q2=32 books, Q3=41 books (3) A: 10 books in order, 5 books; B: 19 books in order, 9.5 books; The dispersion of player B is larger.'

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"Example 70 | Calculation of Mean and Median\nTeam A: \ 4, 7, 8, 6, 9 \\nTeam B: \ 6, 9, 10, 5, 7, 6 \ (scores are in points)\n(1) Calculate the mean of Team A's data and the mean of Team B's data separately. Round to the nearest hundredth.\n(2) Calculate the mean of the combined data of the 11 individuals from Team A and Team B.\n(3) Calculate the median of Team A's data and the median of Team B's data separately."

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'Please tell me about the representative values of the data.'

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'Practice The table on the right is a frequency distribution table of test scores for 8 students. Assume that all scores are integers.'

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'The following data is the scores of a game played by 6 people. Where a is a positive integer. 138, 79, 123, 185, 151, a (in points)\n(1) Investigate how the value of a affects the median.\n(2) When the value of a is unknown, how many possible values can the median of this data have?'

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Join our Discord#### Q.95

"The data will have the minimum average when each value is the minimum value of each class, so \\[\\frac{1}{8}(20 \\times 1+40 \\times 5+60 \\times 2)=\\frac{340}{8}=42.5\\] The data's average will be maximum when each value is the maximum value of each class, so \\[\\frac{1}{8}(39 \\times 1+59 \\times 5+79 \\times 2)=\\frac{492}{8}=61.5\\] Therefore, it will be between 42.5 points and 61.5 points. Another solution [the same as finding the minimum value of the data's average], the data's average will be maximum when each value is 19 points higher than the minimum value, so the average score will also be 19 points higher, \42.5+19=61.5\ Therefore, it will be between 42.5 points and 61.5 points."

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'For the data 1, 3, 4, 10, 12, answer the following questions:\n(1) Calculate the mean of this data.\n(2) Calculate the variance of this data using the following two methods:\n (a) By calculating the mean of the squares of the deviations.\n (b) Using the mean of the squares of each value in the data.\n(3) Calculate the standard deviation of this data. Round your answer to two decimal places.'

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'Calculate the following statistics for the given dataset {2, 9, 2, -9, 1, -5, 6, 1, 2, -3}: 1. Mean 2. Median 3. Mode'

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'To deepen understanding of data organization and measures of central tendency, calculate the mean and median of the following dataset: [5, 8, 6, 9, 7]'

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'For the dataset {1, 3, 4, 10, 12}, calculate the following statistics:\n1. Mean\n2. Variance\n3. Standard Deviation'

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'When A to D change, how do the maximum, minimum values of the data change?'