Center and Variability

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Center and Variability · Grade 6 · Worksheet 1

1. Matiu is analyzing the heights (in centimeters) of 12 different plants in his garden, recorded as follows: 24, 28, 28, 30, 32, 32, 32, 36, 38, 40, 42, 46. Find the mean, median, mode, range, and interquartile range (IQR) of this data set. Answer: Mean: 34, Median: 32, Mode: 32, Range: 22, IQR: 12 Solution: Find the mean. Sum = 24 + 28 + 28 + 30 + 32 + 32 + 32 + 36 + 38 + 40 + 42 + 46 = 408. Count = 12.
   Full step-by-step solution

   Step 1: Find the mean. Sum = 24 + 28 + 28 + 30 + 32 + 32 + 32 + 36 + 38 + 40 + 42 + 46 = 408. Count = 12. Mean = 408 / 12 = 34. Step 2: Find the median. Since there are 12 numbers (even), the median is the average of the 6th and 7th numbers. 6th = 32, 7th = 32. Median = (32 + 32) / 2 = 32. Step 3: Find the mode. The number 32 appears 3 times, more than any other. Mode = 32. Step 4: Find the range. Largest = 46, smallest = 24. Range = 46 - 24 = 22. Step 5: Find the IQR. Split the data into two halves (6 numbers each). Lower half: 24, 28, 28, 30, 32, 32. Median of lower half (Q1) = average of 3rd and 4th values = (28 + 30) / 2 = 29. Upper half: 32, 36, 38, 40, 42, 46. Median of upper half (Q3) = average of 3rd and 4th values = (38 + 40) / 2 = 39. IQR = Q3 - Q1 = 39 - 29 = 10. The answer is Mean: 34, Median: 32, Mode: 32, Range: 22, IQR: 10.

2. Data: 1024, 1088, 1024, 1152, 1120. Find the mean, median, mode, and range. Answer: Mean = 1081.6, Median = 1088, Mode = 1024, Range = 128 Solution: Arrange the data in order: 1024, 1024, 1088, 1120, 1152 Mean = (1024 + 1088 + 1024 + 1152 + 1120) / 5 = 5408 / 5 = 1081.6 Median = the middle number in the ordered list = 1088 Mode = the number that appears most often = 1024 (appears twice) Range = largest - smallest = 1152 - 1024 = 128 The…
   Full step-by-step solution

   Step 1: Arrange the data in order: 1024, 1024, 1088, 1120, 1152 Step 2: Mean = (1024 + 1088 + 1024 + 1152 + 1120) / 5 = 5408 / 5 = 1081.6 Step 3: Median = the middle number in the ordered list = 1088 Step 4: Mode = the number that appears most often = 1024 (appears twice) Step 5: Range = largest - smallest = 1152 - 1024 = 128 The answer is Mean = 1081.6, Median = 1088, Mode = 1024, Range = 128.

3. Hana recorded the number of books she read each month over the past year. The data is shown in the dot plot below: Months: 1 2 3 4 5 6 7 8 9 10 11 12 Books read: 4, 6, 6, 8, 8, 8, 10, 10, 12, 14, 14, 18 What is the range and the interquartile range (IQR) of the number of books Hana read per month? Answer: Range = 14, IQR = 6 Solution: The data is already in order: 4, 6, 6, 8, 8, 8, 10, 10, 12, 14, 14, 18 Find the range: highest value = 18, lowest value = 4. Range = 18 - 4 = 14. Find the median of the entire data set.
   Full step-by-step solution

   Step 1: The data is already in order: 4, 6, 6, 8, 8, 8, 10, 10, 12, 14, 14, 18 Step 2: Find the range: highest value = 18, lowest value = 4. Range = 18 - 4 = 14. Step 3: Find the median of the entire data set. There are 12 values, so the median is the average of the 6th and 7th values. 6th value = 8, 7th value = 10. Median = (8 + 10)/2 = 9. Step 4: Split the data into lower half (first 6 values): 4, 6, 6, 8, 8, 8. Upper half (last 6 values): 10, 10, 12, 14, 14, 18. Step 5: Find the median of the lower half (Q1). Lower half has 6 values, so Q1 is the average of the 3rd and 4th values: (6 + 8)/2 = 7. Step 6: Find the median of the upper half (Q3). Upper half has 6 values, so Q3 is the average of the 3rd and 4th values: (12 + 14)/2 = 13. Step 7: IQR = Q3 - Q1 = 13 - 7 = 6. The answer is Range = 14, IQR = 6.

4. A rectangular city park is drawn on a coordinate plane with corners at (0, 0), (1200, 0), (1200, 800), and (0, 800). A circular fountain with diameter 200 units is placed exactly in the center of the park. What percentage of the park's area is occupied by the fountain? (Use π = 3.14) Answer: 3.27 Solution: Park length = 1200 units, park width = 800 units Park area = length × width = 1200 × 800 = 960,000 square units Fountain diameter = 200 units, so radius = 200 ÷ 2 = 100 units Fountain area = π × radius² = 3.14 × 100² = 3.14 × 10,000 = 31,400 square units Find what percentage the fountain area is…
   Full step-by-step solution

   Step 1: Find the area of the rectangular park Park length = 1200 units, park width = 800 units Park area = length × width = 1200 × 800 = 960,000 square units Step 2: Find the area of the circular fountain Fountain diameter = 200 units, so radius = 200 ÷ 2 = 100 units Fountain area = π × radius² = 3.14 × 100² = 3.14 × 10,000 = 31,400 square units Step 3: Find what percentage the fountain area is of the park area Fraction = fountain area ÷ park area = 31,400 ÷ 960,000 = 0.0327083... Step 4: Convert to percentage Percentage = 0.0327083... × 100 = 3.27083...% Step 5: Round to 2 decimal places 3.27083...% rounds to 3.27% The answer is 3.27.

5. The mean of 124, 138, 115, and x is 130. x = ? Answer: 143 Solution: Write the equation for the mean: (124 + 138 + 115 + x) ÷ 4 = 130 Multiply both sides by 4: 124 + 138 + 115 + x = 130 × 4 Calculate 130 × 4 = 520 Add the known numbers: 124 + 138 + 115 = 377 Substitute into the equation: 377 + x = 520 Solve for x: x = 520 - 377 Calculate: 520 - 377 = 143 The…
   Full step-by-step solution

   Step 1: Write the equation for the mean: (124 + 138 + 115 + x) ÷ 4 = 130 Step 2: Multiply both sides by 4: 124 + 138 + 115 + x = 130 × 4 Step 3: Calculate 130 × 4 = 520 Step 4: Add the known numbers: 124 + 138 + 115 = 377 Step 5: Substitute into the equation: 377 + x = 520 Step 6: Solve for x: x = 520 - 377 Step 7: Calculate: 520 - 377 = 143 The answer is 143.

6. Liam surveyed his classmates about their weekly reading time and recorded these results in hours: 3, 5, 7, 7, 8, 10, 12. What is the mean number of hours his classmates spend reading each week? Answer: 7.42857142857 Solution: To find the mean number of hours, we need to calculate the average of all the recorded values. The mean is found by adding all the numbers together, then dividing by how many numbers there are. 3, 5, 7, 7, 8, 10, 12 There are 7 numbers.
   Full step-by-step solution

   To find the mean number of hours, we need to calculate the average of all the recorded values. The mean is found by adding all the numbers together, then dividing by how many numbers there are. Step 1: List the data values 3, 5, 7, 7, 8, 10, 12 Step 2: Count how many numbers there are There are 7 numbers. Step 3: Add all the numbers together 3 + 5 = 8 8 + 7 = 15 15 + 7 = 22 22 + 8 = 30 30 + 10 = 40 40 + 12 = 52 So the total sum is 52. Step 4: Divide the sum by the count Mean = Total sum / Count Mean = 52 / 7 Step 5: Perform the division 52 ÷ 7 = 7.42857142857... This is a repeating decimal, but we can write it as 7.42857142857. Therefore, the mean number of hours spent reading each week is 7.42857142857.

7. Liam is analyzing the test scores of his classmates. The scores were: 78, 85, 92, 85, 96, 78, 88, 85, 90, and 78. He wants to find which measure of center best represents the typical score. Calculate the mean, median, and mode of the data set. Then, explain which measure you would choose to represent the typical score and why. Answer: Mean: 85.5, Median: 85, Mode: 78 and 85. The median of 85 is the best measure because the data has multiple modes and some very high and low scores that affect the mean. Solution: The mean is the sum of all scores divided by the number of scores. Sum = 78 + 78 + 78 + 85 + 85 + 85 + 88 + 90 + 92 + 96 = 78 × 3 = 234 85 × 3 = 255 234 + 255 = 489 489 + 88 = 577 577 + 90 = 667 667 + 92 = 759 759 + 96 = 855 Number of scores = 10 Mean = 855 / 10 = 85.5 The median is the middle…
   Full step-by-step solution

   First, let's list the test scores in order to make calculations easier: 78, 78, 78, 85, 85, 85, 88, 90, 92, 96. --- **Step 1: Calculate the mean** The mean is the sum of all scores divided by the number of scores. Sum = 78 + 78 + 78 + 85 + 85 + 85 + 88 + 90 + 92 + 96 = 78 × 3 = 234 85 × 3 = 255 234 + 255 = 489 489 + 88 = 577 577 + 90 = 667 667 + 92 = 759 759 + 96 = 855 Number of scores = 10 Mean = 855 / 10 = 85.5 --- **Step 2: Find the median** The median is the middle value when the data is arranged in order. Ordered data: 78, 78, 78, 85, 85, 85, 88, 90, 92, 96 Since there are 10 scores (even number), the median is the average of the 5th and 6th values. 5th value = 85 6th value = 85 Median = (85 + 85) / 2 = 170 / 2 = 85 --- **Step 3: Find the mode** The mode is the number(s) that appear most frequently. 78 appears 3 times 85 appears 3 times 88 appears 1 time 90 appears 1 time 92 appears 1 time 96 appears 1 time Both 78 and 85 occur 3 times, so the data is bimodal: modes are 78 and 85. --- **Step 4: Choose the best measure of center** We have: Mean = 85.5 Median = 85 Mode = 78 and 85 The mean is slightly pulled up by the high scores (92, 96) and pulled down by the low scores (78). The mode gives two different values (78 and 85), and 78 is at the lower end of the data, so it doesn’t represent the “typical” score well. The median is not affected by the extreme low or high values and sits in the middle of the data cluster. Thus, the median of 85 best represents the typical score because the data has multiple modes and some very high and low scores that affect the mean. --- **Final Answer:** Mean: 85.5, Median: 85, Mode: 78 and 85. The median of 85 is the best measure because the data has multiple modes and some very high and low scores that affect the mean.

8. Data: 1015, 1030, 1045, 1060, 1075, 1090. Find the mean and median. Answer: Mean = 1052.5, Median = 1052.5 Solution: List the data in order: 1015, 1030, 1045, 1060, 1075, 1090. Find the mean. Sum = 1015 + 1030 + 1045 + 1060 + 1075 + 1090 = 6315.
   Full step-by-step solution

   Step 1: List the data in order: 1015, 1030, 1045, 1060, 1075, 1090. Step 2: Find the mean. Sum = 1015 + 1030 + 1045 + 1060 + 1075 + 1090 = 6315. Number of values = 6. Mean = 6315 ÷ 6 = 1052.5. Step 3: Find the median. Since there are 6 values (even), the median is the average of the 3rd and 4th values: (1045 + 1060) ÷ 2 = 2105 ÷ 2 = 1052.5. The answer is Mean = 1052.5, Median = 1052.5.