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Center and Variability

Grade 6 · Measurement · Worksheet 3

  1. Mason surveyed his classmates to find how many books each had read over the summer. He recorded the data and displayed it in a stem-and-leaf plot: Stem | Leaf 2 | 2 7 3 | 2 2 7 7 4 | 2 2 2 7 7 7 5 | 2 7 Key: 2 | 2 means 22 books. Find the mean, median, mode, range, and interquartile range (IQR) of this data set. Answer: ______________
  2. Data: 1120, 1340, 1560, 1120, 1480. Find the mean, median, mode, and range. Answer: ______________
  3. Liam is analyzing the test scores of his math class. The scores are: 78, 85, 92, 85, 96, 88, 85, and 90. What is the mode of the test scores? Answer: ______________
  4. √(144 + 25) = ? Answer: ______________
  5. Emma is studying the growth of her bean plants. She measured the height (in cm) of each plant in her garden and recorded the data in a stem-and-leaf plot: Stem | Leaf 2 | 0, 5 3 | 0, 0, 5, 5 4 | 0, 0, 0, 5, 5, 5 5 | 0, 0, 5 Key: 2 | 0 = 20 cm Calculate the mean, median, mode, and range of the plant heights. Answer: ______________
  6. The mean of 12, 15, 18, and x is 16. x = ? Answer: ______________
  7. The mean of 24, 32, 41, and x is 35. x = ? Answer: ______________
  8. Data: 1026, 1044, 1032, 1068, 1026, 1050. Find the mean, median, mode, and range. Answer: ______________
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Answer Key & Explanations

Center and Variability · Grade 6 · Worksheet 3

  1. Mason surveyed his classmates to find how many books each had read over the summer. He recorded the data and displayed it in a stem-and-leaf plot: Stem | Leaf 2 | 2 7 3 | 2 2 7 7 4 | 2 2 2 7 7 7 5 | 2 7 Key: 2 | 2 means 22 books. Find the mean, median, mode, range, and interquartile range (IQR) of this data set. Answer: Mean: 37, Median: 37, Mode: 42 and 47, Range: 35, IQR: 10 Solution: List all data values from the stem-and-leaf plot: 22, 27, 32, 32, 37, 37, 42, 42, 42, 47, 47, 47, 52, 57. Order the data (already in order): 22, 27, 32, 32, 37, 37, 42, 42, 42, 47, 47, 47, 52, 57. There are 14 values.
    Full step-by-step solution

    Step 1: List all data values from the stem-and-leaf plot: 22, 27, 32, 32, 37, 37, 42, 42, 42, 47, 47, 47, 52, 57. Step 2: Order the data (already in order): 22, 27, 32, 32, 37, 37, 42, 42, 42, 47, 47, 47, 52, 57. There are 14 values. Step 3: Calculate the mean. Sum = 22+27+32+32+37+37+42+42+42+47+47+47+52+57 = 518. Mean = 518 / 14 = 37. Step 4: Find the median. Since there are 14 values (even), median is the average of the 7th and 8th values. 7th = 42, 8th = 42. Median = (42+42)/2 = 42. Step 5: Find the mode. The numbers 42 and 47 each appear 3 times, which is more than any other number. Modes: 42 and 47. Step 6: Find the range. Largest = 57, smallest = 22. Range = 57 - 22 = 35. Step 7: Find the IQR. Lower half (first 7 values): 22, 27, 32, 32, 37, 37, 42. Median of lower half (Q1) = 32. Upper half (last 7 values): 42, 42, 47, 47, 47, 52, 57. Median of upper half (Q3) = 47. IQR = Q3 - Q1 = 47 - 32 = 15. The answer is Mean: 37, Median: 42, Mode: 42 and 47, Range: 35, IQR: 15.

  2. Data: 1120, 1340, 1560, 1120, 1480. Find the mean, median, mode, and range. Answer: Mean = 1324, Median = 1340, Mode = 1120, Range = 440 Solution: List the data: 1120, 1340, 1560, 1120, 1480. Find the mean: Sum = 1120 + 1340 + 1560 + 1120 + 1480 = 6620. Number of values = 5.
    Full step-by-step solution

    Step 1: List the data: 1120, 1340, 1560, 1120, 1480. Step 2: Find the mean: Sum = 1120 + 1340 + 1560 + 1120 + 1480 = 6620. Number of values = 5. Mean = 6620 ÷ 5 = 1324. Step 3: Find the median: Order the numbers: 1120, 1120, 1340, 1480, 1560. The middle value (3rd) is 1340. Step 4: Find the mode: 1120 appears twice, more than any other number. Mode = 1120. Step 5: Find the range: Largest = 1560, smallest = 1120. Range = 1560 - 1120 = 440. The answer is Mean = 1324, Median = 1340, Mode = 1120, Range = 440.

  3. Liam is analyzing the test scores of his math class. The scores are: 78, 85, 92, 85, 96, 88, 85, and 90. What is the mode of the test scores? Answer: 85 Solution: List all the test scores. We have: 78, 85, 92, 85, 96, 88, 85, 90. Understand what the mode is.
    Full step-by-step solution

    Step 1: List all the test scores. We have: 78, 85, 92, 85, 96, 88, 85, 90. Step 2: Understand what the mode is. The mode is the number that appears most frequently in the data set. Step 3: Count how many times each score appears. - 78 appears 1 time - 85 appears 3 times - 92 appears 1 time - 96 appears 1 time - 88 appears 1 time - 90 appears 1 time Step 4: Compare the frequencies. The score 85 appears 3 times, which is more than any other score. Step 5: Conclusion. Since 85 occurs most often, the mode is 85.

  4. √(144 + 25) = ? Answer: 13 Solution: Look at the expression inside the square root: 144 + 25. Add the two numbers together. 144 + 25 = 169 The problem now becomes the square root of the result from Step 2.
    Full step-by-step solution

    Step 1: Look at the expression inside the square root: 144 + 25. Step 2: Add the two numbers together. 144 + 25 = 169 Step 3: The problem now becomes the square root of the result from Step 2. So we need to calculate the square root of 169. Step 4: Recall that finding the square root means finding a number which, when multiplied by itself, gives the number inside the root. We ask: "What number multiplied by itself equals 169?" Step 5: We know that 12 * 12 = 144, which is too small. We also know that 13 * 13 = 169. Step 6: Since 13 * 13 = 169, the square root of 169 is 13. Therefore, the final answer is 13.

  5. Emma is studying the growth of her bean plants. She measured the height (in cm) of each plant in her garden and recorded the data in a stem-and-leaf plot: Stem | Leaf 2 | 0, 5 3 | 0, 0, 5, 5 4 | 0, 0, 0, 5, 5, 5 5 | 0, 0, 5 Key: 2 | 0 = 20 cm Calculate the mean, median, mode, and range of the plant heights. Answer: Mean = 39 cm, Median = 40 cm, Mode = 40 cm and 45 cm, Range = 35 cm Solution: List all data values from the stem-and-leaf plot. Stem 2: 20, 25 Stem 3: 30, 30, 35, 35 Stem 4: 40, 40, 40, 45, 45, 45 Stem 5: 50, 50, 55 Total number of plants = 2 + 4 + 6 + 3 = 15 Calculate the mean.
    Full step-by-step solution

    Step 1: List all data values from the stem-and-leaf plot. Stem 2: 20, 25 Stem 3: 30, 30, 35, 35 Stem 4: 40, 40, 40, 45, 45, 45 Stem 5: 50, 50, 55 Total number of plants = 2 + 4 + 6 + 3 = 15 Step 2: Calculate the mean. Sum = 20 + 25 + 30 + 30 + 35 + 35 + 40 + 40 + 40 + 45 + 45 + 45 + 50 + 50 + 55 Sum = 20 + 25 = 45 45 + 30 = 75 75 + 30 = 105 105 + 35 = 140 140 + 35 = 175 175 + 40 = 215 215 + 40 = 255 255 + 40 = 295 295 + 45 = 340 340 + 45 = 385 385 + 45 = 430 430 + 50 = 480 480 + 50 = 530 530 + 55 = 585 Sum = 585 Mean = 585 / 15 = 39 cm Step 3: Calculate the median. Data in order: 20, 25, 30, 30, 35, 35, 40, 40, 40, 45, 45, 45, 50, 50, 55 Since there are 15 values, the median is the 8th value (the middle). Counting: 1st=20, 2nd=25, 3rd=30, 4th=30, 5th=35, 6th=35, 7th=40, 8th=40 Median = 40 cm Step 4: Calculate the mode. Count each value: 20: 1 time 25: 1 time 30: 2 times 35: 2 times 40: 3 times 45: 3 times 50: 2 times 55: 1 time The values 40 and 45 both appear 3 times, which is the most frequent. Mode = 40 cm and 45 cm (bimodal) Step 5: Calculate the range. Highest value = 55 cm Lowest value = 20 cm Range = 55 - 20 = 35 cm The answer is: Mean = 39 cm, Median = 40 cm, Mode = 40 cm and 45 cm, Range = 35 cm.

  6. The mean of 12, 15, 18, and x is 16. x = ? Answer: 19 Solution: We are told the mean of the numbers 12, 15, 18, and x is 16. The mean is the sum of the numbers divided by how many numbers there are. Write the mean formula.
    Full step-by-step solution

    Step 1: Understand the problem. We are told the mean of the numbers 12, 15, 18, and x is 16. The mean is the sum of the numbers divided by how many numbers there are. Step 2: Write the mean formula. Mean = (sum of all numbers) / (number of numbers) Here: (12 + 15 + 18 + x) / 4 = 16 Step 3: Multiply both sides by 4 to remove the denominator. 12 + 15 + 18 + x = 16 * 4 Step 4: Calculate the known values. 12 + 15 + 18 = 45 16 * 4 = 64 So: 45 + x = 64 Step 5: Solve for x. x = 64 - 45 x = 19 Step 6: Check the answer. Numbers: 12, 15, 18, 19 Sum = 12 + 15 + 18 + 19 = 64 Mean = 64 / 4 = 16, which matches the problem. Final answer: x = 19

  7. The mean of 24, 32, 41, and x is 35. x = ? Answer: 43 Solution: Write the equation for the mean: (24 + 32 + 41 + x) ÷ 4 = 35 Multiply both sides by 4: 24 + 32 + 41 + x = 140 Add the known numbers: 24 + 32 + 41 = 97 Substitute: 97 + x = 140 Subtract 97 from both sides: x = 140 - 97 Calculate: x = 43 The answer is 43.
    Full step-by-step solution

    Step 1: Write the equation for the mean: (24 + 32 + 41 + x) ÷ 4 = 35 Step 2: Multiply both sides by 4: 24 + 32 + 41 + x = 140 Step 3: Add the known numbers: 24 + 32 + 41 = 97 Step 4: Substitute: 97 + x = 140 Step 5: Subtract 97 from both sides: x = 140 - 97 Step 6: Calculate: x = 43 The answer is 43.

  8. Data: 1026, 1044, 1032, 1068, 1026, 1050. Find the mean, median, mode, and range. Answer: Mean = 1041, Median = 1038, Mode = 1026, Range = 42 Solution: Order the data: 1026, 1026, 1032, 1044, 1050, 1068 Mean = (1026 + 1044 + 1032 + 1068 + 1026 + 1050) / 6 = 6246 / 6 = 1041 Median: Since there are 6 numbers (even), median = (1032 + 1044) / 2 = 2076 / 2 = 1038 Mode: 1026 appears twice, all others appear once, so mode = 1026 Range = 1068 - 1026 =…
    Full step-by-step solution

    Step 1: Order the data: 1026, 1026, 1032, 1044, 1050, 1068 Step 2: Mean = (1026 + 1044 + 1032 + 1068 + 1026 + 1050) / 6 = 6246 / 6 = 1041 Step 3: Median: Since there are 6 numbers (even), median = (1032 + 1044) / 2 = 2076 / 2 = 1038 Step 4: Mode: 1026 appears twice, all others appear once, so mode = 1026 Step 5: Range = 1068 - 1026 = 42 The answer is Mean = 1041, Median = 1038, Mode = 1026, Range = 42.