Center and Variability

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Answer Key & Explanations

Center and Variability · Grade 6 · Worksheet 2

1. Noah is analyzing the monthly rainfall data for his city over the past year. The rainfall amounts in millimeters were: 45, 52, 38, 67, 52, 59, 41, 52, 63, 48, 55, and 42. He wants to find which measure of center best represents the typical monthly rainfall. Calculate the mean, median, and mode of the data set. Then, explain which measure you would choose to represent the typical rainfall and why. Answer: The mean is 50.8 mm, the median is 51.5 mm, and the mode is 52 mm. I would choose the median to represent the typical monthly rainfall because it is not affected by the unusually high value of 67 mm, which makes the mean slightly higher than most data points. Solution: Add all the rainfall amounts: 45 + 52 + 38 + 67 + 52 + 59 + 41 + 52 + 63 + 48 + 55 + 42 = 610. Divide by the number of months (12): 610 ÷ 12 = 50.833... which rounds to 50.8 mm.
   Full step-by-step solution

   Step 1: First, let's calculate the mean (average). Add all the rainfall amounts: 45 + 52 + 38 + 67 + 52 + 59 + 41 + 52 + 63 + 48 + 55 + 42 = 610. Divide by the number of months (12): 610 ÷ 12 = 50.833... which rounds to 50.8 mm. Step 2: Now find the median. First, arrange the data in order: 38, 41, 42, 45, 48, 52, 52, 52, 55, 59, 63, 67. Since there are 12 values (even number), the median is the average of the 6th and 7th values: (52 + 52) ÷ 2 = 52 ÷ 2 = 52 mm. Step 3: Find the mode. The value that appears most frequently is 52 mm (appears 3 times). Step 4: Compare the measures. The mean is 50.8 mm, median is 52 mm, and mode is 52 mm. The value 67 mm is higher than most other values, which pulls the mean upward slightly. The median and mode are very close and represent the center of the data well. Since the median is not affected by extreme values, it would be the best choice to represent typical monthly rainfall. Final answer: The mean is 50.8 mm, the median is 51.5 mm, and the mode is 52 mm. I would choose the median to represent the typical monthly rainfall because it is not affected by the unusually high value of 67 mm, which makes the mean slightly higher than most data points.

2. Liam is analyzing the test scores of his math class. The scores are: 78, 85, 92, 85, 96, 88, 85, and 90. He wants to understand the data by finding the mean, median, and mode. What is the mode of the test scores? Answer: 85 Solution: The scores are: 78, 85, 92, 85, 96, 88, 85, 90. Arrange the scores in order (not strictly necessary for mode, but helps for checking). Ordered scores: 78, 85, 85, 85, 88, 90, 92, 96.
   Full step-by-step solution

   First, let's list the test scores in order to make it easier to count how many times each score appears. The scores are: 78, 85, 92, 85, 96, 88, 85, 90. Step 1: Arrange the scores in order (not strictly necessary for mode, but helps for checking). Ordered scores: 78, 85, 85, 85, 88, 90, 92, 96. Step 2: Count the frequency of each score. - 78 appears 1 time - 85 appears 3 times - 88 appears 1 time - 90 appears 1 time - 92 appears 1 time - 96 appears 1 time Step 3: Identify the mode. The mode is the number that appears most frequently in the data set. Here, 85 appears 3 times, which is more than any other score. Step 4: Conclusion. Therefore, the mode of the test scores is 85.

3. Data: 1015, 1025, 1035, 1045, 1055. Find the mean and median. Answer: Mean = 1035, Median = 1035 Solution: Find the mean. Add all numbers: 1015 + 1025 + 1035 + 1045 + 1055 = 5175. Divide by 5: 5175 ÷ 5 = 1035.
   Full step-by-step solution

   Step 1: Find the mean. Add all numbers: 1015 + 1025 + 1035 + 1045 + 1055 = 5175. Divide by 5: 5175 ÷ 5 = 1035. Step 2: Find the median. The numbers are already in order: 1015, 1025, 1035, 1045, 1055. The middle number is 1035. The answer is Mean = 1035, Median = 1035.

4. Noah is analyzing the monthly rainfall data for his city over the past year. The rainfall amounts in millimeters were: 45, 52, 38, 29, 41, 67, 82, 75, 58, 43, 36, 48. He wants to find both the median rainfall amount and the range of rainfall amounts for the year. What are the median and range? Answer: median: 46.5 mm, range: 53 mm Solution: Arrange the rainfall data in order from smallest to largest: 29, 36, 38, 41, 43, 45, 48, 52, 58, 67, 75, 82 Find the median.
   Full step-by-step solution

   Step 1: Arrange the rainfall data in order from smallest to largest: 29, 36, 38, 41, 43, 45, 48, 52, 58, 67, 75, 82 Step 2: Find the median. Since there are 12 numbers (even count), the median is the average of the 6th and 7th numbers: (45 + 48) ÷ 2 = 93 ÷ 2 = 46.5 mm Step 3: Find the range by subtracting the smallest value from the largest value: 82 - 29 = 53 mm Step 4: The median rainfall is 46.5 mm and the range is 53 mm.

5. Data: 1017, 1035, 1017, 1053, 1071, 1029. Find the mean, median, mode, and range. Answer: Mean = 1037, Median = 1032, Mode = 1017, Range = 54 Solution: Sort the data: 1017, 1017, 1029, 1035, 1053, 1071 Mean = (1017 + 1035 + 1017 + 1053 + 1071 + 1029) / 6 = 6222 / 6 = 1037 Median: There are 6 numbers, so median is average of 3rd and 4th values: (1029 + 1035) / 2 = 2064 / 2 = 1032 Mode: 1017 appears twice, all others once, so mode = 1017 Range =…
   Full step-by-step solution

   Step 1: Sort the data: 1017, 1017, 1029, 1035, 1053, 1071 Step 2: Mean = (1017 + 1035 + 1017 + 1053 + 1071 + 1029) / 6 = 6222 / 6 = 1037 Step 3: Median: There are 6 numbers, so median is average of 3rd and 4th values: (1029 + 1035) / 2 = 2064 / 2 = 1032 Step 4: Mode: 1017 appears twice, all others once, so mode = 1017 Step 5: Range = 1071 - 1017 = 54 The answer is Mean = 1037, Median = 1032, Mode = 1017, Range = 54.

6. Liam is analyzing the monthly rainfall data for his city over the past year. The rainfall amounts in millimeters were: 45, 52, 38, 29, 41, 67, 82, 75, 58, 43, 39, 48. He wants to understand both the central tendency and variability of the rainfall. What is the median rainfall amount, and what is the range of the rainfall data? Answer: 46.5 mm and 53 mm Solution: Arrange the rainfall amounts in order from smallest to largest: 29, 38, 39, 41, 43, 45, 48, 52, 58, 67, 75, 82 Find the median.
   Full step-by-step solution

   Step 1: Arrange the rainfall amounts in order from smallest to largest: 29, 38, 39, 41, 43, 45, 48, 52, 58, 67, 75, 82 Step 2: Find the median. Since there are 12 values (an even number), the median is the average of the 6th and 7th values: (45 + 48) ÷ 2 = 93 ÷ 2 = 46.5 mm Step 3: Find the range. The highest value is 82 mm and the lowest value is 29 mm: 82 - 29 = 53 mm Step 4: State both answers: The median rainfall is 46.5 mm and the range is 53 mm.

7. The mean of 1245, 1380, 1520, and x is 1450. x = ? Answer: 1655 Solution: The mean is calculated by summing all values and dividing by the number of values. There are 4 values total (1245, 1380, 1520, and x).
   Full step-by-step solution

   Step 1: The mean is calculated by summing all values and dividing by the number of values. Step 2: There are 4 values total (1245, 1380, 1520, and x). Step 3: Multiply the mean (1450) by the number of values (4): 1450 × 4 = 5800 Step 4: Calculate the sum of the known values: 1245 + 1380 + 1520 = 4145 Step 5: Subtract the sum of known values from the total: 5800 - 4145 = 1655 Step 6: Therefore, x = 1655

8. Find the mean of: 12, 15, 18, 21, 24 Answer: 18 Solution: To find the mean of a set of numbers, follow these steps: Add all the numbers together. 12 + 15 = 27 27 + 18 = 45 45 + 21 = 66 66 + 24 = 90 The sum of all the numbers is 90. Count how many numbers are in the set.
   Full step-by-step solution

   To find the mean of a set of numbers, follow these steps: Step 1: Add all the numbers together. 12 + 15 = 27 27 + 18 = 45 45 + 21 = 66 66 + 24 = 90 The sum of all the numbers is 90. Step 2: Count how many numbers are in the set. The numbers are: 12, 15, 18, 21, 24. There are 5 numbers in the set. Step 3: Divide the sum by the count. Mean = (Sum of numbers) / (Count of numbers) Mean = 90 / 5 Step 4: Perform the division. 90 divided by 5 equals 18. Therefore, the mean of the numbers 12, 15, 18, 21, and 24 is 18.