To calculate the mean, median, and mode for the data provided, let's first list the data again:
Data: 78, 82, 85, 84, 80, 79, 88, 82, 85, 83, 81, 86
Step 1: Calculate the Mean
The mean is calculated by adding all the numbers together and dividing by the number of students.
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Sum of the values: 78 + 82 + 85 + 84 + 80 + 79 + 88 + 82 + 85 + 83 + 81 + 86 = 1027
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Number of students: 12
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Mean = Sum / Number of students = 1027 / 12 ≈ 85.5833
Step 2: Calculate the Median
To find the median, we must first organize the data in ascending order:
Sorted Data: 78, 79, 80, 81, 82, 82, 83, 84, 85, 85, 86, 88
Since there are 12 numbers (an even number), the median will be the average of the 6th and 7th values:
- 6th value = 82
- 7th value = 83
Median = (82 + 83) / 2 = 82.5
Step 3: Calculate the Mode
The mode is the number that appears most frequently.
- The frequencies are:
- 78: 1
- 79: 1
- 80: 1
- 81: 1
- 82: 2
- 83: 1
- 84: 1
- 85: 2
- 86: 1
- 88: 1
The numbers 82 and 85 both appear twice, making them the modes.
Summary of Calculated Values:
- Mean: 85.5833
- Median: 82.5
- Modes: 82 and 85
Comparison with Options:
None of the provided options match the calculated mean of 85.5833; however, the median (82.5) and modes (82 and 85) are correctly identified in options b and d:
b. Mean = 82.75 ; Median = 82.5 ; Mode(s) = 82 and 85
d. Mean = 82.0 ; Median = 82.5 ; Mode(s) = 82 and 85
Given the calculations, the correct mean is significantly different from the options, so none of the answer choices are accurate as provided, but options b and d have the accurate median and modes.
Conclusion:
Assuming correct calculation alternative:
- The median is 82.5
- The modes are 82 and 85
None of the options show the correct mean; thus, none of the options are entirely valid based on our calculations.