To analyze the data provided by the students, we can start by calculating some basic statistical measures such as the mean (average), median, mode, range, and standard deviation.
Data Set
The times given are: 42, 41, 35, 45, 72, 63, 60, 64, 49, 53, 65, 55, 60, 57, 52, 50, 62, 67
Step 1: Sort the Data
First, let's sort the data in ascending order: 35, 41, 42, 45, 49, 50, 52, 53, 55, 57, 60, 60, 62, 63, 64, 65, 67, 72
Step 2: Calculate the Mean
To calculate the mean, we sum all the times and divide by the number of data points.
\[ \text{Mean} = \frac{sum \ of \ data}{number \ of \ data \ points} \]
Sum of data = 35 + 41 + 42 + 45 + 49 + 50 + 52 + 53 + 55 + 57 + 60 + 60 + 62 + 63 + 64 + 65 + 67 + 72 = 1013
Number of data points = 18
Mean = \( \frac{1013}{18} \approx 56.28 \)
Step 3: Calculate the Median
The median is the middle value when the numbers are sorted. Since there are 18 values (an even number), the median will be the average of the 9th and 10th values in the sorted list.
9th value = 55
10th value = 57
Median = \( \frac{55 + 57}{2} = 56 \)
Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set. In the sorted list, the number 60 appears twice, while all other numbers appear once.
Mode = 60
Step 5: Calculate the Range
The range is the difference between the maximum and minimum values.
Range = Maximum value - Minimum value = 72 - 35 = 37
Step 6: Calculate the Standard Deviation
To calculate the standard deviation, we'll first find each value's deviation from the mean, square it, sum these squared deviations, divide by the number of values, and then take the square root.
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Calculate deviations from mean and square them:
- \( (35 - 56.28)^2 \)
- \( (41 - 56.28)^2 \)
- \( (42 - 56.28)^2 \)
- \( (45 - 56.28)^2 \)
- \( (49 - 56.28)^2 \)
- \( (50 - 56.28)^2 \)
- \( (52 - 56.28)^2 \)
- \( (53 - 56.28)^2 \)
- \( (55 - 56.28)^2 \)
- \( (57 - 56.28)^2 \)
- \( (60 - 56.28)^2 \)
- \( (60 - 56.28)^2 \)
- \( (62 - 56.28)^2 \)
- \( (63 - 56.28)^2 \)
- \( (64 - 56.28)^2 \)
- \( (65 - 56.28)^2 \)
- \( (67 - 56.28)^2 \)
- \( (72 - 56.28)^2 \)
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The squared deviations sum to approximately \( 1174.38 \).
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Divide by the number of data points: \( \text{Variance} = \frac{1174.38}{18} \approx 65.25 \)
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The standard deviation is the square root of the variance: \( \text{Standard Deviation} \approx \sqrt{65.25} \approx 8.08 \)
Summary
- Mean: approximately 56.28
- Median: 56
- Mode: 60
- Range: 37
- Standard Deviation: approximately 8.08
If you need any further analysis or information, feel free to ask!
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