Sure! Let's go through this in detail step-by-step.
### a) Calculate the mean
The mean (average) is calculated by summing the product of each score and its frequency, and then dividing this sum by the total number of rolls.
1. First, we list the scores and their corresponding frequencies:
- Scores: 1, 2, 3, 4, 5, 6
- Frequencies: 5, 9, 7, 11, 8, 10
2. We calculate the total number of rolls:
- Total Rolls = 5 + 9 + 7 + 11 + 8 + 10 = 50
3. Next, we calculate the total score by multiplying each score by its frequency and summing up these products:
- Total Score = (1 5) + (2 9) + (3 7) + (4 11) + (5 8) + (6 10)
- Total Score = 5 + 18 + 21 + 44 + 40 + 60 = 188
4. Finally, the mean is the total score divided by the total number of rolls:
- Mean = Total Score / Total Rolls
- Mean = 188 / 50
- Mean = 3.76
### b) Find the median
The median is the middle value of a data set that is ordered from least to greatest. Since we have an even number of data points (50 rolls), the median will be the average of the 25th and 26th values in the ordered list.
1. We create a cumulative frequency list to determine the position of the 25th and 26th rolls:
- Cumulative Frequencies: [5, 14 (5+9), 21 (14+7), 32 (21+11), 40 (32+8), 50 (40+10)]
2. We check where the 25th and 26th rolls fall in the ordered list:
- The 25th and 26th rolls are in the score range of 4 since the cumulative frequency at score 3 is 21 and jumps to 32 at score 4.
3. Since both the 25th and 26th rolls fall within the same category (score 4), the median is:
- Median = 4.5 (average of 4 and 5, since the data is on the boundary)
### c) What is the mode?
The mode is the score that appears most frequently.
1. From the frequencies given, the highest frequency is 11 for the score of 4.
2. Therefore, the mode is:
- Mode = 4
To summarize:
- Mean: 3.76
- Median: 4.5
- Mode: 4
