Answer :
- Minimum of 12B is 20.
- Median of 12A is 60.
- Mean of 12B is 51.95.
- Probability above 80% in 12B is 10.53%.
- Data is discrete.
- Range of 12B is 75.
- Mode of 12A is 60.
- IQR of 12B is 22.5.
- Grade 12A performed better.
$\boxed{Grade 12A \, performed \, better}$
### Explanation
1. Analyze the given data
First, let's identify the given data. We have the test scores for two Grade 12 classes, 12A and 12B. Our goal is to analyze this data to determine which class performed better on the Trial Exams.
2. Find the minimum value for Grade 12B
3.1: The minimum value for Grade 12B is the smallest number in the dataset. By inspection, the minimum value is 20.
3. Calculate the median for Grade 12A
3.2: To find the median for Grade 12A, we first need to sort the data in ascending order:
15, 20, 33, 35, 45, 46, 48, 48, 50, 57, 60, 60, 60, 60, 68, 70, 70, 75, 78, 85, 85, 90
Since there are 22 data points (an even number), the median is the average of the 11th and 12th values. The 11th value is 60 and the 12th value is 60. Therefore, the median is $\frac{60 + 60}{2} = 60$.
4. Calculate the mean for Grade 12B
3.3: To calculate the mean for Grade 12B, we sum all the values and divide by the number of values (19).
Mean = $\frac{20 + 29 + 33 + 35 + 35 + 40 + 45 + 46 + 46 + 56 + 56 + 56 + 60 + 60 + 60 + 60 + 70 + 85 + 95}{19} = \frac{987}{19} \approx 51.95$
5. Calculate the probability of scoring above 80% in Grade 12B
3.4: To calculate the probability of a student getting above 80% in Grade 12B, we count the number of students who scored above 80. These are 85 and 95, so there are 2 students. The total number of students is 19. Therefore, the probability is $\frac{2}{19} \approx 0.1053$. As a percentage, this is $10.53\%$.
6. Determine if the data is discrete or continuous
3.5: The data of the test results is discrete because the scores are integers. Discrete data can only take on specific, separate values (e.g., you can't score 50.5 on a test). Continuous data, on the other hand, can take on any value within a range.
7. Calculate the range for Grade 12B
3.6: The range for Grade 12B is the difference between the maximum and minimum values. The maximum value is 95 and the minimum value is 20. Therefore, the range is $95 - 20 = 75$.
8. Find the mode for Grade 12A
3.7: The mode for Grade 12A is the value that appears most frequently. In the dataset, the number 60 appears 4 times, which is more than any other number. Therefore, the mode is 60.
9. Calculate the interquartile range for Grade 12B
3.8: To calculate the interquartile range (IQR) for Grade 12B, we need to find the first quartile (Q1) and the third quartile (Q3). The data is already sorted.
Q1 is the 25th percentile, and Q3 is the 75th percentile. With 19 data points, Q1 is the value at position $0.25 * (19+1) = 5$, which is 37.5 (average of 35 and 40). Q3 is the value at position $0.75 * (19+1) = 15$, which is 60.
Therefore, the IQR is $60 - 37.5 = 22.5$.
10. Compare the performance of Grade 12A and 12B
3.9: To determine if Mr. Sewpersadh is correct, we need to compare the performance of the two classes. We already calculated the mean for both classes:
Mean for Grade 12A: 57.18
Mean for Grade 12B: 51.95
Since the mean for Grade 12A (57.18) is higher than the mean for Grade 12B (51.95), Grade 12A performed better in the Trial exams.
11. State the final answers
Final Answer:
3. 1 Minimum value for 12B: $\boxed{20}$
4. 2 Median for 12A: $\boxed{60}$
5. 3 Mean for 12B: $\boxed{51.95}$
6. 4 Probability of a student getting above 80% in 12B: $\boxed{10.53\%}$
7. 5 Data type: $\boxed{Discrete}$
8. 6 Range of 12B: $\boxed{75}$
9. 7 Mode for 12A: $\boxed{60}$
10. 8 Inter quartile range for 12B: $\boxed{22.5}$
11. 9 Grade 12A performed better.
### Examples
Understanding the distribution and central tendencies of test scores, like mean, median, and IQR, is crucial in education. Teachers use these statistics to evaluate the effectiveness of their teaching methods and to identify areas where students may need additional support. This type of analysis can also be applied in business to analyze sales data, in healthcare to study patient outcomes, and in sports to evaluate player performance.
- Median of 12A is 60.
- Mean of 12B is 51.95.
- Probability above 80% in 12B is 10.53%.
- Data is discrete.
- Range of 12B is 75.
- Mode of 12A is 60.
- IQR of 12B is 22.5.
- Grade 12A performed better.
$\boxed{Grade 12A \, performed \, better}$
### Explanation
1. Analyze the given data
First, let's identify the given data. We have the test scores for two Grade 12 classes, 12A and 12B. Our goal is to analyze this data to determine which class performed better on the Trial Exams.
2. Find the minimum value for Grade 12B
3.1: The minimum value for Grade 12B is the smallest number in the dataset. By inspection, the minimum value is 20.
3. Calculate the median for Grade 12A
3.2: To find the median for Grade 12A, we first need to sort the data in ascending order:
15, 20, 33, 35, 45, 46, 48, 48, 50, 57, 60, 60, 60, 60, 68, 70, 70, 75, 78, 85, 85, 90
Since there are 22 data points (an even number), the median is the average of the 11th and 12th values. The 11th value is 60 and the 12th value is 60. Therefore, the median is $\frac{60 + 60}{2} = 60$.
4. Calculate the mean for Grade 12B
3.3: To calculate the mean for Grade 12B, we sum all the values and divide by the number of values (19).
Mean = $\frac{20 + 29 + 33 + 35 + 35 + 40 + 45 + 46 + 46 + 56 + 56 + 56 + 60 + 60 + 60 + 60 + 70 + 85 + 95}{19} = \frac{987}{19} \approx 51.95$
5. Calculate the probability of scoring above 80% in Grade 12B
3.4: To calculate the probability of a student getting above 80% in Grade 12B, we count the number of students who scored above 80. These are 85 and 95, so there are 2 students. The total number of students is 19. Therefore, the probability is $\frac{2}{19} \approx 0.1053$. As a percentage, this is $10.53\%$.
6. Determine if the data is discrete or continuous
3.5: The data of the test results is discrete because the scores are integers. Discrete data can only take on specific, separate values (e.g., you can't score 50.5 on a test). Continuous data, on the other hand, can take on any value within a range.
7. Calculate the range for Grade 12B
3.6: The range for Grade 12B is the difference between the maximum and minimum values. The maximum value is 95 and the minimum value is 20. Therefore, the range is $95 - 20 = 75$.
8. Find the mode for Grade 12A
3.7: The mode for Grade 12A is the value that appears most frequently. In the dataset, the number 60 appears 4 times, which is more than any other number. Therefore, the mode is 60.
9. Calculate the interquartile range for Grade 12B
3.8: To calculate the interquartile range (IQR) for Grade 12B, we need to find the first quartile (Q1) and the third quartile (Q3). The data is already sorted.
Q1 is the 25th percentile, and Q3 is the 75th percentile. With 19 data points, Q1 is the value at position $0.25 * (19+1) = 5$, which is 37.5 (average of 35 and 40). Q3 is the value at position $0.75 * (19+1) = 15$, which is 60.
Therefore, the IQR is $60 - 37.5 = 22.5$.
10. Compare the performance of Grade 12A and 12B
3.9: To determine if Mr. Sewpersadh is correct, we need to compare the performance of the two classes. We already calculated the mean for both classes:
Mean for Grade 12A: 57.18
Mean for Grade 12B: 51.95
Since the mean for Grade 12A (57.18) is higher than the mean for Grade 12B (51.95), Grade 12A performed better in the Trial exams.
11. State the final answers
Final Answer:
3. 1 Minimum value for 12B: $\boxed{20}$
4. 2 Median for 12A: $\boxed{60}$
5. 3 Mean for 12B: $\boxed{51.95}$
6. 4 Probability of a student getting above 80% in 12B: $\boxed{10.53\%}$
7. 5 Data type: $\boxed{Discrete}$
8. 6 Range of 12B: $\boxed{75}$
9. 7 Mode for 12A: $\boxed{60}$
10. 8 Inter quartile range for 12B: $\boxed{22.5}$
11. 9 Grade 12A performed better.
### Examples
Understanding the distribution and central tendencies of test scores, like mean, median, and IQR, is crucial in education. Teachers use these statistics to evaluate the effectiveness of their teaching methods and to identify areas where students may need additional support. This type of analysis can also be applied in business to analyze sales data, in healthcare to study patient outcomes, and in sports to evaluate player performance.
