46. The Table Below Shows The Scores Of 20 Students In A Further Mathematics Test.$\[ \begin{array}{|l|l|l|l|l|l|l|l|r|} \hline \text{Score} & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{Frequency} & 1 & 0 & 7 & 5 & 2 & 3 & 1 & 1
Introduction
In this analysis, we will be examining the scores of 20 students in a Further Mathematics test. The scores are presented in a table, showing the frequency of each score from 3 to 10. We will be using this data to calculate various statistics and gain insights into the performance of the students.
Table of Scores
| Score | Frequency |
|---|---|
| 3 | 1 |
| 4 | 0 |
| 5 | 7 |
| 6 | 5 |
| 7 | 2 |
| 8 | 3 |
| 9 | 1 |
| 10 | 1 |
Mean Score Calculation
To calculate the mean score, we need to multiply each score by its frequency and then sum up the products.
Mean Score = (3 x 1) + (4 x 0) + (5 x 7) + (6 x 5) + (7 x 2) + (8 x 3) + (9 x 1) + (10 x 1) Mean Score = 3 + 0 + 35 + 30 + 14 + 24 + 9 + 10 Mean Score = 125 Mean Score = 125 / 20 Mean Score = 6.25
Median Score Calculation
To calculate the median score, we need to first arrange the scores in ascending order.
3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10
Since there are 20 scores, the median will be the average of the 10th and 11th scores.
Median Score = (6 + 6) / 2 Median Score = 12 / 2 Median Score = 6
Mode Score Calculation
To calculate the mode score, we need to find the score that appears most frequently.
From the table, we can see that the score 5 appears 7 times, which is the highest frequency.
Mode Score = 5
Range Calculation
To calculate the range, we need to subtract the lowest score from the highest score.
Range = Highest Score - Lowest Score Range = 10 - 3 Range = 7
Interquartile Range (IQR) Calculation
To calculate the IQR, we need to first arrange the scores in ascending order.
3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10
The first quartile (Q1) is the median of the lower half of the scores.
Q1 = Median of (3, 3, 5, 5, 5, 5, 5, 6, 6, 6) Q1 = 5
The third quartile (Q3) is the median of the upper half of the scores.
Q3 = Median of (7, 7, 8, 8, 8, 9, 10, 10, 10) Q3 = 8
IQR = Q3 - Q1 IQR = 8 - 5 IQR = 3
Variance Calculation
To calculate the variance, we need to first calculate the squared differences between each score and the mean.
(3 - 6.25)^2 + (4 - 6.25)^2 + (5 - 6.25)^2 + (6 - 6.25)^2 + (7 - 6.25)^2 + (8 - 6.25)^2 + (9 - 6.25)^2 + (10 - 6.25)^2 = (-3.25)^2 + (-2.25)^2 + (-1.25)^2 + (-0.25)^2 + 0.75^2 + 1.75^2 + 2.75^2 + 3.75^2 = 10.5625 + 5.0625 + 1.5625 + 0.0625 + 0.5625 + 3.0625 + 7.5625 + 14.0625 = 42.5
Variance = Σ(xi - μ)^2 / (n - 1) Variance = 42.5 / (20 - 1) Variance = 42.5 / 19 Variance = 2.2368
Standard Deviation Calculation
To calculate the standard deviation, we need to take the square root of the variance.
Standard Deviation = √Variance Standard Deviation = √2.2368 Standard Deviation = 1.49
Conclusion
Q: What is the purpose of this analysis?
A: The purpose of this analysis is to examine the scores of 20 students in a Further Mathematics test and calculate various statistics to gain insights into the performance of the students.
Q: What are the key statistics calculated in this analysis?
A: The key statistics calculated in this analysis include the mean score, median score, mode score, range, interquartile range (IQR), variance, and standard deviation.
Q: What is the mean score, and how is it calculated?
A: The mean score is 6.25, and it is calculated by multiplying each score by its frequency and then summing up the products.
Q: What is the median score, and how is it calculated?
A: The median score is 6, and it is calculated by arranging the scores in ascending order and finding the average of the 10th and 11th scores.
Q: What is the mode score, and how is it calculated?
A: The mode score is 5, and it is calculated by finding the score that appears most frequently in the data.
Q: What is the range, and how is it calculated?
A: The range is 7, and it is calculated by subtracting the lowest score from the highest score.
Q: What is the interquartile range (IQR), and how is it calculated?
A: The IQR is 3, and it is calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1).
Q: What is the variance, and how is it calculated?
A: The variance is 2.2368, and it is calculated by finding the average of the squared differences between each score and the mean.
Q: What is the standard deviation, and how is it calculated?
A: The standard deviation is 1.49, and it is calculated by taking the square root of the variance.
Q: What are the implications of these statistics for teaching and learning?
A: These statistics provide valuable insights into the performance of the students and can be used to inform teaching and learning strategies. For example, the mean score can be used to identify areas where students may need additional support, while the standard deviation can be used to identify students who may be at risk of falling behind.
Q: How can these statistics be used to improve student outcomes?
A: These statistics can be used to improve student outcomes by identifying areas where students may need additional support and by developing targeted interventions to address these needs. Additionally, these statistics can be used to monitor student progress over time and to make data-driven decisions about teaching and learning strategies.
Q: What are some potential limitations of this analysis?
A: Some potential limitations of this analysis include the small sample size and the fact that the data may not be representative of the larger population of students. Additionally, the analysis may not capture the full range of student performance, as it only includes scores from a single test.
Q: How can these statistics be used to inform policy decisions?
A: These statistics can be used to inform policy decisions by providing a snapshot of student performance and identifying areas where students may need additional support. Additionally, these statistics can be used to monitor student progress over time and to make data-driven decisions about policy initiatives.