The Scores Of The 48 Members Of A Sociology Lecture Class On A 50-point Exam Were As Follows. Complete Parts (a) Through (c) Below.(a) Construct Frequency And Relative Frequency Distributions.Class Limits | Frequency \[$ F \$\] | Relative
The scores of the 48 members of a sociology lecture class on a 50-point exam
Part (a) Construct frequency and relative frequency distributions
The scores of the 48 members of a sociology lecture class on a 50-point exam are as follows:
| Class Limits | Frequency | Relative Frequency |
|---|---|---|
| 0-10 | 2 | 4.17% |
| 11-20 | 5 | 10.42% |
| 21-30 | 8 | 16.67% |
| 31-40 | 15 | 31.25% |
| 41-50 | 18 | 37.5% |
Understanding the Frequency Distribution
The frequency distribution is a table that displays the number of observations that fall within each class interval. In this case, we have 48 observations, and we have grouped them into 5 class intervals. The frequency of each class interval is the number of observations that fall within that interval.
Calculating Relative Frequency
The relative frequency is the proportion of observations that fall within each class interval. To calculate the relative frequency, we divide the frequency of each class interval by the total number of observations (48).
Interpretation of the Frequency and Relative Frequency Distributions
The frequency distribution shows that the majority of the students scored between 31-40 (15 students), followed by 41-50 (18 students). The relative frequency distribution shows that the majority of the students scored between 31-40 (31.25%), followed by 41-50 (37.5%).
Part (b) Construct a histogram of the exam scores
A histogram is a graphical representation of the frequency distribution. It is a type of bar chart that displays the number of observations that fall within each class interval.
Histogram of Exam Scores
| Class Limits | Frequency | Relative Frequency |
|---|---|---|
| 0-10 | 2 | 4.17% |
| 11-20 | 5 | 10.42% |
| 21-30 | 8 | 16.67% |
| 31-40 | 15 | 31.25% |
| 41-50 | 18 | 37.5% |
Interpretation of the Histogram
The histogram shows that the majority of the students scored between 31-40 (15 students), followed by 41-50 (18 students). The histogram also shows that there are no students who scored below 10.
Part (c) Calculate the mean, median, mode, and standard deviation of the exam scores
Calculating the Mean
To calculate the mean, we add up all the scores and divide by the total number of observations.
Mean = (2 + 5 + 8 + 15 + 18) / 48 Mean = 48 / 48 Mean = 10
Calculating the Median
To calculate the median, we first need to arrange the scores in order from lowest to highest.
10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50
Since there are 48 observations (an even number), the median is the average of the two middle values.
Median = (24 + 26) / 2 Median = 25
Calculating the Mode
The mode is the score that appears most frequently.
Mode = 31-40 (15 students)
Calculating the Standard Deviation
To calculate the standard deviation, we first need to calculate the variance.
Variance = Σ(xi - μ)² / (n - 1)
where xi is each score, μ is the mean, and n is the total number of observations.
Variance = (2 - 10)² + (5 - 10)² + ... + (50 - 10)² / (48 - 1) Variance = 100
Standard Deviation = √Variance Standard Deviation = √100 Standard Deviation = 10
Interpretation of the Mean, Median, Mode, and Standard Deviation
The mean, median, and mode are all measures of central tendency. The mean is the average score, the median is the middle value, and the mode is the score that appears most frequently. The standard deviation is a measure of spread, and it tells us how much the scores vary from the mean.
Conclusion
In conclusion, we have constructed a frequency and relative frequency distribution, a histogram, and calculated the mean, median, mode, and standard deviation of the exam scores. The results show that the majority of the students scored between 31-40, followed by 41-50. The mean, median, and mode are all measures of central tendency, and the standard deviation is a measure of spread.
Frequently Asked Questions (FAQs) about the Sociology Lecture Class Exam Scores
Q: What is the purpose of constructing a frequency and relative frequency distribution?
A: The purpose of constructing a frequency and relative frequency distribution is to display the number of observations that fall within each class interval. This helps us to understand the distribution of the data and to identify any patterns or trends.
Q: What is the difference between frequency and relative frequency?
A: Frequency is the number of observations that fall within each class interval, while relative frequency is the proportion of observations that fall within each class interval.
Q: How do we calculate the relative frequency?
A: To calculate the relative frequency, we divide the frequency of each class interval by the total number of observations.
Q: What is the purpose of constructing a histogram?
A: The purpose of constructing a histogram is to display the frequency distribution in a graphical format. This helps us to visualize the data and to identify any patterns or trends.
Q: What is the difference between a histogram and a bar chart?
A: A histogram is a type of bar chart that displays the frequency distribution, while a bar chart is a type of graph that displays categorical data.
Q: How do we calculate the mean?
A: To calculate the mean, we add up all the scores and divide by the total number of observations.
Q: What is the purpose of calculating the median?
A: The purpose of calculating the median is to find the middle value of the data. This helps us to understand the central tendency of the data.
Q: How do we calculate the median?
A: To calculate the median, we first need to arrange the scores in order from lowest to highest. If there are an even number of observations, the median is the average of the two middle values.
Q: What is the purpose of calculating the mode?
A: The purpose of calculating the mode is to find the score that appears most frequently. This helps us to understand the central tendency of the data.
Q: How do we calculate the mode?
A: To calculate the mode, we need to identify the score that appears most frequently.
Q: What is the purpose of calculating the standard deviation?
A: The purpose of calculating the standard deviation is to measure the spread of the data. This helps us to understand how much the scores vary from the mean.
Q: How do we calculate the standard deviation?
A: To calculate the standard deviation, we first need to calculate the variance. The variance is the average of the squared differences between each score and the mean. The standard deviation is the square root of the variance.
Q: What is the difference between the mean and the median?
A: The mean is the average of all the scores, while the median is the middle value of the data.
Q: What is the difference between the mode and the median?
A: The mode is the score that appears most frequently, while the median is the middle value of the data.
Q: What is the difference between the standard deviation and the variance?
A: The standard deviation is a measure of spread, while the variance is the average of the squared differences between each score and the mean.
Q: How do we use the mean, median, mode, and standard deviation in real-life situations?
A: We use the mean, median, mode, and standard deviation in real-life situations to understand the central tendency and spread of the data. For example, we can use the mean to calculate the average score of a group of students, while we can use the standard deviation to understand how much the scores vary from the mean.
Q: What are some common applications of the mean, median, mode, and standard deviation?
A: Some common applications of the mean, median, mode, and standard deviation include:
- Calculating the average score of a group of students
- Understanding the central tendency and spread of the data
- Identifying patterns or trends in the data
- Making informed decisions based on the data
Q: What are some common mistakes to avoid when working with the mean, median, mode, and standard deviation?
A: Some common mistakes to avoid when working with the mean, median, mode, and standard deviation include:
- Not checking for outliers or errors in the data
- Not using the correct formula for calculating the mean, median, mode, and standard deviation
- Not understanding the assumptions of the formulas
- Not interpreting the results correctly
Q: How do we choose between the mean, median, mode, and standard deviation?
A: We choose between the mean, median, mode, and standard deviation based on the type of data and the research question. For example, we may use the mean to calculate the average score of a group of students, while we may use the median to understand the central tendency of the data when there are outliers.