An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Create frequency distribution table: Count the number of scores in each class interval.
Calculate the mean: n ∑ x i = 75.46 .
Calculate the mean deviation: n ∑ ∣ x i − Mean ∣ = 8.435 .
Calculate the variance and standard deviation: Variance = 114.539 , Standard Deviation = 10.702 .
Determine the interquartile range: I QR = Q 3 − Q 1 = 14.0 .
I QR = 14.0
Explanation
Data Analysis First, let's analyze the data. We have the final grades of 50 students. We need to create a frequency distribution table, a cumulative frequency curve, determine the interquartile range (IQR), and calculate the mean, mean deviation, variance, and standard deviation.
Frequency Distribution Next, we create the frequency distribution table using the given class intervals. The intervals are 50-54, 55-59, 60-64, 65-69, 70-74, 75-79, 80-84, 85-89, 90-94, and 95-99. By counting the number of scores falling into each interval, we obtain the following frequencies:
50-54: 0 55-59: 1 60-64: 8 65-69: 6 70-74: 8 75-79: 14 80-84: 2 85-89: 5 90-94: 2 95-99: 4
Calculate the Mean Now, we calculate the mean score. The mean is the sum of all scores divided by the number of scores. The result of this calculation is: Mean = n ∑ x i = 75.46
Calculate the Mean Deviation Next, we calculate the mean deviation. The mean deviation is the average of the absolute differences between each score and the mean. The result of this calculation is: Mean Deviation = n ∑ ∣ x i − Mean ∣ = 8.435
Calculate the Variance Then, we calculate the variance. The variance is the average of the squared differences between each score and the mean. Since we are dealing with sample data, we will use the sample variance formula (dividing by n-1). The result of this calculation is: Variance = n − 1 ∑ ( x i − Mean ) 2 = 114.539
Calculate the Standard Deviation We calculate the standard deviation by taking the square root of the variance. The result of this calculation is: Standard Deviation = Variance = 10.702
Determine the Interquartile Range To determine the interquartile range (IQR), we first need to find the first quartile (Q1) and the third quartile (Q3). Q1 is the 25th percentile, and Q3 is the 75th percentile. Based on the data, we find: Q 1 = 67.25 Q 3 = 81.25 Therefore, the interquartile range is: I QR = Q 3 − Q 1 = 81.25 − 67.25 = 14.0
Final Results Finally, we have calculated all the required statistics. The frequency distribution is listed above, the mean is 75.46, the mean deviation is 8.435, the variance is 114.539, the standard deviation is 10.702, and the interquartile range is 14.0.
Examples
Understanding the distribution and variability of grades, as we've done here, is crucial in education. For example, a teacher can use the mean and standard deviation to understand the overall performance of the class and the spread of scores. The interquartile range can help identify the range of scores for the middle 50% of the class, which can be useful for tailoring instruction. This kind of analysis can also be applied to other areas, such as analyzing the distribution of income in a population or the variability of product quality in manufacturing.
