In a statistic class, 10 scores were randomly selected with the following results were obtained: 74, 73, 87, 57, 71, 68, 65, 77, 67, 66. What are the inner fences? What is the coefficient of variation?
Find the mean and standard deviation for your data. Once you calculate those two values, you can use the following formula to find the coefficient of variation:
CV = (sd/mean) * 100
...where sd = standard deviation and * means to multiply.
I hope this will help.
Note: CV is the standard deviation expressed as a percentage of the mean.
To find the inner fences for a dataset, we need to calculate the first and third quartiles of the data. Here are the steps:
Step 1: Sort the data in ascending order: 57, 65, 66, 67, 68, 71, 73, 74, 77, 87.
Step 2: Calculate the first quartile (Q1):
Position of Q1 = (n + 1) / 4
where n is the number of data points
Position of Q1 = (10 + 1) / 4 = 2.75
Since 2.75 is between the 2nd and 3rd values in the sorted data set, we can interpolate to find Q1:
Q1 = 65 + 0.75 * (66 - 65) = 65.75
Step 3: Calculate the third quartile (Q3):
Position of Q3 = 3 * (n + 1) / 4
Position of Q3 = 3 * (10 + 1) / 4 = 8.25
Again, we interpolate to find Q3:
Q3 = 74 + 0.25 * (77 - 74) = 74.75
Step 4: Calculate the interquartile range (IQR):
IQR = Q3 - Q1 = 74.75 - 65.75 = 9
Step 5: Calculate the inner fences:
Lower inner fence = Q1 - (1.5 * IQR)
Upper inner fence = Q3 + (1.5 * IQR)
Lower inner fence = 65.75 - (1.5 * 9) = 52.25
Upper inner fence = 74.75 + (1.5 * 9) = 88.25
Therefore, the inner fences for the given dataset are 52.25 and 88.25.
Now, let's calculate the coefficient of variation, which measures the relative variability of the data.
Step 1: Calculate the mean (x̄) of the dataset:
Sum of all scores = 74 + 73 + 87 + 57 + 71 + 68 + 65 + 77 + 67 + 66 = 715
Mean (x̄) = Sum of all scores / Number of scores = 715 / 10 = 71.5
Step 2: Calculate the standard deviation (σ) of the dataset:
Squared deviations:
(74 - 71.5)^2 + (73 - 71.5)^2 + (87 - 71.5)^2 + (57 - 71.5)^2 + (71 - 71.5)^2 + (68 - 71.5)^2 + (65 - 71.5)^2 + (77 - 71.5)^2 + (67 - 71.5)^2 + (66 - 71.5)^2 = 1470
Variance (s^2) = Sum of squared deviations / (Number of scores - 1) = 1470 / (10 - 1) = 163.33
Standard deviation (σ) = √(Variance) = √(163.33) = 12.77
Step 3: Calculate the coefficient of variation (CV):
CV = (Standard deviation / Mean) * 100
CV = (12.77 / 71.5) * 100 = 17.86%
Therefore, the coefficient of variation for the given dataset is 17.86%.
To find the inner fences and the coefficient of variation, we need to perform some calculations.
1. Inner Fences:
Inner fences are used to detect any potential outliers in a dataset. They can be calculated using the following formulas:
Lower Inner Fence (LIF) = Q1 - (1.5 * IQR)
Upper Inner Fence (UIF) = Q3 + (1.5 * IQR)
First, we need to find the quartiles and the interquartile range (IQR).
- Quartiles:
To find the quartiles, we need to arrange the data in ascending order.
57, 65, 66, 67, 68, 71, 73, 74, 77, 87
The median (Q2) is the middle value of the dataset, which is 68.
To find Q1, we need to find the median of the lower half of the dataset (from 57 to 68):
Q1 = median of 57, 65, 66 (which is (65+66)/2 = 65.5)
To find Q3, we need to find the median of the upper half of the dataset (from 68 to 87):
Q3 = median of 71, 73, 74 (which is (73+74)/2 = 73.5)
- Interquartile Range (IQR):
IQR = Q3 - Q1 = 73.5 - 65.5 = 8
Now we can calculate the inner fences:
LIF = 65.5 - (1.5 * 8) = 65.5 - 12 = 53.5
UIF = 73.5 + (1.5 * 8) = 73.5 + 12 = 85.5
Therefore, the lower inner fence is 53.5, and the upper inner fence is 85.5.
2. Coefficient of Variation (CV):
The coefficient of variation measures the relative variability of a dataset. It can be calculated using the following formula:
CV = (Standard Deviation / Mean) * 100
First, we need to calculate the mean and standard deviation of the dataset.
- Mean:
Mean = (Sum of scores) / (Number of scores)
Mean = (74 + 73 + 87 + 57 + 71 + 68 + 65 + 77 + 67 + 66) / 10 = 715 / 10 = 71.5
- Standard Deviation:
To calculate the standard deviation, we need to find the deviations of each score from the mean, square them, find the sum of the squared deviations, divide by the number of scores, and finally, take the square root.
Let's perform the calculations step by step:
Deviation from the mean for each score:
74 - 71.5 = 2.5
73 - 71.5 = 1.5
87 - 71.5 = 15.5
57 - 71.5 = -14.5
71 - 71.5 = -0.5
68 - 71.5 = -3.5
65 - 71.5 = -6.5
77 - 71.5 = 5.5
67 - 71.5 = -4.5
66 - 71.5 = -5.5
Squared deviations from the mean:
2.5^2 = 6.25
1.5^2 = 2.25
15.5^2 = 240.25
(-14.5)^2 = 210.25
(-0.5)^2 = 0.25
(-3.5)^2 = 12.25
(-6.5)^2 = 42.25
5.5^2 = 30.25
(-4.5)^2 = 20.25
(-5.5)^2 = 30.25
Sum of squared deviations = 6.25 + 2.25 + 240.25 + 210.25 + 0.25 + 12.25 + 42.25 + 30.25 + 20.25 + 30.25 = 604.5
Standard deviation = sqrt(604.5 / 10) = sqrt(60.45) ≈ 7.78
Now we can calculate the coefficient of variation:
CV = (7.78 / 71.5) * 100 = 10.85%
Therefore, the coefficient of variation is approximately 10.85%.