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?
To find the inner fences, we first need to calculate the lower and upper quartiles.
Step 1: Sort the data in ascending order:
57, 65, 66, 67, 68, 71, 73, 74, 77, 87
Step 2: Find the median:
The median is the middle value of the sorted data set. In this case, we have 10 scores, so the median is the average of the 5th and 6th values:
Median = (68 + 71) / 2 = 69.5
Step 3: Find the lower quartile (Q1):
Q1 is the median of the lower half of the data. In this case, the lower half is the first 5 values. Since we have an odd number of values, the lower quartile is the middle value of this lower half:
Q1 = 66
Step 4: Find the upper quartile (Q3):
Q3 is the median of the upper half of the data. In this case, the upper half is the last 5 values. Again, the upper quartile is the middle value of this upper half:
Q3 = 74
Step 5: Calculate the interquartile range (IQR):
The IQR is the difference between Q3 and Q1:
IQR = Q3 - Q1 = 74 - 66 = 8
Step 6: Calculate the inner fences:
The inner fences help identify potential outliers. They are calculated by subtracting and adding 1.5 times the IQR to Q1 and Q3, respectively:
Lower inner fence = Q1 - 1.5 * IQR
Upper inner fence = Q3 + 1.5 * IQR
Lower inner fence = 66 - 1.5 * 8 = 66 - 12 = 54
Upper inner fence = 74 + 1.5 * 8 = 74 + 12 = 86
Therefore, the inner fences for the given data set are 54 and 86.