Here are the scores of
13
students on an algebra test.
55
,
57
,
60
,
60
,
65
,
68
,
70
,
70
,
80
,
80
,
84
,
89
,
92
Notice that the scores are ordered from least to greatest.
Give the five-number summary and the interquartile range for the data set.
To find the five-number summary and the interquartile range for the given data set, first, let's define what each component represents:
1. Minimum (Min): The smallest value in the data set.
2. First Quartile (Q1): The median of the lower half of the data set.
3. Median (Q2): The middle value when the data set is arranged in ascending order.
4. Third Quartile (Q3): The median of the upper half of the data set.
5. Maximum (Max): The largest value in the data set.
6. Interquartile Range (IQR): The range between the first quartile and the third quartile.
Now, let's calculate each component step by step:
Step 1: Ordering the data set in ascending order:
55, 57, 60, 60, 65, 68, 70, 70, 80, 80, 84, 89, 92
Step 2: Finding the minimum (Min):
The minimum value is 55.
Step 3: Finding the first quartile (Q1):
Since we have an odd number of data points (13), the first quartile (Q1) will be the median of the lower half. The lower half includes the first six values: 55, 57, 60, 60, 65, 68. The median of this lower half is (60 + 60) / 2 = 60.
Step 4: Finding the median (Q2):
The median is the middle value of the ordered data set. In this case, it is 68.
Step 5: Finding the third quartile (Q3):
Again, since we have an odd number of data points, the third quartile (Q3) will be the median of the upper half. The upper half includes the last six values: 70, 70, 80, 80, 84, 89. The median of this upper half is (80 + 80) / 2 = 80.
Step 6: Finding the maximum (Max):
The maximum value is 92.
Step 7: Calculating the interquartile range (IQR):
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). In this case, it is 80 - 60 = 20.
Therefore, the five-number summary for the data set is Min: 55, Q1: 60, Q2: 68, Q3: 80, Max: 92. The interquartile range (IQR) is 20.