To calculate the five-number summary for the dataset of test scores \(90, 95, 89, 84, 74, 99, 97, 86, 93\), we first need to sort the scores in ascending order.
The sorted scores are:
\[74, 84, 86, 89, 90, 93, 95, 97, 99\]
Now we can determine the five-number summary:
-
Minimum: The smallest value in the dataset.
\[\text{Minimum} = 74\] -
Maximum: The largest value in the dataset.
\[\text{Maximum} = 99\] -
Median: The middle value of the dataset. Since there are 9 numbers (an odd dataset), the median is the value at the \( \frac{n+1}{2} \) position.
\[ \text{Median position} = \frac{9+1}{2} = 5\]
The median is the 5th number in the sorted list, which is \(90\).
\[\text{Median} = 90\] -
Q1 (First Quartile): The median of the first half of the dataset (the lower half). The first half is \(74, 84, 86, 89\). There are 4 numbers here, so the median is the average of the 2nd and 3rd numbers:
\[Q1 = \frac{84 + 86}{2} = 85\] -
Q3 (Third Quartile): The median of the second half of the dataset (the upper half). The second half is \(93, 95, 97, 99\). Similarly, the median is the average of the 2nd and 3rd numbers of this half:
\[Q3 = \frac{95 + 97}{2} = 96\]
Putting it all together, the five-number summary is:
- Minimum: \(74\)
- Q1: \(85\)
- Median: \(90\)
- Q3: \(96\)
- Maximum: \(99\)
Final Answers:
- The minimum is 74.
- The value for Q1 is 85.
- The median of the dataset is 90.
- The value for Q3 is 96.
- The maximum of the dataset is 99.