Consider the GPA’s for the following 15 students
3.10, 3.20, 3.49, 2.68, 3.73, 3.39, 3.80, 3.11, 3.10, 3.10, 4.0, 3.89, 2.55, 4.0, and 3.75
a) Compute the max, min, Q1, median 3.9, Q3, IQR, and range 1.45
b) Using the variables from part a), construct a box plot for GPA.
c) What is the shape of the data?
To compute the summary statistics for the given GPA data, follow these steps:
a) Compute the max, min, Q1, median, Q3, IQR, and range:
1. Sort the data in ascending order:
2.55, 2.68, 3.10, 3.10, 3.10, 3.11, 3.20, 3.39, 3.49, 3.73, 3.75, 3.80, 3.89, 4.00, 4.00
2. Find the minimum and maximum values:
Minimum: 2.55
Maximum: 4.00
3. Find the median (middle value) of the dataset:
Median: (3.11 + 3.20)/2 = 3.155
4. Calculate the first quartile (Q1):
- Find the median of the lower half of the data:
- For the first quartile, consider the numbers from the start to the median inclusive.
- Q1: (2.68 + 3.10)/2 = 2.89
5. Calculate the third quartile (Q3):
- Find the median of the upper half of the data:
- For the third quartile, consider the numbers from the median to the end inclusive.
- Q3: (3.80 + 3.89)/2 = 3.845
6. Calculate the interquartile range (IQR):
IQR = Q3 - Q1 = 3.845 - 2.89 = 0.955
7. Calculate the range:
Range = Maximum - Minimum = 4.00 - 2.55 = 1.45
b) To construct a box plot:
A box plot is a graphical representation of the summary statistics. Here's how to construct the box plot using the variables from part a):
1. On a number line or horizontal axis, mark the minimum, Q1, median, Q3, and maximum values in ascending order.
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2.55 | 2.89 | 3.155 | 3.845 | 4.00
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2. Draw a box starting from Q1 and ending at Q3.
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2.55 | | ■ | | 4.00
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3. Draw a line inside the box representing the median (Q2).
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2.55 | | ■■■ | | 4.00
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4. Extend lines (whiskers) from the sides of the box to the minimum and maximum values.
| |
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2.55 4.00
c) The shape of the data can be determined by examining the box plot. If the box plot is symmetrically distributed around the median, it indicates that the data is approximately normally distributed. If there are outliers or the box plot is skewed to one side, it suggests a non-normal distribution. In this case, without further description or examination of the data, it is not possible to definitively determine the shape of the data.