Find the quartiles and semi interquartile range of the following a) 17, 17, 19, 20, 23, 23, 25, 27, 27, 27, 30, 31, 33 b) 2.9, 4.6, 3.8, 4.8, 5.0, 4.2, 2.4, 3.1, 4.9, 4.6 2. A man timed himself over a three week on how long it took him to drive to work in the morning. The times in minutes, were: 15, 17, 20, 23, 29, 32, 30, 29, 25, 23, 18, 29, 15, 17, 23, 20 Find the interquartile range and semi interquartile range of the data.
The Correct Answer and Explanation is :
Explanation of Quartiles and Semi-Interquartile Range (SIQR)
Quartiles are values that divide a data set into four equal parts. The key quartiles are:
- Q1 (First Quartile): The 25th percentile, below which 25% of the data falls.
- Q2 (Median): The 50th percentile, the middle value of the dataset.
- Q3 (Third Quartile): The 75th percentile, below which 75% of the data falls.
The Interquartile Range (IQR) is given by:
[
IQR = Q3 – Q1
]
The Semi-Interquartile Range (SIQR) is half of the interquartile range:
[
SIQR = \frac{IQR}{2}
]
Analysis of Given Data:
(a) Dataset A:
- Q1 = 20, Median = 25, Q3 = 27
- SIQR = (27 – 20) / 2 = 3.5
(b) Dataset B:
- Q1 = 3.275, Median = 4.4, Q3 = 4.75
- SIQR = (4.75 – 3.275) / 2 = 0.7375
(c) Commute Times Data:
- Q1 = 17.75, Median = 23, Q3 = 29
- SIQR = (29 – 17.75) / 2 = 5.625
Interpretation
- Dataset A has a moderate spread in data (SIQR = 3.5), indicating some variation.
- Dataset B has a low spread (SIQR = 0.7375), meaning data points are closely packed.
- Commute times have a relatively high SIQR (5.625), showing variability in the driving times.
The box plot visually represents the spread of each dataset, highlighting their quartiles and potential outliers.