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Chapter 3: Problem 32

Use the given data to construct a boxplot and identify the 5-number summary. Fourteen different second-year medical students at Bellevue Hospital measured the blood pressure of the same person. The systolic readings \((\mathrm{mm} \mathrm{Hg})\) are listed below. \(\begin{array}{lllllllllllllll}138 & 130 & 135 & 140 & 120 & 125 & 120 & 130 & 130 & 144 & 143 & 140 & 130 & 150\end{array}\)

Short Answer

Expert verified
Minimum: 120, Q1: 125, Median: 132.5, Q3: 140, Maximum: 150.

Step by step solution

01

- Arrange Data in Ascending Order

Start by sorting the data. Arrange the systolic readings (138, 130, 135, 140, 120, 125, 120, 130, 130, 144, 143, 140, 130, 150) in ascending order: 120, 120, 125, 130, 130, 130, 130, 135, 138, 140, 140, 143, 144, 150.
02

- Identify the Minimum and Maximum Values

The minimum value is the smallest number in the ordered list and the maximum value is the largest number. In this case:Minimum value (Q0) = 120Maximum value (Q4) = 150
03

- Find the Median (Q2)

The median divides the data into two equal halves. Since there are 14 readings, the median is the average of the 7th and 8th largest numbers. In the ordered list (120, 120, 125, 130, 130, 130, 130, 135, 138, 140, 140, 143, 144, 150): Median (Q2) = (130 + 135)/2 = 132.5
04

- Determine the First Quartile (Q1)

The first quartile is the median of the lower half of the data, excluding the overall median. For the values 120, 120, 125, 130, 130, 130, 130:First Quartile (Q1) = 125
05

- Determine the Third Quartile (Q3)

The third quartile is the median of the upper half of the data, excluding the overall median. For the values 135, 138, 140, 140, 143, 144, 150:Third Quartile (Q3) = 140
06

- Construct the Boxplot

A boxplot is constructed using the minimum value, Q1, median, Q3, and the maximum value. Draw a number line that includes the range of the data. Above this line, draw a box from Q1 (125) to Q3 (140). Mark the median (132.5) inside the box. Finally, draw 'whiskers' from the minimum value (120) to Q1 and from Q3 to the maximum value (150).
07

- Identify the 5-Number Summary

The five-number summary includes:1. Minimum: 1202. First Quartile (Q1): 1253. Median (Q2): 132.54. Third Quartile (Q3): 1405. Maximum: 150

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

5-Number Summary
A 5-number summary provides a quick snapshot of a dataset. It consists of:
  • Minimum Value (Q0): The smallest number in the dataset
  • First Quartile (Q1): The median of the lower half of the dataset
  • Median (Q2): The middle number that divides the dataset into two equal parts
  • Third Quartile (Q3): The median of the upper half of the dataset
  • Maximum Value (Q4): The largest number in the dataset
For the given blood pressure readings, the 5-number summary looks like this:
  • Minimum: 120
  • First Quartile (Q1): 125
  • Median (Q2): 132.5
  • Third Quartile (Q3): 140
  • Maximum: 150
Understanding the 5-number summary is key to grasping the distribution of data quickly.
Quartiles
Quartiles divide your dataset into four equal parts. They help in understanding the spread and center of the data.
There are three quartiles:
  • First Quartile (Q1): It is the median of the lower half of the data. It marks the 25th percentile. For our dataset, Q1 is 125.
  • Second Quartile (Q2 or Median): It divides the data into two equal halves. For our dataset, Q2 is 132.5.
  • Third Quartile (Q3): It is the median of the upper half. It marks the 75th percentile. For our dataset, Q3 is 140.
Quartiles help us find the interquartile range (IQR), which is calculated as Q3 - Q1 and measures the spread of the middle 50% of the data.
Data Visualization
Data visualization makes it easier to see patterns, trends, and outliers. A commonly used method is the boxplot.
A boxplot consists of:
  • A box that extends from Q1 (25th percentile) to Q3 (75th percentile): The box contains the middle 50% of the data.
  • A line inside the box that marks the median (Q2)
  • 'Whiskers' that extend from the box to the minimum and maximum values, unless there are outliers
To construct a boxplot for our data:
  1. Draw a number line that covers the range from the minimum to the maximum value.
  2. Draw a box from Q1 (125) to Q3 (140).
  3. Mark the median (132.5) inside the box.
  4. Extend whiskers from Q1 to the minimum value (120) and from Q3 to the maximum value (150).
Drawing a boxplot helps visually summarize the distribution of the dataset, showing its spread and central tendency.
Median Calculation
The median is the middle value in an ordered dataset. It splits the data into two equal halves.
To find the median:
  1. Arrange the data in ascending order. For our dataset: 120, 120, 125, 130, 130, 130, 130, 135, 138, 140, 140, 143, 144, 150.
  2. Determine whether the number of data points is odd or even. Here, we have 14 readings (an even number).
  3. Since we have an even number of values, take the average of the 7th and 8th values. In this case, (130 + 135)/2 = 132.5.
The median value for our dataset is 132.5. This is crucial because it indicates where the center of the data lies, offering insight into the typical blood pressure reading for the person measured by the medical students.

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Most popular questions from this chapter

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