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Chapter 1: Problem 38

Blood pressure values are often reported to the nearest \(5 \mathrm{mmHg}(100,105,110\), etc.). Suppose the actual blood pressure values for nine randomly selected individuals are \(\begin{array}{lllllll}118.6 & 127.4 & 138.4 & 130.0 & 113.7 & 122.0 & 108.3\end{array}\) \(131.5 \quad 133.2\) a. What is the median of the reported blood pressure values? b. Suppose the blood pressure of the second individual is \(127.6\) rather than \(127.4\) (a small change in a single value). How does this affect the median of the reported values? What does this say about the sensitivity of the median to rounding or grouping in the data?

Short Answer

Expert verified
a. Median is 125. b. After adjustment, median is 130, indicating sensitivity to rounding.

Step by step solution

01

Arrange the Data

First, arrange the given blood pressure values in numerical order to find the median. The actual blood pressure values are: 108.3, 113.7, 118.6, 122.0, 127.4, 130.0, 131.5, 133.2, and 138.4.
02

Identify the Median

With nine data points (an odd number), the median is the middle value in the ordered list. Hence, the fifth value is the median in this list. The fifth value is 127.4.
03

Round Values to Reported Blood Pressure

Round each of the nine values to the nearest 5 mmHg to get the reported values: 110, 115, 120, 120, 125, 130, 130, 135, and 140.
04

Find the Median of Reported Values

With the rounded values: 110, 115, 120, 120, 125, 130, 130, 135, and 140, the median is the fifth value, which is 125.
05

Adjust the Second Individual's Value

Change the second individual's original blood pressure from 127.4 to 127.6. The reordered list becomes: 108.3, 113.7, 118.6, 122.0, 127.6, 130.0, 131.5, 133.2, and 138.4.
06

Round the Adjusted Values

After rounding, the values are: 110, 115, 120, 120, 130, 130, 130, 135, and 140.
07

Recalculate the Median of Reported Values

After the rounding adjustments, the median of these reported values is now 130.
08

Assess Sensitivity to Changes

The median has increased from 125 to 130 after a slight change in the original data, indicating that the median can be sensitive to rounding or grouping of the data in reported measurements.

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Key Concepts

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

Rounding Errors
Rounding errors occur when the true value of a number is adjusted to a nearby approximation. This adjustment introduces small differences, which we call rounding errors. In statistical analysis, these errors might seem trivial, but they can significantly impact measures like the median.
For instance, when blood pressure values are rounded to the nearest 5 mmHg, small rounding differences can alter the dataset. The original data:
  • 118.6 rounds to 120
  • 127.4 rounds to 125, yet 127.6 rounds to 130

Small changes in actual values can lead to different rounding results, impacting the calculated median. Rounding can introduce biases, hence accuracy is vital during data collection and reporting.
Data Sensitivity
Data sensitivity refers to how a statistical measure, like the median, responds to small changes in the dataset. The median is typically considered robust since it resists influence by extreme values or outliers.
However, as seen in the exercise, it does exhibit sensitivity to rounding. A minuscule change in the original blood pressure data (from 127.4 to 127.6) shifted the median of the reported rounded data.
This highlights that while the median is generally stable, it's not entirely insensitive to how data is rounded or grouped.
Numerical Ordering
Numerical ordering involves arranging data from smallest to largest. This step is crucial when determining the median. The order of data helps specify which value is the median鈥攅specially in a list with an odd number of entries.
To find the median, first write the data in ascending numerical order:
  • 108.3, 113.7, 118.6, 122.0, 127.4, 130.0, 131.5, 133.2, 138.4

The middle value here, 127.4, is identified as the median. Proper numerical ordering ensures accuracy when calculating statistical measures such as the median, mean, or mode.
Statistical Measures
Statistical measures help describe, summarize, and understand data. They include the mean (average), median, mode (most frequent value), and range (difference between high and low values).
The median, the focus of this exercise, offers a central point of a data set which is less affected by outliers compared to the mean.
In practical terms, finding the median involves organizing the data and selecting the middle value. Even minor adjustments in numerically close values can affect this measure, as seen in the changes caused by rounding. It's essential to remain conscious of these impacts when analyzing and presenting statistical data.

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Most popular questions from this chapter

The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. Would this be an enumerative or an analytic study? Explain your reasoning.

The article "Can We Really Walk Straight?" (Amer. J. of Physical Anthropology, 1992: 19-27) reported on an experiment in which each of 20 healthy men was asked to walk as straight as possible to a target \(60 \mathrm{~m}\) away at normal speed. Consider the following observations on cadence (number of strides per second): \(\begin{array}{rrrrrrrrrr}.95 & .85 & .92 & .95 & .93 & .86 & 1.00 & .92 & .85 & .81 \\ .78 & .93 & .93 & 1.05 & .93 & 1.06 & 1.06 & .96 & .81 & .96\end{array}\) Use the methods developed in this chapter to summarize the data; include an interpretation or discussion wherever appropriate. [Note: The author of the article used a rather sophisticated statistical analysis to conclude that people cannot walk in a straight line and suggested several explanations for this.]

The amount of flow through a solenoid valve in an automobile's pollution- control system is an important characteristic. An experiment was carried out to study how flow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor were chosen, and a single observation on flow was made for each combination of levels. a. The resulting data set consisted of how many observations? b. Is this an enumerative or analytic study? Explain your reasoning.

Calculate and interpret the values of the sample median, sample mean, and sample standard deviation for the following observations on fracture strength (MPa, read from a graph in "Heat-Resistant Active Brazing of Silicon Nitride: Mechanical Evaluation of Braze Joints," Welding J., August, 1997): \(\begin{array}{llllllllll}87 & 93 & 96 & 98 & 105 & 114 & 128 & 131 & 142 & 168\end{array}\)

Let \(\bar{x}_{n}\) and \(s_{n}^{2}\) denote the sample mean and variance for the sample \(x_{1}, \ldots, x_{n}\) and let \(\bar{x}_{n+1}\) and \(s_{n+1}^{2}\) denote these quantities when an additional observation \(x_{n+1}\) is added to the sample. a. Show how \(\bar{x}_{n+1}\) can be computed from \(\bar{x}_{n}\) and \(x_{n+1}\). b. Show that $$ n s_{n+1}^{2}=(n-1) s_{n}^{2}+\frac{n}{n+1}\left(x_{n+1}-\bar{x}_{n}\right)^{2} $$ so that \(s_{n+1}^{2}\) can be computed from \(x_{n+1}, \bar{x}_{n}\), and \(s_{n}^{2}\) c. Suppose that a sample of 15 strands of drapery yarn has resulted in a sample mean thread elongation of \(12.58 \mathrm{~mm}\) and a sample standard deviation of \(.512 \mathrm{~mm}\). A 16 th strand results in an elongation value of \(11.8\). What are the values of the sample mean and sample standard deviation for all 16 elongation observations?
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