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Chapter 2: Problem 139

Bad statistic A teacher summarizes grades on an exam by \(\operatorname{Min}=26, \mathrm{Q} 1=67, \mathrm{Q} 2=80, \mathrm{Q} 3=87, \operatorname{Max}=100\), Mean \(=76,\) Mode \(=100,\) Standard deviation \(=76\) \(\mathrm{IQR}=20\) She incorrectly recorded one of these. Which one do you think it was? Why?

Short Answer

Expert verified
The Standard Deviation is likely incorrect as it does not match the tight grade distribution suggested by other statistics.

Step by step solution

01

Understand the Summary Statistics

The teacher provides key statistics including Minimum, Quartiles, Maximum, Mean, Mode, Standard Deviation, and Interquartile Range. Ensure you understand each term: Minimum is the lowest score; Maximum is the highest; Q1, Q2 (Median), and Q3 are quartiles; Mean is the average; Mode is the most frequent score; Standard Deviation (SD) measures data spread; IQR is the range between Q1 and Q3.
02

Analyze the Mean vs Mode and Maximum

Check if the Mean and Mode make sense together with the provided data. Mode is 100, which implies that many students scored 100. If the Mode value is frequent, the Mean should not be far from the Mode unless very low values skew the data. The given Mean is 76, which is unrealistically low if 100 is the Mode and indicates many scores around 100.
03

Evaluate the Standard Deviation

Standard Deviation indicates data spread. A very large SD suggests wide variance in scores. The provided Standard Deviation is 76, which is very high and implies extreme spread. Most scores would need to be far from 76 to create such variability, contradicting the presence of Mode = 100 with many repetitions.
04

Assess the Consistency of the Statistics

Combine insights: The Mode suggests a peak at 100. If many scores are at 100 and others are lower, the SD would be smaller. Considering high SD of 76 with Mode 100 is illogical unless there are several much lower scores, but Quartiles are relatively close (67 to 87).
05

Conclude the Error

Given the analysis, the Standard Deviation value of 76 is suspect as it contradicts the rest of the statistics. With comparatively clustered Quartiles and Mode of 100, the data isn't widely spread enough to justify such a high SD.

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Key Concepts

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

Summary Statistics
Understanding summary statistics is essential for interpreting the data's overall structure. Summary statistics provide a snapshot of different aspects of a dataset. They often include:
  • Minimum and Maximum: The smallest and largest values, respectively. They show the data range.
  • Mean: The average of all the data points, a central value that provides a general sense of the dataset.
  • Mode: The most frequently appearing value in the dataset. It highlights the most common result.
  • Quartiles: These divide the data into four equal parts. Q1 (first quartile) is the score below which 25% of the data falls, Q2 (second quartile) is the median, and Q3 (third quartile) cuts off the lowest 75% of the data.
By analyzing these values, we can infer the dataset's distribution and central tendencies. In the exercise, the given values suggest a generally high performance, yet they are inconsistent, notably the mode and the mean.
Standard Deviation
The standard deviation (SD) measures how spread out numbers are in a dataset. A higher SD means the data is more spread out from the mean, whereas a lower SD means data points are closer to the mean. The formula for standard deviation is:
\[ SD = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \bar{x})^2} \]
Where:
  • \( x_i \) represents each data point,
  • \( \bar{x} \) is the mean,
  • \( N \) is the number of data points.
A very high standard deviation, such as 76 in this context, indicates wide variability. Given the clustered quartiles and high mode, such an SD is not logical. It suggests an error since the scores aren't spread enough to justify such a number, possibly due to an error in recording.
Quartiles
Quartiles are statistical data points that divide the data into quarters after it has been sorted in ascending order. They play a critical role in understanding the distribution's central orientation and variability.
  • Q1 (First Quartile): Separates the lowest 25% of the data set. Located at the 25th percentile.
  • Q2 (Second Quartile, Median): Divides the data set into two equal halves. Located at the 50th percentile and is a critical measure of central tendency.
  • Q3 (Third Quartile): Separates the lowest 75% from the highest 25%. Located at the 75th percentile.
The interquartile range (IQR) is the difference between Q3 and Q1, providing a measure of variability around the median. In our scenario, the quartiles indicate a relatively consistent data spread. This contradicts the unusually high standard deviation of 76, suggesting the numbers aren't as dispersed as such an SD would require. This emphasizes the importance of the quartiles in identifying potential anomalies or errors within a dataset.

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Most popular questions from this chapter

European Union unemployment rates The 2007 unemployment rates of countries in the European Union shown in Exercise 2.64 ranged from 3.2 to \(8.7,\) with \(\mathrm{Q} 1=4.5\), median \(=6.7, \mathrm{Q} 3=7.8\), a mean of \(6.3,\) and standard deviation of 1.8 . a. In a box plot, what would be the values at the outer edges of the box, and what would be the values to which the whiskers extend? b. Greece had the highest unemployment rate of 8.7 . Is it an outlier according to the 3 standard deviation criterion? Explain. c. What unemployment value for a country would have a \(z\) -score equal to \(0 ?\)

Categorical or quantitative? Identify each of the following variables as either categorical or quantitative. a. Choice of diet (vegetarian, nonvegetarian) b. Time spent in previous month attending a place of religious worship c. Ownership of a personal computer (yes, no) d. Number of people you have known who have been elected to a political office

Vacation days \(\quad\) National Geographic Traveler magazine recently presented data on the annual number of vacation days averaged by residents of eight different countries. They reported 42 days for Italy, 37 for France, 35 for Germany, 34 for Brazil, 28 for Britain, 26 for Canada, 25 for Japan, and 13 for the United States. a. Report the median. b. By finding the median of the four values below the median, report the first quartile. c. Find the third quartile. d. Interpret the values found in parts \(a-c\) in the context of these data.

Controlling asthma A study of 13 children suffering from asthma (Clinical and Experimental Allergy, vol. 20 , pp. \(429-432,1990\) ) compared single inhaled doses of formoterol (F) and salbutamol (S). Each child was evaluated using both medications. The outcome measured was the child's peak expiratory flow (PEF) eight hours following treament. Is there a difference in the PEF level for the two medications? The data on PEF follow: $$ \begin{array}{ccc} \hline \text { Child } & \text { F } & \text { S } \\ \hline 1 & 310 & 270 \\ 2 & 385 & 370 \\ 3 & 400 & 310 \\ 4 & 310 & 260 \\ 5 & 410 & 380 \\ 6 & 370 & 300 \\ 7 & 410 & 390 \\ 8 & 320 & 290 \\ 9 & 330 & 365 \\ 10 & 250 & 210 \\ 11 & 380 & 350 \\ 12 & 340 & 260 \\ 13 & 220 & 90 \\ \hline \end{array} $$ a. Construct plots to compare formoterol and salbutamol. Write a short summary comparing the two distributions of the peak expiratory flow. b. Consider the distribution of differences between the PEF levels of the two medications. Find the 13 differences and construct and interpret a plot of the differences. If on the average there is no difference between the PEF level for the two brands, where would you expect the differences to be centered?

Graphing exam scores A teacher shows her class the scores on the midterm exam in the stem-and-leaf plot shown: $$ \begin{array}{l|l} 6 & 588 \\ 7 & 01136779 \\ 8 & 1223334677789 \\ 9 & 011234458 \end{array} $$ a. Identify the number of students and their minimum and maximum scores. b. Sketch how the data could be displayed in a dot plot. c. Sketch how the data could be displayed in a histogram with four intervals.
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