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Chapter 3: Problem 28

The number of incidents in which police were needed for a sample of 9 schools in Allegheny County is \(7,37,3,8,48,11,6,0,10 .\) Find the first and third quartiles for the data.

Short Answer

Expert verified
The first quartile is 4.5, and the third quartile is 24.

Step by step solution

01

Order the Data

Begin by organizing the data in ascending order. The original data set is \(\{7, 37, 3, 8, 48, 11, 6, 0, 10\}\). When ordered, it becomes \(\{0, 3, 6, 7, 8, 10, 11, 37, 48\}\).
02

Identify Quartile Positions

To find the quartiles, we first determine their positions. With 9 data points: \(Q_1 = (\frac{1(9+1)}{4}) = 2.5\) and \(Q_3 = (\frac{3(9+1)}{4}) = 7.5\).
03

Calculate the First Quartile (Q1)

Since \(Q_1\) is at position 2.5, we find the average of the values at positions 2 and 3 in the ordered list: \(\frac{3 + 6}{2} = 4.5\). Thus, \(Q_1 = 4.5\).
04

Calculate the Third Quartile (Q3)

For \(Q_3\), positioned at 7.5, average the values at positions 7 and 8 in the ordered list: \(\frac{11 + 37}{2} = 24\). Thus, \(Q_3 = 24\).

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Key Concepts

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

Data Ordering
Before diving into calculating quartiles, we need to sort our data. Why? Ordering data from smallest to largest helps us find positions accurately.
Think of it as arranging books on a shelf from shortest to tallest; it makes finding the right one much easier. In our example, the original data \( \{7, 37, 3, 8, 48, 11, 6, 0, 10\} \) becomes \( \{0, 3, 6, 7, 8, 10, 11, 37, 48\} \) after ordering.
  • Ensure each number is sorted.
  • Double-check for any missed data entries.
  • Revisit sorted list if your results seem odd; you might have missed a step!
By organizing data, placing quartiles becomes straightforward.
Quartile Positions
Quartiles are values that split your data into four equal parts, helping describe the spread and center of your data.
Finding their positions aids in calculating their values. Imagine dividing a pizza into equal slices; quartile positions tell you where to cut.Here's a quick breakdown:- **Position of First Quartile (Q1):** Use the formula \( \frac{1(n+1)}{4} \), where \(n\) is the number of data points. For our nine data points, this becomes \(2.5\).- **Position of Third Quartile (Q3):** Employ \(\frac{3(n+1)}{4}\), leading to \(7.5\) in our example.Knowing these positions tells us where to go next when calculating the quartile values.
First Quartile (Q1)
Now, let's compute the first quartile, or \(Q_1\). The position found was 2.5, indicating that \(Q_1\) lies between the second and third values in our ordered data list.To find \(Q_1\):- Average the numbers at positions 2 and 3: \( \frac{3 + 6}{2} = 4.5 \).Thus, \(Q_1\) equals 4.5 in this data set. It reflects the value below which 25% of data falls, highlighting lower data spread.
Third Quartile (Q3)
Calculating the third quartile, \(Q_3\), is just as straightforward. With \(Q_3\)'s position at 7.5, we look between the seventh and eighth values.Here's how to find \(Q_3\):
  • Identify values at these positions (11 and 37).
  • Average these two: \( \frac{11 + 37}{2} = 24 \).
Thus, \(Q_3 = 24\).
This means 75% of the data values lie below this point, providing insight into the upper half of your data set.
Statistical Analysis
Understanding quartiles is essential in statistical analysis. They provide insights into data variability and distribution. Here's how: - **Data Spread:** Quartiles present a data summary, showing distribution. - **Comparison Aid:** Allows comparison across different datasets. - **Outlier Detection:** Helps isolate unusual data points. By using quartiles, you gain a clearer view of data behavior.
Analysts often present this as a boxplot, visually conveying the data's distribution.
Recognizing where your data stands aids in building informed conclusions.

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Most popular questions from this chapter

Listed below are the enrollments for selected independent religiously controlled 4-year colleges that offer bachelor's degrees only. Construct a grouped frequency distribution with six classes and find the mean and modal class. $$\begin{array}{llllllllll}1013 & 1867 & 1268 & 1666 & 2309 & 1231 & 3005 & 2895 & 2166 & 1136 \\\1532 & 1461 & 1750 & 1069 & 1723 & 1827 & 1155 & 1714 & 2391 & 2155 \\\1412 & 1688 & 2471 & 1759 & 3008 & 2511 & 2577 & 1082 & 1067 & 1062 \\\1319 & 1037 & 2400 & & & & & & &\end{array}$$

A four-month record for the number of tornadoes in \(2013-2015\) is given here. $$\begin{array}{lrrr} & 2013 & 2014 & 2015 \\\\\hline \text { April } & 80 & 130 & 170 \\\\\text { May } & 227 & 133 & 382 \\\\\text { June } & 126 & 280 & 184 \\\\\text { July } & 69 & 90 & 116\end{array}$$ a. Which month had the highest mean number of tornadoes for this 3 -year period? b. Which year has the highest mean number of tornadoes for this 4 -month period? c. Construct three boxplots and compare the distributions.

An employment evaluation exam has a variance of \(250 .\) Two particular exams with raw scores of 142 and 165 have \(z\) scores of -0.5 and \(0.955,\) respectively. Find the mean of the distribution.

The increases (in cents) in cigarette taxes for 17 states in a 6 -month period are \(60,20,40,40,45,12,34,51,30,70,42,31,69,32,8,18,50\) Find the range, variance, and standard deviation for the data. Use the range rule of thumb to estimate the standard deviation. Compare the estimate to the actual standard deviation.

The national average for mathematics SATs in 2014 was 538 . Suppose that the distribution of scores was approximately bell-shaped and that the standard deviation was approximately \(48 .\) Within what boundaries would you expect \(68 \%\) of the scores to fall? What percentage of scores would be above \(634 ?\)
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