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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

Arrange Data in Ascending Order

First, we need to sort the data in ascending order to identify quartiles. The given data is: 7, 37, 3, 8, 48, 11, 6, 0, 10. When sorted, it becomes: 0, 3, 6, 7, 8, 10, 11, 37, 48.
02

Find the First Quartile (Q1)

The first quartile is the value below which 25% of the data falls. In a dataset of 9 numbers, the first quartile can be found at the position \( Q1 = \frac{9 + 1}{4} = 2.5 \). This means we need the average of the 2nd and 3rd data points. The 2nd point is 3 and the 3rd point is 6. Thus, \( Q1 = \frac{3 + 6}{2} = 4.5 \).
03

Find the Third Quartile (Q3)

The third quartile is the value below which 75% of the data falls. In the same dataset of 9 numbers, the third quartile is found at position \( Q3 = \frac{3(9 + 1)}{4} = 7.5 \). We need the average of the 7th and 8th data points. The 7th data point is 11 and the 8th is 37. Therefore, \( Q3 = \frac{11 + 37}{2} = 24 \).

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Key Concepts

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

Quartiles
Quartiles are all about dividing your dataset into four equal parts, making it easier to understand and interpret. The first quartile, known as Q1, represents the 25th percentile of the data. This means 25% of the data values are below Q1, while 75% are above it.
The third quartile, Q3, marks the 75th percentile, which signifies that 75% of the data falls below this point, and only 25% is above.
To quickly find these quartiles:
  • Arrange your data in ascending order.
  • Use the formula for the first quartile: \( Q1 = \frac{n+1}{4} \), where \( n \) is the number of data points.
  • Apply the formula for the third quartile: \( Q3 = \frac{3(n+1)}{4} \).
It's that simple! The quartiles provide a simplified view of the dataset, giving insight into its spread and identifying any outliers. Knowing quartiles helps students understand the distribution of data beyond just mean and median.
Data Analysis
When it comes to statistics, data analysis is essential for interpreting different trends, patterns, and outliers within a dataset. It involves organizing, transforming, and modeling the raw data to extract useful information. In practical terms, data analysis has several key stages:
  • Data Collection: Gathering reliable data is the first step. This could come from surveys, experiments, or databases.
  • Data Cleaning: Remove any inconsistencies or errors from the dataset. This could include fixing typos or dealing with missing values.
  • Exploratory Data Analysis (EDA): This part involves summarizing the main characteristics with visual aids like graphs and tables.
  • Data Modeling: This stage involves applying statistical models to test hypotheses or forecast future trends.
For students studying educational statistics, understanding data analysis is crucial. It equips them to critically interpret statistical results and make data-driven decisions.
Educational Statistics
Educational statistics focuses on applying statistical methods to challenges in the educational domain. It encompasses the data-driven analysis of various educational problems to improve learning outcomes and policy making.
An important aspect of educational statistics is ensuring data is presented in an actionable and comprehensible format. Some common applications include:
  • Assessing Student Performance: Using data to identify trends in achievement and areas for intervention.
  • Resource Allocation: Analyzing data to efficiently distribute educational resources across districts or schools.
  • Program Evaluation: Measuring the effectiveness of educational programs by using statistical techniques.
With the application of educational statistics, educators can foster environments that improve student learning and well-being. It empowers educators with the tools needed to make informed decisions that lead to educational innovation.

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Most popular questions from this chapter

The harmonic mean (HM) is defined as the number of values divided by the sum of the reciprocals of each value. The formula is $$\mathrm{HM}=\frac{n}{\Sigma(1 / X)}$$ For example, the harmonic mean of \(1,4,5,\) and 2 is $$\mathrm{HM}=\frac{4}{1 / 1+1 / 4+1 / 5+1 / 2} \approx 2.051$$ This mean is useful for finding the average speed. Suppose a person drove 100 miles at 40 miles per hour and returned driving 50 miles per hour. The average miles per hour is not 45 miles per hour, which is found by adding 40 and 50 and dividing by 2. The average is found as shown. Since $$\text { Time }=\text { distance } \div \text { rate }$$ then $$\begin{array}{l}{\text { Time } 1=\frac{100}{40}=2.5 \text { hours to make the trip }} \\ {\text { Time } 2=\frac{100}{50}=2 \text { hours to return }}\end{array}$$ Hence, the total time is 4.5 hours, and the total miles driven are \(200 .\) Now, the average speed is $$\text { Rate }=\frac{\text { distance }}{\text { time }}=\frac{200}{4.5} \approx 44.444 \text { miles per hour }$$ This value can also be found by using the harmonic mean formula $$\mathrm{HM}=\frac{2}{1 / 40+1 / 50} \approx 44.444$$ Using the harmonic mean, find each of these. a. A salesperson drives 300 miles round trip at 30 miles per hour going to Chicago and 45 miles per hour returning home. Find the average miles per hour. b. A bus driver drives the 50 miles to West Chester at 40 miles per hour and returns driving 25 miles per hour. Find the average miles per hour. c. A carpenter buys \(\$ 500\) worth of nails at \(\$ 50\) per pound and \(\$ 500\) worth of nails at \(\$ 10\) per pound. Find the average cost of 1 pound of nails.

The number of annual precipitation days for one-half of the 50 largest U.S. cities is listed below. Find the range, variance, and standard deviation of the data. \(\begin{array}{llllllllll}{135} & {128} & {136} & {78} & {116} & {77} & {111} & {79} & {44} & {97} \\ {116} & {123} & {88} & {102} & {26} & {82} & {156} & {133} & {107} & {35} \\ {112} & {98} & {45} & {122} & {125} & {}\end{array}\)

Starting teachers鈥� salaries (in equivalent U.S. dollars) for upper secondary education in selected countries are listed. Find the range, variance, and standard deviation for the data. Which set of data is more variable? (The U.S. average starting salary at this time was $\$ 29,641.) $$\begin{array}{llll}{\text { Sweden }} & {\$ 48,704} & {\text { Korea }} & {\$ 26,852} \\ {\text { Germany }} & {41,441} & {\text { Japan }} & {23,493} \\\ {\text { Spain }} & {32,679} & {\text { India }} & {18,247} \\ {\text { Finland }} & {32,136} & {\text { Malaysia }} & {13,647} \\ {\text { Denmark }} & {30,384} & {\text { Philippines }} & {9,857} \\ {\text { Netherlands }} & {29,326} & {\text { Thailand }} & {5,862} \\ {\text { Scotland }} & {27,789} & {}\end{array}$$

The mean and standard deviation of the bonuses that the employees of a company received 10 years ago were, respectively, \(\$ 2,000\) and \(\$ 325 .\) Today the amount of the bonuses is 5 times what it was 10 years ago. Find the mean and standard deviation of the new bonuses.

The mean price of the fish in a pet shop is \(\$ 2.17\), and the standard deviation of the price is \(\$ 0.55 .\) If the owner decides to triple the prices, what will be the mean and standard deviation of the new prices?
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