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Chapter 3: Problem 1

Consider a sample with data values of \(10,20,12,17,\) and \(16 .\) Compute the mean and median.

Short Answer

Expert verified
Mean = 15; Median = 16.

Step by step solution

01

Define the Data Set

We are given the data values: 10, 20, 12, 17, and 16. We will use these numbers to compute the mean and median.
02

Calculate the Mean

To find the mean (average), sum up all the numbers and then divide by the number of values. Sum = 10 + 20 + 12 + 17 + 16 = 75. The number of values = 5.Mean = \( \frac{75}{5} = 15 \).
03

Arrange Data in Ascending Order

Order the numbers from least to greatest to find the median: 10, 12, 16, 17, 20.
04

Calculate the Median

The median is the middle value in an ordered data set. Since there are 5 numbers, the median is the third number: 16.

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Key Concepts

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

Mean
In descriptive statistics, understanding the concept of the mean is crucial. The mean, frequently referred to as the average, is a measure of central tendency used to summarize a set of numbers. To find the mean, follow these simple steps:
  • First, add up all the numbers in your data set. For instance, with the values 10, 20, 12, 17, and 16, the total sum is 75.
  • Next, count how many numbers are in your data set. Here, there are 5 numbers.
  • Finally, divide the total sum by the number of numbers. So, the mean is calculated as \( \frac{75}{5} = 15 \).
Mean provides a quick snapshot of the data, representing its overall average. However, always consider outliers as they can skew the mean considerably. In our sample, no number is unusually higher or lower than others, so the mean of 15 accurately reflects our data set's center.
Median
The median is another important concept in descriptive statistics. It helps find the middle value of a data set, which can be particularly useful when your data contains outliers that might distort the mean.
To determine the median, follow these steps:
  • Firstly, arrange the numbers in your data set in ascending order. Using our example, order 10, 12, 16, 17, 20 in increasing sequence.
  • Next, locate the middle number. Because our data set contains five numbers, the median is the third number, which is 16 in this case.
The median provides a robust measure of central tendency, especially valuable when the data is skewed. Unlike the mean, it is not affected by extremely large or small values.
Data Analysis
Data analysis is a powerful tool for extracting useful insights from a data set. Understanding measures like the mean and median allows you to summarize and elucidate patterns within your data.
The process of data analysis can involve several crucial steps:
  • Collecting Data: Gather relevant data that you want to analyze.
  • Organizing Data: Arrange your data in a structured format, often in tables or lists.
  • Analyzing Data: Use statistical methods like calculating the mean and median to find insights.
  • Interpreting Data: Draw conclusions about the information represented by your data, looking for trends or outliers.
By applying these elementary concepts of mean, median, and basic data analysis, you can transform raw data into meaningful intelligence, aiding in decision-making processes throughout diverse fields.

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Most popular questions from this chapter

The national average for the verbal portion of the College Board's Scholastic Aptitude Test (SAT) is 507 (The World Almanac, 2006 ). The College Board periodically rescales the test scores such that the standard deviation is approximately \(100 .\) Answer the following questions using a bell-shaped distribution and the empirical rule for the verbal test scores. a. What percentage of students have an SAT verbal score greater than \(607 ?\) b. What percentage of students have an SAT verbal score greater than \(707 ?\) c. What percentage of students have an SAT verbal score between 407 and \(507 ?\) d. What percentage of students have an SAT verbal score between 307 and \(607 ?\)

Consider a sample with data values of \(27,25,20,15,30,34,28,\) and \(25 .\) Provide the fivenumber summary for the data.

Consider a sample with data values of \(10,20,12,17,\) and \(16 .\) Compute the \(z\) -score for each of the five observations.

Suppose the data have a bell-shaped distribution with a mean of 30 and a standard deviation of \(5 .\) Use the empirical rule to determine the percentage of data within each of the following ranges. a. 20 to 40 b. 15 to 45 c. 25 to 35

How do grocery costs compare across the country? Using a market basket of 10 items including meat, milk, bread, eggs, coffee, potatoes, cereal, and orange juice, Where to Retire magazine calculated the cost of the market basket in six cities and in six retirement areas across the country (Where to Retire, November/December 2003 ). The data with market basket cost to the nearest dollar are as follows: $$\begin{array}{lclr} \text { City } & \text { cost } & \text { Retirement Area } & \text { cost } \\\ \text { Buffalo, NY } & \$ 33 & \text { Biloxi-Gulfport, MS } & \$ 29 \\ \text { Des Moines, IA } & 27 & \text { Asheville, NC } & 32 \\ \text { Hartford, CT } & 32 & \text { Flagstaff, AZ } & 32 \\ \text { Los Angeles, CA } & 38 & \text { Hilton Head, SC } & 34 \\ \text { Miami, FL } & 36 & \text { Fort Myers, FL } & 34 \\ \text { Pittsburgh, PA } & 32 & \text { Santa Fe, NM } & 31\end{array}$$ a. Compute the mean, variance, and standard deviation for the sample of cities and the sample of retirement areas. b. What observations can be made based on the two samples?
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