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Chapter 6: Problem 12

Find the first, second, and third quartiles of the data. $$20,73,31,53,22,64,47$$

Short Answer

Expert verified
The first quartile (Q1) is 22, the second quartile (Q2 or the median) is 47, and the third quartile (Q3) is 64.

Step by step solution

01

Arrange the data in ascending order

The first step is important because the quartiles are based on the order of the data. The arranged data would be \(20, 22, 31, 47, 53, 64, 73\).
02

Find the Median (Q2)

The median is the middle value of an arranged data set. If the set has an odd number of observations, the median is the middle number. If the set has an even number of observations, the median is calculated by averaging the two middle numbers. Since this set has 7 numbers, the median is the 4th number which is 47. So, Q2 or the median is 47.
03

Find Q1

Q1 is the median of the lower half of data (not including Q2 if the number of data points is odd). The lower half of our data is 20, 22, 31. So, Q1 is the second number in the lower half which is 22.
04

Find Q3

Q3 is the median of the upper half of data (not including Q2 if the number of data points is odd). The upper half of our data is 53, 64, 73. So, Q3 is the second number in the upper half which is 64.

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Key Concepts

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

Median
The median is a central concept in statistics, helping us find the middle of a data set. Imagine you have a number of boxes of different sizes. To find the median box size, you would first line them up from smallest to largest.
Once arranged, the median is simply the box in the middle.
This ensures that half of the boxes are smaller and half are larger.

For a set of numbers, like in our exercise, the concept is the same.
  • If the data set has an odd number of observations, the median is exactly the middle number.
  • If there is an even number, you find the average of the two central numbers.
This middle value represents the typical sample in statistics, giving us a sense of the center of our dataset without being affected by extremely high or low values.
Data Arrangement
Data arrangement is the foundation for several statistical calculations. Before you can analyze or find values like the median, your data needs to be in a logical order.
In most cases, this means placing numbers from smallest to largest.

This ordering helps identify patterns and make sense of the data.
  • An unordered list can mislead or make it more challenging to understand the dataset characteristics.
  • Arranging data properly is crucial for accurate calculations of other statistics like quartiles or percentiles.
In the exercise, arranging the numbers in ascending order is a vital step that precedes finding the median and quartiles. This is a common starting point in statistical evaluations.
Statistics
Statistics is like the science toolbox for handling data. It includes a variety of techniques and methods to collect, analyze, interpret, and present data.
Understanding statistical concepts allows us to make informed decisions based on data.

In our exercise, we used several statistical concepts to find quartiles.
  • Statistics teaches us how to order data to find measures like the median.
  • It provides methods for summarizing extensive datasets into understandable insights through values like the mean or standard deviation.
Every concept used is part of the larger statistical landscape that aids in processing and interpreting information, helping us run accurate analyses and draw meaningful conclusions.

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