Problem 29 - Thinking Mathematically

Question

In Exercises 25-32, find the mode for each group of data items. If there is no mode, so state. \(100,40,70,40,60\)

Step-by-Step Solution

Answer

The mode of the data set \(100,40,70,40,60\) is \(40\).

Step 1: Identify the Data Set

The data set given in the exercise is: \(100, 40, 70, 40, 60\).

Step 2: Analyse the Frequency of Each Number

Analyse the frequency of each number in the data set. The frequencies are: \(100 - 1\), \(40 - 2\), \(70 - 1\), \(60 - 1\)

Step 3: Determine the Mode

The mode is the number with the highest frequency in the data set. The number \(40\) appears twice which is the maximum frequency. Therefore, \(40\) is the mode of this data set.

Key Concepts

Statistical Modes

Understanding statistical modes can be as easy as finding out which item is your friend's favorite in a collection of toys - the one they play with most often! In statistical terms, the mode is simply the data point that appears the most frequently in a given set. This can be particularly useful when analyzing categorical data, such as survey results about favorite colors, where numbers might not make sense.

When you look at a list of numbers, like in our exercise with the data set of \(100, 40, 70, 40, 60\), finding the mode gives us insight into the 'popularity' of certain numbers. If a number repeats, it's a contender for being the mode. If there's only one 'winner' that repeats more than any other number, that's the mode, plain and simple. In this case, \(40\) appears twice, making it the statistical mode of our data set.

Frequency Distribution

Imagine having a jar of mixed candies, and you want to know which candy is the most common. You'd sort them out and count each type - that's essentially what frequency distribution is about! It is a summary that shows how often each value in a set of data occurs. By organizing data into frequency tables or charts, we can see at a glance which values are the 'candies' that pop up most often and least often.

To create a frequency distribution, list each data point and then tally how many times that point appears. In our previous example, we listed and counted occurrences of the numbers: \(100 - 1\), \(40 - 2\), \(70 - 1\), and \(60 - 1\). The number \(40\), with a frequency of 2, stands out from the rest. This distribution quickly shows which number is the most frequent, which is key when searching for the mode.

Data Set Analysis

Data set analysis is like being a detective, taking a close look at clues (data points) to solve a puzzle. It involves examining, cleaning, transforming, and modeling data with the goal of highlighting useful information. Through analysis, we can make decisions, uncover patterns, or even just describe and summarize the data's characteristics.

Starting with a data set like \(100, 40, 70, 40, 60\), we may first clean it by removing errors or duplicates that don't belong. Next, we organize it for better understanding, perhaps by putting it in numerical order or categorizing similar values. Then we analyze it by calculating various statistics, like the average or median, and in our case, identifying the mode. These steps in data set analysis help us interpret what the numbers are telling us, revealing the story hidden in the raw data.