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Chapter 58: Problem 1

Determine the mean, median and mode for the set: $$ \\{2,3,7,5,5,13,1,7,4,8,3,4,3\\} $$

Short Answer

Expert verified
Mean: 5, Median: 4, Mode: 3

Step by step solution

01

Arrange the Data

First, let's arrange the data set in ascending order: \[ \{1, 2, 3, 3, 3, 4, 4, 5, 5, 7, 7, 8, 13\} \]
02

Calculate the Mean

To find the mean, sum all the numbers in the data set and divide by the count of numbers. Sum: \[ 1 + 2 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 7 + 7 + 8 + 13 = 65 \]Count of numbers: 13Mean: \[ \frac{65}{13} = 5 \]
03

Determine the Median

The median is the middle value when the data set is ordered. Since there are 13 numbers (an odd number), the median is the 7th value. Median: \[ 4 \]
04

Identify the Mode

The mode is the number that appears most frequently in the data set. The numbers 3 and 5 each appear three times, but 3 appears first.Mode: \[ 3 \]

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Key Concepts

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

Mean Calculation
The mean is a way to find the average of a data set. To calculate it, you sum all the numbers in the set, then divide the total by the number of values. For the set \(\{2,3,7,5,5,13,1,7,4,8,3,4,3\}\), the sum is 65. The total number of elements is 13, so the mean is calculated as follows:
\[ \frac{65}{13} = 5 \]
Using the mean allows for a single value representation of the entire data set, giving you an idea of the central tendency.
Median Calculation
The median represents the middle value of a data set once it is ordered. When arranging numbers from smallest to largest, you can easily find the middle element. If the number of data points is odd, like the 13 elements in our set, the median is the 7th value from the sorted list: \( \{1, 2, 3, 3, 3, 4, 4, 5, 5, 7, 7, 8, 13\} \).
The median here is \(4\), a value that divides the data into two equal halves, providing insight into the data's distribution.
Mode Determination
The mode indicates the value that appears most frequently in a data set. It helps identify the most common element. In our example, we have two numbers, 3 and 5, each repeating three times. When you encounter this situation, you can list both as modes if necessary. However, traditionally, the first occurring number is often prioritized if one number needs to be chosen. Hence, the mode of this set is \(3\).
Recognizing the mode is beneficial in understanding common trends within the data.
Data Arrangement
Arranging data involves sorting the elements in a sequence, usually from the smallest to largest value. This step is crucial in calculating both the median and mode. For our data, the sorted order is \( \{1, 2, 3, 3, 3, 4, 4, 5, 5, 7, 7, 8, 13\} \).
Sorting provides a clear picture and aids in identifying patterns or anomalies. It's a foundational step in many statistical analyses, helping you navigate complex data sets with ease.

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Most popular questions from this chapter

. The following set of data refers to the amount of money in pounds taken by a news vendor for sixdays. Determine the mean, median and modal values of the set: $$ \\{27.90,34.70,54.40,18.92,47.60,39.68\\} $$

Determine the standard deviation from the mean of the set of numbers: \(\\{5,6,8,4,10,3\\}\) correct to 4 significant figures.

Determine the numbers contained in the (a) 41 st to 50 th percentile group, and (b) eighth decile group of the set of numbers shown below: \(\begin{array}{llllllllll}14 & 22 & 17 & 21 & 30 & 28 & 37 & 7 & 23 & 32 \\\ 24 & 17 & 20 & 22 & 27 & 19 & 26 & 21 & 15 & 29\end{array}\)
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