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Chapter 3: Problem 4

Find the mean, median, and mode of the data set \(\begin{array}{lllll}10 & 12 & 20 & 15 & 20\end{array}\)

Short Answer

Expert verified
Mean: 15.4, Median: 15, Mode: 20.

Step by step solution

01

Organize the Data Set

Start by organizing the numbers in the data set in ascending order. This helps in finding the median and makes identifying the mode easier.Data in ascending order: \(10, 12, 15, 20, 20\)
02

Calculate the Mean

The mean is the average of all the numbers in the data set. Add all the numbers and divide by the total number of values.\[\text{Mean} = \frac{10 + 12 + 15 + 20 + 20}{5} = \frac{77}{5} = 15.4\]
03

Find the Median

The median is the middle number in the ordered data set. With an odd number of values (5 in this case), the median is the third number.Median: \(15\)
04

Identify the Mode

The mode is the number that appears most frequently in the data set. Look for the most repeated number in the organized list.Mode: \(20\) since it appears twice.

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Key Concepts

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

Mean
In statistics, the mean is commonly known as the average. It provides a simple way of determining the central tendency of a data set. Here鈥檚 how you calculate the mean from a set of numbers:
  • Step 1: Add up all the numbers in the data set. For our example data set \( \{10, 12, 15, 20, 20\} \), the sum is \(10 + 12 + 15 + 20 + 20 = 77\).
  • Step 2: Count the number of elements in the data set. This data set has 5 numbers.
  • Step 3: Divide the total sum by the number of elements. That means we have: \( \frac{77}{5} = 15.4 \).
The mean offers a quick snapshot of the overall data, useful for comparing different data sets or tracking changes over time.
Median
The median represents the middle value in an ordered data set. It is a way to describe the center of a distribution that is not skewed by extreme scores. Here's how you find the median:
  • Step 1: Arrange the numbers in ascending order. In our case, the numbers are rearranged as \(10, 12, 15, 20, 20\).
  • Step 2: Identify the middle number. Since our data set contains 5 elements (an odd number), the median is the third number: \(15\).
The median is particularly useful in skewed distributions because it is not affected by extremely high or low values, providing a more accurate central point of the data.
Mode
The mode is the number that appears most frequently in a data set. Unlike the mean and the median, the mode must be a real number from the data set. Let鈥檚 find the mode for our data:
  • Step 1: Look at the organized list of numbers: \(10, 12, 15, 20, 20\).
  • Step 2: Identify the number that appears most frequently. In our set, \(20\) appears twice, which is more than any other number.
You should note that a data set can have one mode, more than one mode, or even no mode at all. In this case, the mode gives us insight into the most common value in the data set. This can be particularly useful in identifying patterns or concentrations in survey or experimental data.

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Most popular questions from this chapter

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Consider the following ordered data: \(\begin{array}{ccccccccc}2 & 5 & 5 & 6 & 7 & 7 & 8 & 9 & 10\end{array}\) (a) Find the low, \(Q_{1}\), median, \(Q_{3}\), high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.

Each of the following data sets has a mean of \(\bar{x}=10 .\) \(\begin{array}{llllll}\text { (i) } 8 & 9 & 10 & 11 & 12\end{array}\) \(\begin{array}{llllll}\text { (ii) } 7 & 9 & 10 & 11 & 13\end{array}\) \(\begin{array}{llllll}\text { (iii) } 7 & 8 & 10 & 12 & 13\end{array}\) (a) Without doing any computations, order the data sets according to increasing value of standard deviations. (b) Why do you expect the difference in standard deviations between data sets (i) and (ii) to be greater than the difference in standard deviations between data sets (ii) and (iii)? Hint: Consider how much the data in the respective sets differ from the mean.
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