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Chapter 11: Problem 2

You are given ten measurements: 5,7,8,6,12,7,8,11,4,10 a) Find \(\bar{x}\) b) Find the median. c) Find the mode.

Short Answer

Expert verified
a) \(\bar{x} = 7.8\), b) Median = 7.5, c) Modes = 7 and 8.

Step by step solution

01

Calculate the Mean

The mean (denoted as \( \bar{x} \)) is calculated by summing all the values and then dividing by the total number of values. Add the given numbers: \(5 + 7 + 8 + 6 + 12 + 7 + 8 + 11 + 4 + 10 = 78\). Then divide by the number of measurements, which is 10: \(\bar{x} = \frac{78}{10} = 7.8\).
02

Arrange Numbers in Order

To find the median, sort the numbers in ascending order: 4, 5, 6, 7, 7, 8, 8, 10, 11, 12.
03

Find the Median

The median is the middle number in an ordered list. Since there are 10 numbers (an even count), the median is the average of the 5th and 6th numbers: \(\text{Median} = \frac{7 + 8}{2} = 7.5\).
04

Determine the Mode

The mode is the number that appears most frequently in the list. In the ordered list from Step 2, the number 7 and 8 both appear twice, which makes the modes 7 and 8.

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Key Concepts

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

Mean Calculation
The mean, often referred to as the average, is a measure of central tendency that provides a single value representing the center of a data set. To calculate the mean, follow these simple steps:

- Add together all of the numbers in your data set. In this exercise, you have the numbers 5, 7, 8, 6, 12, 7, 8, 11, 4, and 10. When added together, these numbers equal 78.
- Divide the sum by the total number of values in the data set. Here, you have 10 numbers. Thus, the mean is calculated as follows:
\[ \bar{x} = \frac{78}{10} = 7.8 \] This means that the average value of this collection of numbers is 7.8. The mean gives us a quick snapshot of the data's overall tendency.
Median Calculation
The median is another measure of central tendency that indicates the middle value of a data set when it is well organized. This is particularly useful for understanding how data values are spread out.

Follow these steps to find the median:

- First, sort the data in ascending order. Your ordered list should be: 4, 5, 6, 7, 7, 8, 8, 10, 11, 12.
- Since there are 10 numbers in this data set, we need to find the middle two numbers because the number of data points is even.
- The 5th and 6th numbers in this list are 7 and 8. So, you calculate the median by taking the average of these two numbers:
\[ \text{Median} = \frac{7 + 8}{2} = 7.5 \] Hence, the median of this data set is 7.5, signifying the middle point of the ordered data.
Mode Identification
The mode in a set of numbers is the value or values that appear most frequently. It helps identify the most common occurrence within a dataset.

To determine the mode, check how many times each number appears in your list:

- From the ordered list: 4, 5, 6, 7, 7, 8, 8, 10, 11, 12, it鈥檚 clear that both 7 and 8 appear twice, whereas other numbers appear only once.
- A data set can have more than one mode or no mode at all if no number repeats.
In this case, the dataset is bimodal, meaning it has two modes: 7 and 8. This information is useful to identify potential patterns or repeated occurrences in data, emphasizing the most common values.

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Most popular questions from this chapter

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