Question

For the data sets in Exercises \(1-4\) calculate the mean, the median, and the mode. Locate these measures on a dotplot. \(n=5\) measurements: 0,5,1,1,3

Short answer

Answer: The mean of the given data set is 2, the median is 1, and the mode is 1. On a dotplot, the mean is located between 1 and 3 on the x-axis, the median is located at the position with the highest frequency (2 dots on the same position), and the mode is also located at the position with the highest frequency.

Step-by-step solution

Calculate the mean

To calculate the mean, we sum up all the measurements and divide the result by the total number of measurements (\(n=5\)). The formula for the mean is: Mean \(= \frac{\sum x}{n}\) Here, \(x\) represents the measurements and \(n\) is the total number of measurements. Using the given data set: Mean \(= \frac{0+5+1+1+3}{5} = \frac{10}{5} = 2\) So, the mean of the data set is 2.

Calculate the median

To calculate the median, we need to arrange the data in ascending order: 0, 1, 1, 3, 5 Now, we find the middle value. Since there are 5 measurements (odd), the middle value is the 3rd value: Median = 1 So the median of the data set is 1.

Calculate the mode

To find the mode, we look for the most frequent value in the data set. In this case, the number 1 appears twice, which is more frequent than any other value: Mode = 1 So, the mode of the data set is 1.

Create a dotplot and locate the measures

We will now create a dotplot with the given data set and locate the mean, median, and mode on the plot. The x-axis represents the measurements and the y-axis represents the frequency. A dotplot for the data set would look something like this: ``` . | | | | | | --------------- 0 1 2 3 4 5 ``` On the dotplot: - The mean (2) is located between 1 and 3 on the x-axis. - The median (1) is located at the position with the highest frequency (2 dots on the same position). - The mode (1) is also located at the position with the highest frequency. In summary, the mean is 2, the median is 1, and the mode is 1. These values are located on a dotplot as described.

Key concepts

Understanding the Mean

The mean is a fundamental concept in descriptive statistics that helps us understand the central tendency of a data set. To calculate the mean, you sum up all the values in your data set and then divide by the number of values.
For example, if we have the data set 0, 5, 1, 1, 3, the mean is calculated as follows:

• Add all the numbers: 0 + 5 + 1 + 1 + 3 = 10.
• Divide the sum by the number of values, which is 5: \( \frac{10}{5} = 2 \).

The mean provides a single value that represents the average of all measurements.
It is important to note that the mean can be heavily influenced by outliers or very large/small numbers in your data set.

Decoding the Median

The median is another measure of central tendency that is particularly useful when your data set contains outliers or is skewed.
To find the median, arrange your data in ascending order and identify the middle value.

• For the data set 0, 5, 1, 1, 3, arrange it as 0, 1, 1, 3, 5.
• Since there are 5 numbers, the median is the third value in the ordered list: 1.

The median is less influenced by outliers than the mean and is a good indicator of the center when your data is not symmetrically distributed.

Exploring the Mode

The mode is the value that appears most frequently in your data set. It is particularly useful for identifying the most common value.
If we examine the data set 0, 5, 1, 1, 3, we see that the number 1 appears twice, more than any other number.
Therefore, the mode of this data set is 1.
Some data sets may have more than one mode if multiple values appear with the same highest frequency. Others might not have a mode at all if all values appear with the same frequency.

Visualizing with a Dotplot

A dotplot is a simple graphical representation that displays the frequency of each number in a data set.
Dotplots help visualize where values occur most often, which is beneficial for understanding the distribution of data.
To create a dotplot for the data set 0, 5, 1, 1, 3:

• Draw a horizontal line (x-axis) with numbers, starting from the smallest value to the largest (0 to 5).
• Above each number on the x-axis, draw dots corresponding to the frequency of each value.

For this data set, you'll have one dot for 0, two dots for 1, no dots for 2, one dot for 3, and one dot for 5.
The dotplot allows you to easily locate the mean, median, and mode visually.
For instance, the mode (1) corresponds to the highest grouping of dots, giving a clear visualization of the most frequent value.