Question

For the data sets calculate the mean, the median, and the mode. Locate these measures on a dotplot. \(n=7\) measurements: 3,6,4,0,3,5,2

Short answer

Question: Calculate the mean, median, and mode of the given data set, and represent these measures on a dotplot: 3, 6, 4, 0, 3, 5, 2. Answer: The mean is 3 2/7, the median is 3, and the mode is 3. The dotplot representing the data set and these measures is as follows: 0 2 3 4 5 6 M Md Mo ● ● ●● ● ● ●

Step-by-step solution

1. Calculate the Mean

To calculate the mean, add up all of the measurements and divide the sum by the total number of measurements. In this case: Mean = \((3 + 6 + 4 + 0 + 3 + 5 + 2) / 7 = 23 / 7\). Mean = \(3\frac{2}{7}\).

2. Calculate the Median

First arrange the data set in ascending order: 0, 2, 3, 3, 4, 5, 6. The median is the middle value in the ordered data set. In this case, there are 7 measurements, so the median is the fourth value (since it is the middle of the remaining six values): Median = 3.

3. Calculate the Mode

The mode is the value(s) that occur most frequently in the data set. In this case, the value 3 appears twice, and no other value appears more than once. Therefore, the mode is 3.

4. Create a Dotplot and Locate the Mean, Median, and Mode

To create a dotplot, draw a number line ranging from the lowest to the highest value in the data set. Place dots above each value to represent the corresponding measurement. 0 2 3 4 5 6 ● ● ●● ● ● ● Mean: \(3\frac{2}{7}\) (indicated with an "M" above the dotplot): 0 2 3 4 5 6 M ● ● ●● ● ● ● Median: 3 (indicated with a "Md" above the dotplot): 0 2 3 4 5 6 M Md ● ● ●● ● ● ● Mode: 3 (indicated with a "Mo" above the dotplot): 0 2 3 4 5 6 M Md Mo ● ● ●● ● ● ● In conclusion, the mean is \(3\frac{2}{7}\), the median is 3, the mode is also 3, and the dotplot representing the data set and the measures is shown above.

Key concepts

Mean Calculation

The mean, also known as the average, is a central value of a dataset. To find it, you sum all the numbers and divide by the total count of numbers. For the given data set: 3, 6, 4, 0, 3, 5, and 2, you begin by adding them together: \[3 + 6 + 4 + 0 + 3 + 5 + 2 = 23\].Next, divide this sum by the number of values, which is 7, so \[ \text{Mean} = \frac{23}{7} = 3\frac{2}{7} \].This mean provides a single number representing the central point of the data.

• Summing brings together all the data points.
• Dividing by the count of data points illustrates the average.

Understanding the mean helps in identifying the overall trend of the dataset.

Median Calculation

Finding the median involves identifying the middle value in a data set after sorting it in order. The values must first be arranged from smallest to largest. In this example, arranging the given numbers results in: 0, 2, 3, 3, 4, 5, 6. With 7 numbers, the median is the fourth number, which is the middle. So, the median here is 3.

• Ordering values is crucial for accurate median calculation.
• The median provides a straightforward central value.
• Unlike the mean, outliers have minimal impact on the median.

The median gives insight into the distribution by focusing on positional value.

Mode Calculation

The mode is the most frequently occurring number in a dataset. Identifying it helps to comprehend data repetition, which can indicate trends. In the data: 3, 6, 4, 0, 3, 5, 2, the number 3 appears more than any other, occurring twice. As no other numbers repeat, the mode is clearly 3.

• Mode highlights the most common data points.
• Useful in spotting the most typical value in a set.
• A dataset might have no mode, one mode, or multiple modes.

Focusing on the mode allows understanding of commonality and frequency within data.

Dotplot Visualization

A dotplot is a simple visual representation of data that shows frequency and distribution along a number line. To create a dotplot for the data set, place dots above each number corresponding to how often each number appears:0, 2, 3, 3, 4, 5, 6 becomes:0 2 3 4 5 6● ● ●● ● ● ●Locating the mean, median, and mode on a dotplot involves marking these calculated values on the number line:- Mean (\(3\frac{2}{7}\)) can be inferred as slightly to the right of 3.- Median is at 3, directly above this value.- Mode is also at 3, marked similarly above this position.Dotplots make quantitative data easy to interpret visually. They:

• Show frequency distribution clearly.
• Help in identifying clusters and gaps.
• Aid in easily spotting mean, median, and mode.

This visualization tool supports a quick grasp of how often data points occur and how they're spread.