Question

For the data sets calculate the mean, the median, and the mode. Locate these measures on a dotplot. \(n=8\) measurements: 3,2,5,6,4,4,3,5

Short answer

Question: Calculate and locate the mean, median, and mode of the following measurements: 3, 2, 5, 6, 4, 4, 3, 5. Represent the data and the central tendencies on a dotplot. Answer: The mean, median, and mode of the given measurements are 4, 4, and 3, 4, 5 respectively. The dotplot with the central tendencies is represented as: 2 3 4 5 6 ------------------- * * * * * * * * * * Mean: 4 (located at third column) Median: 4 (located at third column) Mode: 3, 4, 5 (located at second, third, and fourth columns)

Step-by-step solution

Calculate the mean

To calculate the mean of the 8 measurements, we will sum them up and then divide by the number of measurements. The formula for the mean is: Mean = \(\frac{\sum x}{n}\) where \(x\) represents individual measurements and \(n\) is the total number of measurements. Mean = \(\frac{3 + 2 + 5 + 6 + 4 + 4 + 3 + 5}{8}\) Mean = \(\frac{32}{8}\) Mean = \(4\) The mean of the data set is 4.

Find the median

To find the median, we should first order the measurements from smallest to largest: 2, 3, 3, 4, 4, 5, 5, 6 Since there is an even number of measurements, the median will be the average of the two middle values. The middle values here are 4 and 4. We'll use this formula: Median = \(\frac{middle \:value_{1} + middle \:value_{2}}{2}\) Median = \(\frac{4 + 4}{2}\) Median = \(\frac{8}{2}\) Median = \(4\) The median of the data set is 4.

Identify the mode

To identify the mode, we will look for the most frequently occurring measurement(s). Counting the occurrences of each measurement, we see that the numbers 3, 4, and 5 each appear twice. Therefore, the data set has three modes: Mode = 3, 4, 5

Create a dotplot and locate the mean, median, and mode

Now we will create a dotplot of the measurements. A dotplot is a graphical representation showing each data point as a dot on a number line. Plot the dots for each measurement, and then locate the mean, median, and mode. Dotplot: 2 3 4 5 6 ------------------- * * * * * * * * * * Mean: 4 (located at third column) Median: 4 (located at third column) Mode: 3, 4, 5 (located at second, third, and fourth columns) The dotplot displays the measurements along with the mean, median, and mode.

Key concepts

mean calculation

The calculation of the mean, or more commonly known as the average, is a fundamental aspect of descriptive statistics. It provides a measure of the central value of a dataset. To compute the mean, you add up all the numbers in the set and then divide by the total count of the numbers.

For instance, in a dataset of 8 measurements such as 3, 2, 5, 6, 4, 4, 3, and 5, the sum is 32. Dividing 32 by the total number of measurements, which is 8, yields a mean of \(4\). This process is encapsulated in the equation:

\[ \text{Mean} = \frac{\sum x}{n} \]
where \(\sum x\) is the sum of all measurements, and \(n\) is the total number of measurements. The mean provides a useful overall indicator of the data set, but it's important to remember that it can be influenced by outliers or extremely high or low values.

median calculation

The median is the middle value in a data set when it's arranged in ascending order. Unlike the mean, it's not affected by extremely high or low values and can thus provide a more accurate representation of the data's center for skewed distributions.

To find the median, you order the data set from the lowest to the highest value. If there is an odd number of observations, the median is the middle number. However, for an even number of observations, such as our dataset (2, 3, 3, 4, 4, 5, 5, 6), you must determine the average of the two central numbers.

For our example, both central values are 4, so the calculation is:

\[ \text{Median} = \frac{middle \:value_{1} + middle \:value_{2}}{2} = \frac{4 + 4}{2} = 4 \]
This dataset's median is hence also 4, conveniently coinciding with the mean in this case.

mode identification

Identification of the mode in a set of data refers to finding the value or values that appear most frequently. A dataset can have one mode (unimodal), two modes (bimodal), several modes (multimodal), or no mode at all if no number repeats.

In our dataset (3, 2, 5, 6, 4, 4, 3, 5), the numbers 3, 4, and 5 each appear twice, making this a trimodal dataset with modes at 3, 4, and 5. Determining the mode is as simple as:

\[ \text{Mode} = \text{frequently occurring values} \]
Remember that the mode is the only measure of central tendency that can be used with nominal data, which are categories or names that cannot be logically ordered or ranked.

dotplot representation

A dotplot is an uncomplicated yet powerful way to visually display data. In a dotplot, each data value is represented by a dot above a number line, allowing you to see the distribution and frequency of the dataset at a glance.

Creating a dotplot involves placing a dot for each occurrence of a number in the data set. It's particularly useful for small to moderately sized datasets. In our previous example, each measurement is plotted over a number line, so you can easily count and compare the frequencies of different values.

The mode can be visually identified as the number(s) with the highest stacks of dots. The mean and median won't always be indicated by dots when the dataset's values are all unique, because the mean is an arithmetic average that may not be a value in the dataset, but for this dataset, both mean and median are represented by a dot at value 4. The dotplot provides an intuitive sense of the data's spread, central tendency, and the location of clusters or gaps.