Problem 83 - Algebra 1

Question

Find the mean, the median, and the mode of the collection of numbers. $$-6,20,-8,-18,10$$

Step-by-Step Solution

Answer

The mean of this set of numbers is \(-0.4\), the median is \(-6\), and there's no mode because all numbers occur only once.

Step 1: Calculating the Mean

First, one must add all the numbers together and then divide by the quantity of numbers available. For this collection of numbers, \(-6 + 20 - 8 - 18 + 10 = -2\). There are five numbers in total, so the mean is found by dividing -2 by 5, which yields \(-0.4\).

Step 2: Calculating the Median

First, sort the collection of numbers in ascending order, which gives: \(-18, -8, -6, 10, 20\). For datasets with an odd number of observations, the median is the middling number. In this case, the median number is \(-6\).

Step 3: Calculating the Mode

The mode of a dataset is defined as the number that appears most often. In this dataset, all numbers appear only once, implying that there's no mode.

Key Concepts

Mean

The mean, sometimes referred to as the average, is a measure of central tendency that provides an idea of the overall trend of a data set. To calculate the mean, sum up all the numbers in the data set and then divide by the number of elements in the set. In our example, we have the numbers

• -6
• 20
• -8
• -18
• 10

Adding these numbers together gives us \[-6 + 20 - 8 - 18 + 10 = -2,\]which is the total sum of these numbers. Since there are five numbers, divide this sum by 5:\[\frac{-2}{5} = -0.4.\]So, the mean of this collection is -0.4. The mean can tell us whether the numbers are generally small, large, or perhaps around zero, as is in this case.

Median

The median is a measure of central tendency that represents the middle point of a data set when the numbers are arranged in either ascending or descending order. It is particularly useful when a dataset has outliers that could skew the mean.
For data with an odd number of entries, the median is the number that is exactly in the middle. In our exercise, the numbers to sort are:

• -18
• -8
• -6
• 10
• 20

After sorting, we find that -6 is the middle number in the sequence.
Thus, the median for this dataset is -6. This value can provide a more accurate representation of the dataset's center when there are extreme values.

Mode

The mode is the value that appears most frequently in a dataset. It's a descriptive statistic that highlights what is most common in the data set. However, not every dataset has a mode. If every number appears with the same frequency, like in our example, then the dataset is considered "mode-less."
In this dataset, the numbers

• -6
• 20
• -8
• -18
• 10

all occur only once.
Thus, there is no mode present. It's essential to remember that while mean and median give us central tendency information, mode can be more about spotting frequent, repeated occurrences in larger data sets.