Problem 61 - Algebra 2

Question

Find the mean, median, and mode for each set of values. $$ \begin{array}{llllllllllll}{45} & {42} & {39} & {35} & {41} & {45} & {49} & {42} & {43} & {48} & {32} & {51} & {42}\end{array} $$

Step-by-Step Solution

Answer

The mean of the given set of numbers is 42.62, the median is 42 and the mode is 42.

Step 1: Calculate the Mean

First, add up all the numbers in the given set: \(45 + 42 + 39 + 35 + 41 + 45 + 49 + 42 + 43 + 48 + 32 + 51 + 42 = 554\). Then, divide this sum by the total count of numbers in the set which is 13. As a result, the mean of the set is \(\frac{554}{13} = 42.62\) (rounded to two decimal places).

Step 2: Find the Median

To find the median, sort the numbers in the set in ascending order: \(32, 35, 39, 41, 42, 42, 42, 43, 45, 45, 48, 49, 51\). Since the set has an odd count of numbers (13), the median is the number directly in the middle of the sorted set, which is 42.

Step 3: Determine the Mode

The mode is the number(s) that appears most frequently in the set. Looking at the sorted set of numbers, it can be seen that the number 42 appears three times, more frequently than any other number. Therefore, the mode of the set is 42.

Key Concepts

Understanding the Mean

The mean, often referred to as the arithmetic average, is a significant measure in statistics that represents the central tendency of a data set. To find the mean, you need to add all the numbers in the set together and then divide the sum by the number of values present.

• Addition of all values: Here, the sum of the values \(45, 42, 39, 35, 41, 45, 49, 42, 43, 48, 32, 51, 42\) is calculated to be 554.
• Number of values: The set contains 13 values.

Once these calculations are done, the mean can be obtained by dividing the total sum (554) by the number of values (13), resulting in an approximate mean of \(42.62\).
The mean is a useful statistic because it takes into account every value in the dataset, providing a complete measure of central tendency. However, it can sometimes be affected by extreme outliers in the data set.

Decoding the Median

The median offers a valuable insight into the distribution of values. It effectively divides the data set into two equal halves. To figure out the median, first, sort the numbers in ascending order.
In our example, the ordered data set reads: \(32, 35, 39, 41, 42, 42, 42, 43, 45, 45, 48, 49, 51\).

• Check the total number of values: Our set contains 13 numbers.
• With an odd count, the median is the value placed exactly in the middle.

For our sorted set, the 7th number is 42, which is the median.
The median is particularly useful as it is not influenced by outliers. It provides a more robust center of the dataset, especially for skewed distributions.

Making Sense of the Mode

The mode is the statistic that identifies the most frequently occurring number in a given set of values. It is the only measure of central tendency that can have more than one result (or none, if no number repeats).
In the dataset, the frequency of appearance of each number shows that 42 appears more often than any other number, occurring three times. Thus, the mode is 42.

• If two numbers repeat with the same highest frequency, the set is called bimodal, with two modes.
• If more than two numbers are tied, the set might even be multimodal.

The mode is exceptionally useful in understanding which value appears most frequently in a dataset, providing insights into commonality or popularity of certain items in the dataset. Unlike mean and median, it can be used with non-numeric data, especially helpful in categorical data analysis.