Problem 57 - Algebra 1

Question

Find the mean, the median, and the mode of the collection of numbers shown below. \(57,40,57,57,30,56,40,30,56\)

Step-by-Step Solution

Answer

The mean is \(47\), the median is \(56\), and the mode is \(57\).

Step 1: Calculate the Mean

The mean is calculated by summing up all the numbers and then dividing by the number of numbers. So, for our data set, sum up \(57,40,57,57,30,56,40,30,56\) to get \(423\) and then divide that by the number of numbers, which is \(9\). So, the mean is \(423/9 = 47\).

Step 2: Calculate the Median

First, the numbers need to be arranged in increasing order: \(30,30,40,40,56,56,57,57,57\). Since the number of values is odd, the median is the middle number, which in this case is \(56\).

Step 3: Calculate the Mode

The mode of a set is the most frequently occurring number. In our data set, the number \(57\) occurs most frequently, three times, so our mode is \(57\).

Key Concepts

Mean

In statistics, the mean is one of the most common measures of central tendency. It provides a way to describe the average or expected value of a dataset. To calculate the mean, also known as the average, you sum up all the numbers in a dataset and then divide by the count of those numbers.
For example, let's use our dataset: 57, 40, 57, 57, 30, 56, 40, 30, 56. First, you add all the numbers together:

• 57 + 40 + 57 + 57 + 30 + 56 + 40 + 30 + 56 = 423

Then, you divide the total by the number of items to find the mean:

• 423 divided by 9 equals 47

Thus, the mean for this dataset is 47. The mean is useful because it uses every number in your data set, which can provide a balanced view of your data. However, it can be affected by extremely high or low numbers, which can skew your results.

Median

The median is another important measure of central tendency. It represents the middle value in a dataset when the numbers are arranged in order. To find the median, you first need to organize your numbers from smallest to largest.
Using our dataset: 57, 40, 57, 57, 30, 56, 40, 30, 56. We reorder these numbers as:

• 30, 30, 40, 40, 56, 56, 57, 57, 57

Since there are 9 numbers, and 9 is an odd number, the median is the number that appears in the fifth position, which is 56. Unlike the mean, the median is less affected by outliers and skewed data, since it solely depends on the middle value. This makes it a reliable indicator of the central tendency in datasets with extreme values.

Mode

The mode is the measure of central tendency that identifies the most frequently occurring value in a dataset. In some cases, there can be more than one mode if multiple values appear with the same highest frequency, while in others, there may be no mode if no number repeats.

• For the dataset 57, 40, 57, 57, 30, 56, 40, 30, 56, we look at how often each number appears.

After counting occurrences:

• 57 appears 3 times
• 40 appears 2 times
• 56 appears 2 times
• 30 appears 2 times

Thus, the mode is 57 because it appears more frequently than any other number. The mode is especially helpful in cases where the most typical item needs to be identified, such as determining the most common shoe size sold in a store. However, it may not always provide the best understanding of your data if the frequency of values is low or highly varied.