Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Heights: \(60,62,63,63,64,65,66,68,68,68,70,75\)

Step-by-Step Solution

Answer

Mean: 66, Median: 65.5, Mode: 68

Step 1: Arrange the Data

First, ensure the data is arranged in increasing order. Our data set for heights, which is already sorted, is: 60, 62, 63, 63, 64, 65, 66, 68, 68, 68, 70, 75.

Step 2: Calculate the Mean

To find the mean, add all the values together and divide by the number of values. The sum of the heights is \(60 + 62 + 63 + 63 + 64 + 65 + 66 + 68 + 68 + 68 + 70 + 75 = 792\). There are 12 heights, so the mean is \(\frac{792}{12} = 66\).

Step 3: Find the Median

The median is the middle value when the data is ordered. Since there are 12 values (an even number), the median is the average of the 6th and 7th values. These values are 65 and 66, so the median is \(\frac{65+66}{2} = 65.5\).

Step 4: Determine the Mode

The mode is the number that appears most frequently in a data set. In the list, 68 occurs three times, which is more frequent than any other number, making the mode 68.

Key Concepts

Mean

The concept of mean, often referred to as the "average," is central to understanding how datasets behave in statistics. The mean of a dataset gives us an idea of the central value or the equilibrium point of the data.

To calculate the mean, you sum up all the values in the dataset and then divide by the total number of values. This method provides a simple way to quantify the "central" point in a set of numbers. For example, if you have the heights dataset provided, the calculation would look like this:

• Sum = 60 + 62 + 63 + 63 + 64 + 65 + 66 + 68 + 68 + 68 + 70 + 75 = 792
• Count = 12 heights
• Mean = \( \frac{792}{12} = 66 \)

By determining the mean, we gain insight into what height might be considered typical or average within this group. It's important to note that means can be skewed by extremely high or low values (outliers), which may not adequately reflect all aspects of the dataset.

Median

The median represents the middle value in a dataset when the numbers are sorted in order. This measure of central tendency is particularly useful because it isn't affected by outliers or extreme values.

For example, in a dataset like our set of heights, even if one or two values are exceptionally high or low, the median remains the central point. Here's how you find it in our sorted dataset:

• Our heights in increasing order are: 60, 62, 63, 63, 64, 65, 66, 68, 68, 68, 70, 75.
• With 12 data points, we have an even number of values.
• This means the median is calculated by averaging the 6th and 7th values.
• These middle numbers are 65 and 66, so the median is \( \frac{65+66}{2} = 65.5 \).

Utilizing the median allows us to understand the central tendency of data in a way that remains robust against any skew or outlier-dominated datasets.

Mode

The mode is the value that appears most frequently in a dataset. This measure of central tendency helps identify the most common item or number.

This is beneficial for understanding what's typical in a dataset, especially if the data can have repeated values.

Let's apply this concept to our list of heights:

• The heights recorded are: 60, 62, 63, 63, 64, 65, 66, 68, 68, 68, 70, 75.
• Among these, the number 68 appears more frequently than any other number, with three occurrences.
• Therefore, the mode of this dataset is 68.

Recognizing the mode in a dataset can be particularly helpful in fields like retail, where businesses can understand which product size or color is sold most often, thereby making informed decisions based on consumer preferences.