Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the mean, the median, and the mode of each set of data. Weights: \(110,112,113,115,15,116,118,118,125,134,145,148\)

Step-by-Step Solution

Answer

Mean: 114.92, Median: 117, Mode: 118.

Step 1: Arrange Data

First, arrange the data set in ascending order. The given weights are: 110, 112, 113, 115, 115, 116, 118, 118, 125, 134, 145, 148.

Step 2: Calculate the Mean

Add all the values in the dataset and divide by the number of values. \[\text{Sum of data} = 110 + 112 + 113 + 115 + 115 + 116 + 118 + 118 + 125 + 134 + 145 + 148 = 1379\]There are 12 values, so the mean is:\[\text{Mean} = \frac{1379}{12} \approx 114.92\]

Step 3: Determine the Median

The median is the middle value of a data set. Since there are 12 values, we take the average of the 6th and 7th values.The 6th value is 116, and the 7th value is 118:\[\text{Median} = \frac{116 + 118}{2} = 117\]

Step 4: Identify the Mode

The mode is the most frequently occurring value in the data set. In this set, the value 118 occurs twice. No other value appears more than once. Therefore, the mode is 118.

Key Concepts

Understanding the Mean

The mean is one of the most common measures of central tendency. To find the mean, you add together all values in a data set and divide by the number of values. This gives you the average value, which can provide a sense of the overall level or center of the data.
For example, in the data set provided, the sum of all values is 1379, and since there are 12 values in total, the mean is calculated by dividing 1379 by 12, which is approximately 114.92.
Here are some key points about the mean:

• The mean is sensitive to outliers. Large or unusual values can skew the mean and may give an impression that does not reflect the typical data point.
• It's most useful when data values are evenly distributed without extreme values.
• The mean can be used to compare different data sets of similar sizes to see which one tends to be larger or smaller overall.

Exploring the Median

The median is the middle value when a data set is ordered from least to greatest. It provides a useful measure of central tendency, particularly for skewed distributions or distributions with outliers.
In the provided data set of weights, there are 12 values, meaning the median is found by taking the average of the 6th and 7th values. These values are 116 and 118, so the median is calculated as (116 + 118)/2, which equals 117.
Points to consider about the median:

• The median is not affected by outliers, making it a more reliable measure of central tendency in certain cases.
• It divides the data into two halves, so half of the data points are above the median, and half are below.
• In data sets with an odd number of values, the median is simply the middle number.

Identifying the Mode

The mode represents the most frequently occurring value in a data set. It's particularly useful for categorical data where mean and median don't apply.
For the given weights, the value 118 appears twice, more than any other value, so the mode is 118.
Important aspects of the mode to keep in mind:

• A data set can have more than one mode. If two or more values occur with the same highest frequency, the data set is multimodal.
• Alternatively, a data set may have no mode if no number repeats.
• The mode helps in understanding the most common trend within a data set.