Problem 5 - Prealgebra

Question

Find the mean, median, and mode for each set of data. If necessary, round to the nearest tenth. The number of books sold during the past week is shown below. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. $$53,61,46,59,61,55,49$$ (TABLE NOT COPY)

Step-by-Step Solution

Answer

Mean: 54.9, Median: 55, Mode: 61; Mode best represents frequency in data as it shows the most common sale number.

Step 1: Calculate the Mean

To find the mean, add up all the numbers in the data set and then divide by the total number of data points.The data set is: 53, 61, 46, 59, 61, 55, 49.First, calculate the sum:\[53 + 61 + 46 + 59 + 61 + 55 + 49 = 384\]Next, divide the sum by the number of data points (which is 7):\[\text{Mean} = \frac{384}{7} \approx 54.9\]So, the mean is 54.9 when rounded to the nearest tenth.

Step 2: Find the Median

To find the median, first arrange the data set in ascending order. Then find the middle number.The ordered data set is: 46, 49, 53, 55, 59, 61, 61.Since there are 7 numbers in the data set, the median is the fourth number:\[\text{Median} = 55\]Therefore, the median is 55.

Step 3: Determine the Mode

The mode is the number that appears most frequently in the data set.From the data set: 53, 61, 46, 59, 61, 55, 49, the number 61 appears twice, while all others appear once.\[\text{Mode} = 61\]Hence, the mode is 61.

Step 4: Select the Best Measure of Central Tendency

To determine the best measure of central tendency, look at the characteristics of each measure in the context of the data. - **Mean**: 54.9 (sensitive to outliers, gives a central value) - **Median**: 55 (less affected by outliers, provides a balanced central point) - **Mode**: 61 (shows the most common value) In this data set, since the values are relatively close and there aren’t extreme outliers, the mean and median are similar. However, considering that 61 occurs twice, the mode can be considered representative for understanding the frequency distribution if the context of "popularity" is important. Thus, if the goal is to understand frequency, the mode 61 is best.

Key Concepts

Mean Calculation

The mean, often called the average, represents a central value for a data set. It is calculated by adding all the numbers in the set and then dividing this sum by the total number of values.
For example, with the data set: 53, 61, 46, 59, 61, 55, 49, the first step is to sum up these numbers. This sum is 384.
Then, you divide 384 by the number of numbers in the set, which is 7.
This calculation results in a mean of about 54.9 when rounded to the nearest tenth.

The mean provides a single value representing the entire data set, which is useful when all numbers are similar.

• Calculation Formula: \[\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}\]
• Example: 53 + 61 + 46 + 59 + 61 + 55 + 49 = 384, then divide by 7 = 54.9

Median Determination

The median is another way to find a central point in a data set. Unlike the mean, the median is the middle number when the data is ordered from smallest to largest.
For our example, we first arrange the numbers: 46, 49, 53, 55, 59, 61, 61. Since there are seven numbers, the median is the fourth number in this ordered list, which is 55.

The median is especially useful because it's less affected by outliers or extremely high or low values that can skew the mean.

• Ordering the set helps find the center value directly.
• Example for odd numbers of data: The middle number in 46, 49, 53, 55, 59, 61, 61 is 55.

Mode Identification

Mode is the value that appears most frequently in your data set. For example, among our numbers: 53, 61, 46, 59, 61, 55, 49, the number 61 appears twice, more than any other number.
Therefore, 61 is the mode of this set.

This measure is excellent at showing what is most common or popular in the data set.

• Mode finding involves counting the frequency of individual numbers.
• Example: In the set, 61 appears twice, making it the mode.

While not providing an idea of how each value aligns with the others (like mean or median does), the mode is useful for understanding patterns in larger data sets.

Data Analysis in Statistics

Understanding which measure of central tendency best represents your data set is critical in data analysis. Often, the context dictates the choice of mean, median, or mode.
The data in our example lacks extreme outliers, resulting in the mean (54.9) and median (55) being quite close.
However, the mode (61) indicates the most frequent occurrence.

To choose a measure:

• The **mean** is best if you're looking for an overall idea of your data set, but beware of potential distortion by outliers.
• The **median** works well for mid-value representation when dealing with skewed data.
• The **mode**, reflecting frequency, is helpful when understanding what occurs most often is necessary for decision making.

In summary, choose your measure based on the representational need: overall average, central point, or the most frequent occurrence.