Problem 54 - Prealgebra

Question

Jonelle's tips as a server are shown in the table. On Friday, her tips were \(\$ 74 .\) Which measure of central tendency will change the most as a result? Explain. $$\begin{array}{|l|c|} \hline \text { DAY } & \text { True } \\ \hline \text { Monday } & \$ 36 \\ \hline \text { Tuesday } & \$ 32 \\ \hline \text { Wednesday } & \$ 40 \\ \hline \text { Thursday } & \$ 36 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The mean changes the most.

Step 1: Understanding the Given Information

The table shows Jonelle's tips from Monday to Thursday with the amounts $36, $32, $40, and $36 respectively. On Friday, she received $74 in tips.

Step 2: Identify Measures of Central Tendency

The measures of central tendency we will consider are the mean, median, and mode.

Step 3: Calculate Initial Mean and Updated Mean

The initial mean is calculated as \((36 + 32 + 40 + 36)/4 = 36\). With Friday's tips added, the new mean becomes \((36 + 32 + 40 + 36 + 74)/5 = 43.6\).

Step 4: Evaluate the Change in Mean

The mean changes from \(36\) to \(43.6\), which is an increase of \(7.6\).

Step 5: Calculate Initial and Updated Median

The initial median, with tips \(32, \)36, \(36, and \)40, is \(36\). The updated set \([32, 36, 36, 40, 74]\) gives a median of \(36\).

Step 6: Evaluate the Change in Median

The median does not change and remains as \(36\).

Step 7: Identify the Mode

The mode initially is \(36\), since it appears twice. Adding \(74 does not change the mode, as \)36 remains the most frequent value.

Step 8: Determine Maximum Change

The mean changes by \(7.6\), while the median and mode do not change at all. Therefore, the mean changes the most.

Key Concepts

Mean in Prealgebra

The mean, often called the average, is a crucial measure of central tendency, especially in prealgebra. To find the mean, you add up all the numbers in your dataset and then divide by how many numbers there are. In Jonelle's example, her tips amounted to \(36, \)32, \(40, and \)36 from Monday to Thursday. You calculate this initial mean by doing \(\frac{36 + 32 + 40 + 36}{4} = 36\).
Now, when her Friday tips of \(74 are added, the calculation for the new mean becomes \(\frac{36 + 32 + 40 + 36 + 74}{5} = 43.6\). You notice a significant increase here. The mean changed by 7.6, which illustrates its sensitivity to changes in a dataset. This property means that if you have an unusually high or low number, like Jonelle's Thursday tips of \)74, it can significantly affect the mean.
Understanding how to calculate and interpret the mean helps in recognizing outliers in data and understanding where most of the values tend to concentrate in practical situations.

Median in Prealgebra

The median is another essential measure of central tendency you often encounter in prealgebra. It helps to find the middle of a data set, providing a valuable insight into the dataset's central point. To determine the median, you must first order the numbers from smallest to largest. For Jonelle's tips, the values \(32, 36, 36,\) and \(40\) yield an initial median of 36 because it's right in the middle of this ordered set, with two numbers on each side.
Adding Friday's tip of $74 results in the new dataset: \(32, 36, 36, 40, 74\). Despite this addition, the median remains 36, as it still occupies the central position in the ordered list. This demonstrates how the median is less affected by extreme values than the mean. The median offers a robust metric of central tendency, especially when dealing with skewed data or outliers, because it focuses on the middle rather than being influenced considerably by any extremes at either end of the dataset.

Mode in Prealgebra

The mode is the third basic measure of central tendency and refers to the number that appears most frequently in a dataset. In prealgebra, finding the mode is as simple as spotting the number that repeats the most.
For Jonelle's weekly tips, the values were $36, $32, $40, and $36, making the mode 36 because it appears more times than any other number. Adding the $74 from Friday didn't change the mode, as it doesn't repeat enough to influence the existing frequency of values. The mode remains 36, highlighting its stability in situations like this.
The mode is particularly useful when you need to identify the most common item in a dataset. It's indispensable in various scenarios, from understanding popular choices in surveys to recognizing the most frequently occurring errors in numerical data.