Question

1. how does the addition of the value 30 to the data set affect the median and the mode?

Explanation:

To answer this, we need to recall the definitions of median and mode:

• Median: The middle value (or average of two middle values) when data is ordered.
• Mode: The most frequently occurring value(s) in a data set.

Step 1: Analyze the Mode

The mode depends on the frequency of values. Adding a new value (30) will only change the mode if 30 becomes the most frequent value (or ties for most frequent). If 30 was not previously the mode, and its frequency after addition does not exceed the current mode’s frequency, the mode remains unchanged.

Step 2: Analyze the Median

The median depends on the order of data. Let’s assume the original data set has \( n \) values. After adding 30, the new data set has \( n + 1 \) values.

• If \( n \) was odd: Original median is the \( \frac{n + 1}{2} \)-th value. After adding 30, the new median is the \( \frac{(n + 1) + 1}{2} = \frac{n + 2}{2} \)-th value. This shifts the median (unless 30 is equal to the original median or the value adjacent to it).
• If \( n \) was even: Original median is the average of the \( \frac{n}{2} \)-th and \( \frac{n}{2} + 1 \)-th values. After adding 30, the new median is the \( \frac{n + 1 + 1}{2} = \frac{n + 2}{2} \)-th value (now a single value, not an average, if \( n + 1 \) is odd).

Key Takeaway

• Mode: Unchanged (unless 30 matches the current mode’s frequency or becomes the new most frequent).
• Median: Changes (shifts) because the number of data points and their order are altered by adding 30.

Answer:

The mode remains unchanged (unless 30 becomes the most frequent value), and the median changes (shifts) due to the addition of 30, as the order and count of data points are modified.