Question

the box plot represents the distribution of the number of points scored by a cross country team at 12 meets.
a. if possible, find the mean. if not possible, explain why not.
b. if possible, find the median. if not possible, explain why not.
c. did the cross country team ever score 30 points at a meet?

Explanation:

Part a

Step1: Recall mean calculation

The mean is calculated as the sum of all data points divided by the number of data points ($\text{Mean} = \frac{\sum x}{n}$).

Step2: Analyze box plot data

A box plot shows the five - number summary (minimum, first quartile, median, third quartile, maximum) and the distribution shape, but it does not provide the individual data values or the sum of the data values. We know $n = 12$ (number of meets), but we don't have the sum of the points scored at each meet. So we can't calculate the mean.

Step1: Recall median definition

For a dataset with $n$ values, if $n$ is even, the median is the average of the $\frac{n}{2}$-th and $(\frac{n}{2}+ 1)$-th ordered values. For $n = 12$ (even), the median is the average of the 6th and 7th ordered values.

Step2: Analyze box plot for median

A box plot shows the median (the line inside the box). From the box plot, we can identify the median value. Looking at the number line, the median (the line in the box) is at 34 (assuming the box is centered around 34, but more precisely, the median is the middle value of the ordered data. Since $n = 12$, the median is the average of the 6th and 7th values, and the box plot displays the median as the vertical line inside the box. From the plot, the median is 34.

Step1: Analyze box plot components

A box plot shows the range of the data (minimum to maximum) and the inter - quartile range (IQR, between $Q_1$ and $Q_3$). The whiskers extend from the minimum to $Q_1$ and from $Q_3$ to maximum, and the box is between $Q_1$ and $Q_3$.

Step2: Check for 30 points

The minimum value (from the left whisker) is 22, and the left end of the box (first quartile $Q_1$) is at 32 (looking at the number line: 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42). The data between the minimum and $Q_1$ is in the left whisker. Since 30 is less than $Q_1 = 32$, and the box plot shows the distribution of the data, but it does not mean that every value in the range from minimum to maximum is present. However, the left whisker goes from 22 to 32 (the start of the box). So 30 is within the range of the data (from 22 to 42), but we can't be sure if 30 was actually scored? Wait, no. Wait, the box plot's whiskers represent the minimum and maximum, and the box represents $Q_1$ to $Q_3$. The data points are within the minimum and maximum. But to check if 30 was scored: the left whisker is from 22 to $Q_1$ (32). So the values from 22 up to 32 are part of the data (in the left whisker and the box starts at 32). So 30 is within the range of the data (minimum = 22, maximum = 42), but does that mean it was scored? Wait, no. Wait, the box plot shows the distribution, but not the exact values. However, the key is: the left whisker is from 22 to $Q_1$ (32). So the data points in the left whisker are from 22 to 32. So 30 is within that interval, so it is possible that they scored 30? Wait, no, actually, the box plot's whiskers are drawn from the minimum to $Q_1$ (for the lower whisker) and from $Q_3$ to maximum (for the upper whisker). The data points are all within the minimum and maximum. But we can't be certain if 30 was a score, but wait, the question is "Did the cross country team ever score 30 points at a meet?". The box plot shows that the minimum is 22 and the first quartile is 32. So the data from minimum (22) to $Q_1$ (32) are the lower 25% of the data. So there are data points between 22 and 32. So 30 is within that range, so yes, they could have scored 30? Wait, no, actually, the box plot's lower whisker is from 22 to 32 (the start of the box). So the values in the lower whisker are from 22 to 32, so 30 is in that range, so it is possible that 30 was a score. But wait, maybe I made a mistake. Wait, the number line: 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42. The left whisker starts at 22 and ends at 32 (the left side of the box). So the data points from 22 to 32 are part of the dataset (the lower 25% of the 12 data points: 3 data points in the lower whisker? Wait, $n = 12$, so the first quartile ($Q_1$) is the 3rd value? No, for $n = 12$, the positions are: minimum (1st), $Q_1$ (3rd), median (6th and 7th), $Q_3$ (9th), maximum (12th). Wait, no, the formula for quartiles: for $n$ data points, the first quartile is the median of the lower half, the second quartile is the median of the whole data, and the third quartile is the median of the upper half. For $n = 12$, the lower half is the first 6 data points, so $Q_1$ is the median of the first 6, which is the average of the 3rd and 4th. The upper half is the last 6 data points, so $Q_3$ is the average of the 9th and 10th. The median is the average of the 6th and 7th. But in the box plot, the left whisker is from minimum to $Q_1$, the box is from $Q_1$ to $Q_3$, and the right whisker is from $Q_3$ to maximum. So the data points in the left whisker are from minimum (22) to $Q_1$…

Answer:

It is not possible to find the mean. A box plot only shows the five - number summary (minimum, $Q_1$, median, $Q_3$, maximum) and the distribution of the data, not the individual data values or their sum, which is required to calculate the mean ($\text{Mean}=\frac{\sum x}{n}$).

Part b