Question

green valley middle school recorded the number of students in each homeroom.
number of students
20 24 22 24 30 26 27 29 24 23 26 26 28 24 28 28
which box plot represents the data?
number of students
number of students

Explanation:

Step1: Order the data

First, we order the data set: \(20, 22, 23, 24, 24, 24, 24, 26, 26, 26, 27, 28, 28, 28, 29, 30\)

Step2: Find the minimum, Q1, median, Q3, maximum

- Minimum: The smallest value is \(20\).
- Maximum: The largest value is \(30\).
- Median (middle value of ordered data): Since there are \(16\) values, the median is the average of the 8th and 9th values. The 8th value is \(26\) and the 9th value is \(26\), so median \(=\frac{26 + 26}{2}=26\).
- Q1 (median of the lower half): The lower half is the first 8 values: \(20, 22, 23, 24, 24, 24, 24, 26\). The median of these 8 values is the average of the 4th and 5th values. The 4th value is \(24\) and the 5th value is \(24\), so \(Q1 = \frac{24+24}{2}=24\).
- Q3 (median of the upper half): The upper half is the last 8 values: \(26, 26, 27, 28, 28, 28, 29, 30\). The median of these 8 values is the average of the 4th and 5th values. The 4th value is \(28\) and the 5th value is \(28\), so \(Q3=\frac{28 + 28}{2}=28\).

Step3: Analyze the box - plots

- The box in a box - plot goes from \(Q1\) to \(Q3\), the line inside the box is the median, and the whiskers go from the minimum to \(Q1\) and from \(Q3\) to the maximum.
- For our data: minimum \( = 20\), \(Q1 = 24\), median \(=26\), \(Q3 = 28\), maximum \(=30\).
- Looking at the first box - plot: The left whisker should go from \(20\) to \(Q1 = 24\)? Wait, no. Wait, the first box - plot's left whisker starts at \(20\) and the box starts at \(22\)? No, our \(Q1\) is \(24\). Wait, no, let's re - check. Wait, in the first box - plot, the left whisker is from \(20\) to \(22\)? No, our \(Q1\) is \(24\). Wait, no, I think I made a mistake. Wait, the lower half: first 8 values: \(20,22,23,24,24,24,24,26\). The median of the lower half: the 4th and 5th values are \(24\) and \(24\), so \(Q1 = 24\). The first box - plot: the left side of the box is at \(22\)? No, that's wrong. Wait, the second box - plot: the left side of the box is at \(24\), which matches our \(Q1 = 24\). The right side of the box is at \(28\) (matches \(Q3 = 28\)), the median line is at \(26\) (matches our median), the left whisker goes from \(20\) to \(24\) (since minimum is \(20\) and \(Q1 = 24\)), and the right whisker goes from \(28\) to \(30\) (since \(Q3 = 28\) and maximum is \(30\)). Wait, no, the first box - plot: let's look at the number line. The first box - plot has the box from \(22\) to \(28\)? No, our \(Q1\) is \(24\) and \(Q3\) is \(28\). Wait, I think I messed up the initial ordering. Wait, the data is: \(20,22,23,24,24,24,24,26,26,26,27,28,28,28,29,30\). So the lower half is \(20,22,23,24,24,24,24,26\) (8 values). The median of the lower half: positions 1 - 8, median is between 4th and 5th. 4th value: \(24\), 5th value: \(24\), so \(Q1 = 24\). The upper half is \(26,26,26,27,28,28,28,29,30\)? Wait, no, 16 values, so upper half is values 9 - 16: \(26,26,27,28,28,28,29,30\) (8 values). Median of upper half: between 4th and 5th values. 4th value: \(28\), 5th value: \(28\), so \(Q3 = 28\). So the box should be from \(Q1 = 24\) to \(Q3 = 28\), median at \(26\), minimum \(20\), maximum \(30\). Now looking at the two box - plots:

First box - plot: The left whisker is from \(20\) to \(22\) (box starts at \(22\)), which would mean \(Q1 = 22\), but our \(Q1 = 24\).

Second box - plot: The left whisker is from \(20\) to \(24\) (box starts at \(24\)), which matches \(Q1 = 24\), box ends at \(28\) (matches \(Q3 = 28\)), median at \(26\), right whisker from \(28\) to \(30\). So the second box - plot is correct? Wait, no, the first box - plot in the image: let's check the number line. The first box - plot has the left whisker at \(20\), box from \(22\) to \(28\), median at \(26\), right whisker at \(30\). The second box - plot has left whisker at \(20\), box from \(24\) to \(28\), median at \(26\), right whisker at \(30\). Wait, our \(Q1\) is \(24\), so the box should start at \(24\). So the second box - plot (the one with the box starting at \(24\)) is correct. Wait, but let's re - calculate Q1. The lower half: data points below the median. The median is \(26\), so the lower half is all values less than or equal to \(26\)? No, for even number of data points, the median is between the 8th and 9th values. The 8th value is \(26\), 9th is \(26\), so the lower half is the first 8 values: \(20,22,23,24,24,24,24,26\), upper half is the last 8 values: \(26,26,27,28,28,28,29,30\). So Q1 is the median of the lower half: (24 + 24)/2 = 24. Q3 is the median of the upper half: (28+28)/2 = 28. So the box should be from 24 (Q1) to 28 (Q3), median at 26, minimum 20, maximum 30. So the second box - plot (the one where the box starts at 24) is correct. Wait, but in the first box - plot, the box starts at 22, which would imply Q1 = 22, but our Q1 is 24. So the correct box - plot is the second one (the one with the box from 24 to 28, median at 26, left whisker from 20 to 24, right whisker from 28 to 30).

Answer:

The second box - plot (the one with the box starting at 24, ending at 28, median at 26, left whisker from 20 to 24, right whisker from 28 to 30)