Question

answer the statistical measures and create a box and whiskers plot for the following set of data. enter all values as whole numbers or decimals. 6,6,6,10,11,11,12,12,14,15,15,16,16 min: 6 q1: 8 med: 12 q3: 15 max: 16 create the box plot by dragging the lines: box plot diagram answer attempt 2 out of 10

Explanation:

Step1: Find Minimum

The smallest value in the data set \(6,6,6,10,11,11,12,12,14,15,15,16,16\) is \(6\).

Step2: Find Q1 (First Quartile)

The data set has \(n = 13\) values. The lower half (excluding median if \(n\) is odd) has the first \(6\) values: \(6,6,6,10,11,11\). The median of this lower half is the average of the 3rd and 4th values: \(\frac{6 + 10}{2}=\frac{16}{2} = 8\).

Step3: Find Median (Med)

For \(n = 13\) (odd), the median is the \(\frac{13 + 1}{2}=7\)-th value. The 7th value in the ordered data is \(12\).

Step4: Find Q3 (Third Quartile)

The upper half (excluding median) has the last \(6\) values: \(14,15,15,16,16\) (wait, correction: ordered data is \(6,6,6,10,11,11,12,12,14,15,15,16,16\), upper half (after 7th value) is \(14,15,15,16,16\)? No, wait, \(n=13\), so upper half is from 8th to 13th: \(12,14,15,15,16,16\)? Wait, no, original data: positions 1 - 13. Median at position 7 (value 12). Lower half: positions 1 - 6: \(6,6,6,10,11,11\). Upper half: positions 8 - 13: \(12,14,15,15,16,16\)? Wait, no, the 8th value is 12? Wait, no, ordered data: 1:6, 2:6, 3:6, 4:10, 5:11, 6:11, 7:12, 8:12, 9:14, 10:15, 11:15, 12:16, 13:16. Oh! I made a mistake earlier. Upper half is positions 8 - 13: \(12,14,15,15,16,16\)? No, position 8 is 12, 9:14, 10:15, 11:15, 12:16, 13:16. So the upper half data is \(12,14,15,15,16,16\)? Wait, no, the median is at position 7 (value 12). So lower half: positions 1 - 6: values \(6,6,6,10,11,11\) (correct). Upper half: positions 8 - 13: values \(12,14,15,15,16,16\)? Wait, no, position 8 is 12? Wait, original data: \(6,6,6,10,11,11,12,12,14,15,15,16,16\). So position 7:12, position 8:12. So upper half is positions 8 - 13: \(12,14,15,15,16,16\)? No, that can't be, because the median is 12, so upper half should be values greater than or equal to median? Wait, no, quartiles: for odd \(n\), we can include the median in both halves or not. The correct way: when \(n\) is odd, to find Q1 and Q3, we can take the lower half as the first \(\lfloor\frac{n}{2}
floor = 6\) values and upper half as the last \(\lfloor\frac{n}{2}
floor = 6\) values. So lower half: first 6 values: \(6,6,6,10,11,11\) (positions 1 - 6), upper half: last 6 values: \(14,15,15,16,16\)? No, last 6 values of 13 are positions 8 - 13: \(12,14,15,15,16,16\)? Wait, position 8 is 12, which is the median. So maybe the correct upper half (excluding median) is positions 8 - 13, but median is at 7, so upper half is positions 8 - 13: values \(12,14,15,15,16,16\)? But then the median of upper half: positions 8 - 13 (6 values), median is average of 10th and 11th? Wait, no, 6 values: positions 8 (12), 9 (14), 10 (15), 11 (15), 12 (16), 13 (16). The median of these 6 is the average of 3rd and 4th (positions 10 and 11? Wait, 6 values: indices 0 - 5 (if 0 - based), so 3rd and 4th. So values: 12,14,15,15,16,16. 3rd value (0 - based index 2) is 15, 4th (index 3) is 15. So average is \(\frac{15 + 15}{2}=15\). So Q3 is 15.

Step5: Find Maximum

The largest value in the data set is \(16\).

For the box - and - whisker plot:

• The left whisker goes from Min (\(6\)) to Q1 (\(8\)).
• The box goes from Q1 (\(8\)) to Q3 (\(15\)), with a line at the median (\(12\)) inside the box.
• The right whisker goes from Q3 (\(15\)) to Max (\(16\)).

To create the plot:

• Plot the minimum at \(x = 6\) (left end of left whisker).
• Plot Q1 at \(x = 8\) (left end of the box).
• Plot the median at \(x = 12\) (line inside the box).
• Plot Q3 at \(x = 15\) (right end of the box).
• Plot the maximum at \(x = 16\) (right end of right whisker).

Answer:

Min: \(6\), Q1: \(8\), Med: \(12\), Q3: \(15\), Max: \(16\)

For the box - and - whisker plot:

• Left whisker: from \(6\) to \(8\).
• Box: from \(8\) to \(15\), with a vertical line at \(12\).
• Right whisker: from \(15\) to \(16\).