Question

unit 4 test: graphs and describing distributions multiple cl
for the following data set below, find the 5 number summary and determine the skew of the data.
{5, 6, 7, 9, 15, 18, 21, 24, 32, 36, 11, 19, 32, 25}
min: q1: med: q3: max:
left-skewed, symmetric or right-skewed? (only type ls, s or rs):

Explanation:

Step1: Sort the data

Sort the data set \(\{5,6,7,9,15,18,21,24,32,36,11,19,32,25\}\) in ascending order: \(\{5,6,7,9,11,15,18,19,21,24,25,32,32,36\}\)

Step2: Find Min and Max

The minimum (Min) is the first element of the sorted data set: \(5\)
The maximum (Max) is the last element of the sorted data set: \(36\)

Step3: Find the median (Med)

Since \(n = 14\) (even number of data points), the median is the average of the \(\frac{n}{2}\) - th and \((\frac{n}{2}+ 1)\) - th values. \(\frac{n}{2}=7\), \(\frac{n}{2}+1 = 8\). The \(7\) - th value is \(18\) and the \(8\) - th value is \(19\). So, \(Med=\frac{18 + 19}{2}=18.5\)

Step4: Find Q1

The lower half of the data is \(\{5,6,7,9,11,15,18\}\). Since \(n_1=7\) (odd number of data points in the lower half), \(Q1\) is the \(\frac{n_1 + 1}{2}\) - th value. \(\frac{7+1}{2}=4\). The \(4\) - th value of the lower half is \(9\)

Step5: Find Q3

The upper half of the data is \(\{19,21,24,25,32,32,36\}\). Since \(n_2 = 7\) (odd number of data points in the upper half), \(Q3\) is the \(\frac{n_2+1}{2}\) - th value. \(\frac{7 + 1}{2}=4\). The \(4\) - th value of the upper half is \(25\)

Step6: Determine the skew

We use the formula for the skewness based on the five - number summary. A rough rule: if \(Q3 - Med>Med - Q1\), the data is right - skewed. \(Q3 - Med=25-18.5 = 6.5\), \(Med - Q1=18.5 - 9=9.5\). Wait, no. Another way: look at the spread. The distance from \(Q1\) to \(Med\) is \(18.5 - 9=9.5\), the distance from \(Med\) to \(Q3\) is \(25 - 18.5 = 6.5\). Also, the tail on the right (from \(Med\) to \(Max\): \(36-18.5 = 17.5\)) is longer than the tail on the left (from \(Min\) to \(Med\): \(18.5 - 5=13.5\)). So the data is right - skewed.

Answer:

Min: \(5\)
Q1: \(9\)
Med: \(18.5\)
Q3: \(25\)
Max: \(36\)
Left - Skewed, Symmetric or Right - Skewed? \(RS\)