Question

here are the scores of 12 students on a science test.
61, 64, 68, 72, 74, 75, 77, 79, 82, 82, 86, 87
notice that the scores are ordered from least to greatest.
give the median, lower quartile, and upper quartile for the data set.
(a) median:
(b) lower quartile:
(c) upper quartile:

Explanation:

Step1: Find the median

The data set has \(n = 12\) values. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{n}{2}=6\), \(\frac{n}{2}+1 = 7\). The 6th value is \(75\) and the 7th value is \(77\). Median\(=\frac{75 + 77}{2}\)

$$=\frac{152}{2}=76$$

Step2: Find the lower quartile

The lower half of the data is the first \(6\) values: \(61,64,68,72,74,75\). The lower quartile (\(Q_1\)) is the average of the 3rd and 4th values. The 3rd value is \(68\) and the 4th value is \(72\). \(Q_1=\frac{68+72}{2}\)

$$=\frac{140}{2}=70$$

Step3: Find the upper quartile

The upper half of the data is the last \(6\) values: \(79,82,82,86,87\). The upper quartile (\(Q_3\)) is the average of the 9th and 10th values. The 9th value is \(82\) and the 10th value is \(82\). \(Q_3=\frac{82 + 82}{2}\)

$$=\frac{164}{2}=82$$

Answer:

(a) Median: \(76\)
(b) Lower quartile: \(70\)
(c) Upper quartile: \(82\)