Question

here are the scores of 13 students on a geography test.
62, 67, 71, 71, 80, 85, 87, 88, 89, 93, 93, 94, 95
notice that the scores are ordered from least to greatest.
give the five - number summary and the interquartile range for the data set.
five - number summary
minimum:
lower quartile:
median:
upper quartile:
maximum:
interquartile
range:

Explanation:

Step1: Find Minimum and Maximum

The data is ordered: 62, 67, 71, 71, 80, 85, 87, 88, 89, 93, 93, 94, 95.
Minimum is the first value: \( 62 \).
Maximum is the last value: \( 95 \).

Step2: Find Median (Q2)

There are \( n = 13 \) values. The median is at position \( \frac{n + 1}{2} = \frac{13 + 1}{2} = 7 \).
The 7th value is \( 87 \), so Median \( = 87 \).

Step3: Find Lower Quartile (Q1)

Lower half: values before the median (positions 1–6): 62, 67, 71, 71, 80, 85.
There are \( n_{\text{lower}} = 6 \) values. Q1 is at position \( \frac{6 + 1}{2} = 3.5 \) (average of 3rd and 4th values).
3rd value: \( 71 \), 4th value: \( 71 \).
Q1 \( = \frac{71 + 71}{2} = 71 \).

Step4: Find Upper Quartile (Q3)

Upper half: values after the median (positions 8–13): 88, 89, 93, 93, 94, 95.
There are \( n_{\text{upper}} = 6 \) values. Q3 is at position \( \frac{6 + 1}{2} = 3.5 \) (average of 3rd and 4th values).
3rd value: \( 93 \), 4th value: \( 93 \).
Q3 \( = \frac{93 + 93}{2} = 93 \).

Step5: Find Interquartile Range (IQR)

IQR \( = Q3 - Q1 = 93 - 71 = 22 \).

Answer:

Five - number summary:
Minimum: \( 62 \)
Lower quartile: \( 71 \)
Median: \( 87 \)
Upper quartile: \( 93 \)
Maximum: \( 95 \)

Interquartile range: \( 22 \)