The table below shows the distribution of scores achieved by 20 students on a final exam: | Score | Frequency | |---|---| | 60 | 3 | | 70 | 5 | | 80 | 6 | | 90 | 4 | | 100 | 2 | What is the interquartile range (IQR) of the scores?

The table below shows the distribution of scores achieved by 20 students on a final exam: … — GRE General Test

The table below shows the distribution of scores achieved by 20 students on a final exam:

Score	Frequency
60	3
70	5
80	6
90	4
100	2

What is the interquartile range (IQR) of the scores?

Cevap: 20

Cevap

The interquartile range (IQR) of the scores is 20.

For a set of 20 ordered scores, the first quartile

Q_1

is the median of the first 10 scores (average of the 5th and 6th values), and the third quartile

Q_3

is the median of the last 10 scores (average of the 15th and 16th values). Using the cumulative frequencies, the 5th and 6th scores are both 70 (

Q_1 = 70

), and the 15th and 16th scores are both 90 (

Q_3 = 90

). Thus, the interquartile range is

IQR = Q_3 - Q_1 = 90 - 70 = 20

Adım Adım Çözüm

Determine the cumulative frequency to locate quartile positions.

Score 60 occupies positions 1 to 3; Score 70 occupies positions 4 to 8; Score 80 occupies positions 9 to 14; Score 90 occupies positions 15 to 18; Score 100 occupies positions 19 to 20.

Tracking data positions in a frequency distribution allows efficient determination of medians and quartiles without expanding the raw list.

Calculate the first quartile (

Q_1

Q_1 = \frac{70 + 70}{2} = 70

With 20 total values, the lower half consists of the first 10 values (positions 1 through 10). The median of these 10 values is the average of the 5th and 6th values, which are both 70.

Calculate the third quartile (

Q_3

Q_3 = \frac{90 + 90}{2} = 90

The upper half consists of the last 10 values (positions 11 through 20). The median of these 10 values is the average of the 15th and 16th values, which are both 90.

Subtract

Q_1

from

Q_3

to find the interquartile range.

IQR = Q_3 - Q_1 = 90 - 70 = 20

The interquartile range represents the spread of the middle 50% of the dataset.

Anahtar Kavram

Interquartile Range (IQR) from a Frequency Distribution

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