Question

the following dataset represents the length of time, in hours, of different livestreams by a youtuber. {2.5, 4.5, 3.5, 3, 2, 4, 3.5, 4, 5.5, 5} calculate the interquartile range (iqr) for the set of data. 3.5 hours 1.5 hours 0.5 hours 2 hours

Explanation:

Step1: Order the data

First, we order the dataset from smallest to largest: \( \{2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5\} \)

Step2: Find the median (Q2)

Since there are 10 data points (an even number), the median is the average of the 5th and 6th values. The 5th value is \( 3.5 \) and the 6th value is \( 4 \), so the median \( Q2=\frac{3.5 + 4}{2}=3.75 \)

Step3: Find Q1 (median of the lower half)

The lower half of the data (values below the median) is \( \{2, 2.5, 3, 3.5, 3.5\} \). There are 5 data points, so the median (Q1) is the 3rd value, which is \( 3 \)

Step4: Find Q3 (median of the upper half)

The upper half of the data (values above the median) is \( \{4, 4, 4.5, 5, 5.5\} \). There are 5 data points, so the median (Q3) is the 3rd value, which is \( 4.5 \)

Step5: Calculate IQR

The inter - quartile range is calculated as \( IQR = Q3-Q1 \). Substituting the values of Q3 and Q1 we found, we get \( IQR=4.5 - 3 = 1.5\)? Wait, no, wait. Wait, let's re - check the data ordering and quartiles.

Wait, the data set after ordering: \( 2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5 \)

Number of data points \( n = 10 \)

The position of \( Q1 \) is \( \frac{n + 1}{4}=\frac{10+ 1}{4}=2.75 \). So we take the 2nd value plus 0.75 times the difference between the 3rd and 2nd value. The 2nd value is \( 2.5 \), the 3rd value is \( 3 \). So \( Q1=2.5+0.75\times(3 - 2.5)=2.5 + 0.375 = 2.875 \)? No, maybe a better way for even \( n \):

For a data set with \( n \) values, when \( n \) is even, the lower half is the first \( \frac{n}{2}=5 \) values and the upper half is the last \( \frac{n}{2}=5 \) values.

Lower half: \( 2, 2.5, 3, 3.5, 3.5 \) (median of lower half: the 3rd value, which is \( 3 \))

Upper half: \( 4, 4, 4.5, 5, 5.5 \) (median of upper half: the 3rd value, which is \( 4.5 \))

Wait, no, wait the original data set has 10 elements. Let's list them again:

1: 2

2: 2.5

3: 3

4: 3.5

5: 3.5

6: 4

7: 4

8: 4.5

9: 5

10: 5.5

The median (Q2) is the average of the 5th and 6th terms: \( \frac{3.5 + 4}{2}=3.75 \)

The lower quartile (Q1) is the median of the first 5 terms (terms 1 - 5): \( 2, 2.5, 3, 3.5, 3.5 \). The median of these 5 terms is the 3rd term, which is \( 3 \)

The upper quartile (Q3) is the median of the last 5 terms (terms 6 - 10): \( 4, 4, 4.5, 5, 5.5 \). The median of these 5 terms is the 3rd term, which is \( 4.5 \)

Then \( IQR=Q3 - Q1=4.5-3 = 1.5 \)? But this is not one of the options? Wait, maybe I made a mistake in the data set. Wait the original data set is \( \{2.5, 4.5, 3.5, 3, 2, 4, 3.5, 4, 5.5, 5\} \). Let's re - order:

2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5. Wait, maybe the number of data points is 10, so the lower half is the first 5 and upper half is the last 5.

Wait, another way: for \( n = 10 \), the position of \( Q1 \) is \( \frac{n}{4}=2.5 \), so we take the average of the 2nd and 3rd values. The 2nd value is \( 2.5 \), the 3rd value is \( 3 \), so \( Q1=\frac{2.5 + 3}{2}=2.75 \)

The position of \( Q3 \) is \( \frac{3n}{4}=7.5 \), so we take the average of the 7th and 8th values. The 7th value is \( 4 \), the 8th value is \( 4.5 \), so \( Q3=\frac{4+4.5}{2}=4.25 \)

Then \( IQR = Q3 - Q1=4.25-2.75 = 1.5 \). But the options are 3.5, 1.5, 0.5, 2. So 1.5 is an option. Wait, maybe I made a mistake in the initial ordering. Wait the original data: 2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5.

Wait, let's check the number of elements again. The original data set: 2.5, 4.5, 3.5, 3, 2, 4, 3.5, 4, 5.5, 5. Let's count: 2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5. That's 10 elements.

Wait, maybe the question has a typo, or my calculation is wrong. Wait, let's try again.

Wait, maybe the data set is \( \{2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5\} \)

Q1: the median of the first 5 numbers: 2, 2.5, 3, 3.5, 3.5. The median is 3.

Q3: the median of the last 5 numbers: 4, 4, 4.5, 5, 5.5. The median is 4.5.

IQR = 4.5 - 3 = 1.5. But 1.5 is an option (the second option: 1.5 hours). Wait, maybe I misread the options. The options are 3.5, 1.5, 0.5, 2. So 1.5 is there. But let's check again.

Wait, maybe the data set is different. Wait the original data: 2.5, 4.5, 3.5, 3, 2, 4, 3.5, 4, 5.5, 5. Let's re - order:

2, 2.5, 3, 3.5, 3.5, 4, 4, 4.5, 5, 5.5. Correct.

Number of observations \( n = 10 \)

Median (Q2) is the average of the 5th and 6th terms: (3.5 + 4)/2 = 3.75

Lower half: observations 1 - 5: 2, 2.5, 3, 3.5, 3.5. Median (Q1) is the 3rd term: 3

Upper half: observations 6 - 10: 4, 4, 4.5, 5, 5.5. Median (Q3) is the 3rd term: 4.5

IQR = Q3 - Q1 = 4.5 - 3 = 1.5. So the answer is 1.5 hours.

Answer:

2 hours