Question

in the data set below, what is the interquartile range?
2 2 2 3 5 7 7 8 9 9

Explanation:

Step1: Find the median (Q2)

The data set has 10 values, so the median is the average of the 5th and 6th values. The 5th value is 5, the 6th is 7. Median $=\frac{5 + 7}{2}=6$.

Step2: Split data into lower and upper halves

Lower half: \(2, 2, 2, 3, 5\) (values below the median). Upper half: \(7, 7, 8, 9, 9\) (values above the median).

Step3: Find Q1 (median of lower half)

Lower half has 5 values, so Q1 is the 3rd value: \(2\).

Step4: Find Q3 (median of upper half)

Upper half has 5 values, so Q3 is the 3rd value: \(8\).

Step5: Calculate interquartile range (IQR)

IQR = Q3 - Q1 = \(8 - 2 = 6\)? Wait, no, wait. Wait, lower half: 2,2,2,3,5. Median (Q1) is the 3rd term? Wait, no, for odd number of terms in lower half (5 terms), the median is the \((5 + 1)/2 = 3\)rd term? Wait, no, wait, the data set is 10 numbers: positions 1 - 10. Median is between 5th and 6th. So lower half is positions 1 - 5: 2,2,2,3,5. Upper half is positions 6 - 10: 7,7,8,9,9. Now, Q1 is the median of lower half (positions 1 - 5). For 5 numbers, median is the 3rd number: 2. Q3 is the median of upper half (positions 6 - 10). For 5 numbers, median is the 3rd number of the upper half. The upper half is 7 (pos6), 7 (pos7), 8 (pos8), 9 (pos9), 9 (pos10). So 3rd term is 8. Then IQR = Q3 - Q1 = 8 - 2 = 6? Wait, but let's check again. Wait, maybe I made a mistake in splitting. Wait, the data set is [2,2,2,3,5,7,7,8,9,9]. The median (Q2) is between 5th (5) and 6th (7), so Q2 = (5 + 7)/2 = 6. Then lower quartile (Q1) is the median of the lower half: [2,2,2,3,5]. The median of this is the 3rd term, which is 2. Upper quartile (Q3) is the median of the upper half: [7,7,8,9,9]. The median of this is the 3rd term, which is 8. So IQR = 8 - 2 = 6? Wait, but let's use another method. Alternatively, using the formula for quartiles in a data set with n elements. For n = 10, the position of Q1 is (n + 1)/4 = 11/4 = 2.75. So we take the 2nd term + 0.75*(3rd term - 2nd term). The 2nd term is 2, 3rd term is 2. So Q1 = 2 + 0.75*(2 - 2) = 2. Position of Q3 is 3*(n + 1)/4 = 33/4 = 8.25. So 8th term + 0.25*(9th term - 8th term). 8th term is 8, 9th term is 9. So Q3 = 8 + 0.25*(9 - 8) = 8.25. Then IQR = 8.25 - 2 = 6.25? Wait, now I'm confused. Wait, maybe the first method is wrong. Let's check with the two methods. Method 1: splitting into lower and upper halves (each with 5 elements) when n is even. Wait, n = 10, so lower half is first 5, upper half is last 5. Then Q1 is median of lower half (5 elements), Q3 is median of upper half (5 elements). So lower half: [2,2,2,3,5], median is 2 (3rd term). Upper half: [7,7,8,9,9], median is 8 (3rd term). So IQR = 8 - 2 = 6. Method 2: using the percentile formula. Q1 is the 25th percentile, Q3 is the 75th percentile. For 25th percentile: index = (25/100)*(n - 1) + 1 = 0.25*9 + 1 = 2.25 + 1 = 3.25? Wait, no, different formulas. Wait, maybe the problem is using the method of splitting into lower and upper halves (for even n, split into two halves of n/2 each). So n = 10, split into lower 5 and upper 5. Then Q1 is median of lower 5, Q3 is median of upper 5. So lower 5: [2,2,2,3,5], median is 2. Upper 5: [7,7,8,9,9], median is 8. So IQR = 8 - 2 = 6. But when using the linear interpolation method (for percentiles), we get a different result. But maybe in this problem, they expect the first method (splitting into two halves). Let's check the data again. The data set is 2,2,2,3,5,7,7,8,9,9. Let's list the positions: 1:2, 2:2, 3:2, 4:3, 5:5, 6:7, 7:7, 8:8, 9:9, 10:9. Median (Q2) is between 5 and 6: (5 + 7)/2 = 6. Lower half: positions 1 - 5: values 2,2,2,3,5. The median of this is the 3rd value (since 5 is odd), which is 2. Upper half: positions 6 - 10: values 7,7,8,9,9. Median is 3rd value, which is 8. So IQR = 8 - 2 = 6. Wait, but when I use the other method (position (n + 1)/4), for n = 10, (10 + 1)/4 = 2.75, so Q1 is at 2.75th position. So value at 2nd position is 2, 3rd position is 2. So Q1 = 2 + 0.75*(2 - 2) = 2. Q3 is at 3*(10 + 1)/4 = 8.25th position. Value at 8th position is 8, 9th position is 9. So Q3 = 8 + 0.25*(9 - 8) = 8.25. Then IQR = 8.25 - 2 = 6.25. But maybe the problem is using the first method (splitting into two halves). Let's check with an example. Suppose the data set is [1,2,3,4,5,6,7,8,9,10]. Lower half: [1,2,3,4,5], median Q1 = 3. Upper half: [6,7,8,9,10], median Q3 = 8. IQR = 5. Using the percentile method: Q1 at (10 + 1)/4 = 2.75, so Q1 = 2 + 0.75*(3 - 2) = 2.75. Q3 at 8.25, so Q3 = 8 + 0.25*(9 - 8) = 8.25. IQR = 5.5. So different methods give different results. But in the problem, the data set is [2,2,2,3,5,7,7,8,9,9]. Let's count the number of elements: 10. Let's use the first method (splitting into lower 5 and upper 5). Lower 5: 2,2,2,3,5. Median (Q1) is 2 (3rd term). Upper 5:7,7,8,9,9. Median (Q3) is 8 (3rd term). So IQR = 8 - 2 = 6. So I think that's the expected answer here.

Answer:

6