Question

in the data set below, what are the lower quartile, the median, and the upper quartile?
10 15 19 19 19 33 42 42 55 56
lower quartile =
median =
upper quartile =

Explanation:

Step1: Order the data (already ordered)

The data set is: \(10, 15, 19, 19, 19, 33, 42, 42, 55, 56\) (n = 10, even number of observations).

Step2: Find the median (Q2)

For even n, median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th values.
\(\frac{n}{2} = 5\), so the 5th value is \(19\), 6th value is \(33\).
Median \(= \frac{19 + 33}{2} = 26\)? Wait, no—wait, n = 10, positions are 1 to 10. Wait, no, correction: when n is even, median is average of \(\frac{n}{2}\) and \(\frac{n}{2}+1\) terms. Wait, n = 10, so \(\frac{10}{2} = 5\), so 5th and 6th terms. 5th term: 19 (index 5, 0-based or 1-based? Wait, 1-based: positions 1:10, 2:15, 3:19, 4:19, 5:19, 6:33, 7:42, 8:42, 9:55, 10:56. So 5th term is 19, 6th is 33. Median = (19 + 33)/2 = 26? Wait, no, wait—wait, the data set is 10 numbers. Wait, maybe I made a mistake. Wait, no, let's recheck:

Wait, the data is: 10, 15, 19, 19, 19, 33, 42, 42, 55, 56. So n = 10. The median (Q2) is the average of the 5th and 6th values (since 10/2 = 5, so 5th and 6th in 1-based). 5th value: 19, 6th: 33. So median = (19 + 33)/2 = 26? Wait, but wait, maybe the method for quartiles: when n is even, we split the data into lower half (first 5 values: 10,15,19,19,19) and upper half (last 5 values: 33,42,42,55,56).

Step3: Find Q1 (lower quartile)

Lower half: 10,15,19,19,19 (n1 = 5, odd). Q1 is the median of the lower half. For n1 = 5 (odd), median is the 3rd value (1-based: 3rd term). 3rd term: 19.

Step4: Find Q3 (upper quartile)

Upper half: 33,42,42,55,56 (n2 = 5, odd). Q3 is the median of the upper half. 3rd term: 42.

Wait, wait, maybe my initial median calculation was wrong. Wait, no—wait, the standard method for quartiles when n is even: split into lower half (first n/2 terms) and upper half (last n/2 terms). For n = 10, lower half: first 5 terms (10,15,19,19,19), upper half: last 5 terms (33,42,42,55,56). Then Q1 is median of lower half, Q3 is median of upper half, and median (Q2) is average of the two middle terms.

So lower half (n=5): median is the 3rd term (1-based: 3rd term is 19). So Q1 = 19.

Upper half (n=5): median is the 3rd term (33,42,42,55,56: 3rd term is 42). So Q3 = 42.

Median (Q2): average of 5th and 6th terms (19 and 33) → (19 + 33)/2 = 26? Wait, but wait, let's confirm with another method. Alternatively, using the formula for quartiles:

For a data set with n observations:

- Q1: value at position \(\frac{n + 1}{4}\) (if not integer, interpolate). But n = 10, so \(\frac{10 + 1}{4} = 2.75\). So Q1 is the 2nd term + 0.75*(3rd term - 2nd term). 2nd term: 15, 3rd term: 19. So Q1 = 15 + 0.75*(19 - 15) = 15 + 3 = 18? Wait, now I'm confused. There are different methods for quartiles (Tukey's method, Mendenhall and Sincich, etc.).

Wait, let's check the data set again: 10, 15, 19, 19, 19, 33, 42, 42, 55, 56.

Using Tukey's method (common in education):

- Median (Q2): for n even, average of n/2 and n/2 + 1 terms. n = 10, so 5th and 6th terms: 19 (5th) and 33 (6th). Median = (19 + 33)/2 = 26.

- Lower quartile (Q1): median of the lower half (values below Q2). The lower half is the first 5 terms: 10, 15, 19, 19, 19. Median of this (n=5, odd) is the 3rd term: 19.

- Upper quartile (Q3): median of the upper half (values above Q2). The upper half is the last 5 terms: 33, 42, 42, 55, 56. Median of this (n=5, odd) is the 3rd term: 42.

Ah, so that's the method. So Q1 = 19, median = 26, Q3 = 42? Wait, but wait, let's check with another approach. Wait, maybe the problem expects using the "median of the lower half" when n is even. Let's confirm:

Data set: 10,15,19,19,19,33,42,42,55,56 (n=10).

Split into lower half (first 5): [10,15,19,19,19], upper half (last 5): [33,42,42,55,56].

Q1: median of lower half: 19 (3rd term).

Median: average of 5th and 6th terms: (19 + 33)/2 = 26.

Q3: median of upper half: 42 (3rd term).

Yes, that's consistent with Tukey's method (used in many introductory stats courses).

So:

Lower quartile (Q1) = 19,

Median (Q2) = 26,

Upper quartile (Q3) = 42.

Wait, but let's check with the data:

Wait, the 5th term is 19 (index 5, 1-based), 6th is 33. So median is (19 + 33)/2 = 26. Correct.

Lower half: first 5 terms (1-5): 10,15,19,19,19. Median is 19 (3rd term).

Upper half: last 5 terms (6-10): 33,42,42,55,56. Median is 42 (3rd term, which is the 8th term in the original data? Wait, 6th term:33, 7th:42, 8th:42, 9th:55, 10th:56. So 3rd term in upper half is 8th term in original data? Wait, no: upper half is [33,42,42,55,56], so positions 1:33, 2:42, 3:42, 4:55, 5:56. So median is 3rd term: 42. Correct.

So:

Lower quartile = 19,

Median = 26,

Upper quartile = 42.

Answer:

lower quartile = \(\boldsymbol{19}\)
median = \(\boldsymbol{26}\)
upper quartile = \(\boldsymbol{42}\)