Question 1176858: Use the appropriate formula to find the quartiles of the following,do not forget to state your interpretations:
Raw scores of 48 students in a mathematics 8 test
SCORES: f: lb: cumulative frequency cf:
46-50 1 45.5 48
41-45 5 40.5 47
36-40 11 35.5 42
31-35 12 30.5 31
26-30 11 25.5 19
21-25 5 20.5 8
16-20 2 15.5 3
11-15 1 10.5 1
____Σr = 48
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's find the quartiles for the given grouped data and interpret the results.
**1. Understanding Quartiles**
* **Q1 (First Quartile):** The value below which 25% of the data falls.
* **Q2 (Second Quartile):** The median, the value below which 50% of the data falls.
* **Q3 (Third Quartile):** The value below which 75% of the data falls.
**2. Formulas for Grouped Data**
The formula for finding quartiles in grouped data is:
* Qk = L + [(kN/4 - cf) / f] * w
Where:
* Qk = the kth quartile
* L = lower boundary of the quartile class
* N = total number of data points
* cf = cumulative frequency of the class before the quartile class
* f = frequency of the quartile class
* w = class width
**3. Calculate Quartiles**
* N = 48
* Class width (w) = 5 (e.g., 50 - 46 + 1 = 5)
**a) Q1 (First Quartile)**
* kN/4 = (1 * 48) / 4 = 12
* The 12th value falls in the 26-30 class (cf = 19, which is > 12).
* L = 25.5
* cf = 8
* f = 11
* Q1 = 25.5 + [(12 - 8) / 11] * 5
* Q1 = 25.5 + (4 / 11) * 5
* Q1 = 25.5 + 20 / 11
* Q1 = 25.5 + 1.818
* Q1 ≈ 27.32
**b) Q2 (Second Quartile)**
* kN/4 = (2 * 48) / 4 = 24
* The 24th value falls in the 31-35 class (cf = 31, which is > 24).
* L = 30.5
* cf = 19
* f = 12
* Q2 = 30.5 + [(24 - 19) / 12] * 5
* Q2 = 30.5 + (5 / 12) * 5
* Q2 = 30.5 + 25 / 12
* Q2 = 30.5 + 2.083
* Q2 ≈ 32.58
**c) Q3 (Third Quartile)**
* kN/4 = (3 * 48) / 4 = 36
* The 36th value falls in the 36-40 class (cf = 42, which is > 36).
* L = 35.5
* cf = 31
* f = 11
* Q3 = 35.5 + [(36 - 31) / 11] * 5
* Q3 = 35.5 + (5 / 11) * 5
* Q3 = 35.5 + 25 / 11
* Q3 = 35.5 + 2.273
* Q3 ≈ 37.77
**4. Interpretations**
* **Q1 ≈ 27.32:** Approximately 25% of the students scored 27.32 or below on the mathematics test.
* **Q2 ≈ 32.58:** Approximately 50% of the students scored 32.58 or below on the mathematics test. This is also the median score.
* **Q3 ≈ 37.77:** Approximately 75% of the students scored 37.77 or below on the mathematics test.
**Summary**
* Q1 ≈ 27.32
* Q2 ≈ 32.58
* Q3 ≈ 37.77
|
|
|
