SOLUTION: SCORES 46-50 41-45 36-40 31-35 26-30 21-25 FREQUENCY 5 11 8 16 3 7 A. Calculate the 1st quartile b. " 2nd quartile c. " 3rd quartile d. " 7th decile e. " 3rd

Question 1179330: SCORES
46-50
41-45
36-40
31-35
26-30
21-25
FREQUENCY
5
11
8
16
3
7
A. Calculate the 1st quartile
b. " 2nd quartile
c. " 3rd quartile
d. " 7th decile
e. " 3rd decile
f. " 25th percentile
g. " 60th percentile
I tried to solve this but, my answers were not correct
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's calculate the quartiles, deciles, and percentiles for the given grouped frequency distribution.
1. Calculate Cumulative Frequencies:
Scores Frequency Cumulative Frequency
46-50 5 5
41-45 11 16
36-40 8 24
31-35 16 40
26-30 3 43
21-25 7 50
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Total frequency (n) = 50
2. Calculate Quartiles:
Q1 (1st Quartile):
Position of Q1 = (1/4) * n = (1/4) * 50 = 12.5
12.5 falls in the 41-45 bin (cumulative frequency 16).
Lower boundary (L) = 40.5
Class width (w) = 5
Frequency of Q1 class (f) = 11
Cumulative frequency before Q1 class (cf) = 5
Q1 = L + [(12.5 - cf) / f] * w = 40.5 + [(12.5 - 5) / 11] * 5 = 40.5 + (7.5 / 11) * 5 ≈ 40.5 + 3.41 ≈ 43.91
Q2 (2nd Quartile/Median):
Position of Q2 = (2/4) * n = (1/2) * 50 = 25
25 falls in the 31-35 bin (cumulative frequency 40).
Lower boundary (L) = 30.5
Class width (w) = 5
Frequency of Q2 class (f) = 16
Cumulative frequency before Q2 class (cf) = 24
Q2 = L + [(25 - cf) / f] * w = 30.5 + [(25 - 24) / 16] * 5 = 30.5 + (1 / 16) * 5 ≈ 30.5 + 0.31 ≈ 30.81
Q3 (3rd Quartile):
Position of Q3 = (3/4) * n = (3/4) * 50 = 37.5
37.5 falls in the 31-35 bin (cumulative frequency 40).
Lower boundary (L) = 30.5
Class width (w) = 5
Frequency of Q3 class (f) = 16
Cumulative frequency before Q3 class (cf) = 24
Q3 = L + [(37.5 - cf) / f] * w = 30.5 + [(37.5 - 24) / 16] * 5 = 30.5 + (13.5 / 16) * 5 ≈ 30.5 + 4.22 ≈ 34.72
3. Calculate Deciles:
D7 (7th Decile):
Position of D7 = (7/10) * n = (7/10) * 50 = 35
35 falls in the 31-35 bin (cumulative frequency 40).
Lower boundary (L) = 30.5
Class width (w) = 5
Frequency of D7 class (f) = 16
Cumulative frequency before D7 class (cf) = 24
D7 = L + [(35 - cf) / f] * w = 30.5 + [(35 - 24) / 16] * 5 = 30.5 + (11 / 16) * 5 ≈ 30.5 + 3.44 ≈ 33.94
D3 (3rd Decile):
Position of D3 = (3/10) * n = (3/10) * 50 = 15
15 falls in the 41-45 bin (cumulative frequency 16).
Lower boundary (L) = 40.5
Class width (w) = 5
Frequency of D3 class (f) = 11
Cumulative frequency before D3 class (cf) = 5
D3 = L + [(15 - cf) / f] * w = 40.5 + [(15 - 5) / 11] * 5 = 40.5 + (10 / 11) * 5 ≈ 40.5 + 4.55 ≈ 45.05
4. Calculate Percentiles:
P25 (25th Percentile):
Same as Q1. P25 ≈ 43.91
P60 (60th Percentile):
Position of P60 = (60/100) * n = (60/100) * 50 = 30
30 falls in the 31-35 bin (cumulative frequency 40).
Lower boundary (L) = 30.5
Class width (w) = 5
Frequency of P60 class (f) = 16
Cumulative frequency before P60 class (cf) = 24
P60 = L + [(30 - cf) / f] * w = 30.5 + [(30 - 24) / 16] * 5 = 30.5 + (6 / 16) * 5 ≈ 30.5 + 1.88 ≈ 32.38
Answers (rounded to two decimal places):
a. Q1 ≈ 43.91
b. Q2 ≈ 30.81
c. Q3 ≈ 34.72
d. D7 ≈ 33.94
e. D3 ≈ 45.05
f. P25 ≈ 43.91
g. P60 ≈ 32.38
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