The test scores of 40 students are listed below. Find *P*65.

Test Scores

30

35

43

44

47

48

54

55

56

57

59

62

63

65

66

68

69

69

71

72

72

73

74

76

77

77

78

79

80

81

81

82

83

85

89

92

93

94

97

98
 
Group of answer choices

78

78.5

77

77.5

To find the 65th percentile (\(P_{65}\)) of a dataset, we can use the following formula:

\[
P_k = \left(\frac{k}{100}\right) \times (n + 1)
\]

Where:
- \(P_k\) is the kth percentile.
- \(k = 65\) (for the 65th percentile).
- \(n = 40\) (since there are 40 students).

### Step 1: Calculate the Position of \(P_{65}\)
\[
\text{Position} = \left(\frac{65}{100}\right) \times (40 + 1) = 0.65 \times 41 = 26.65
\]
So, \(P_{65}\) is located between the 26th and 27th values in the sorted list.

### Step 2: Find the 26th and 27th Scores
From the sorted list, the 26th value is 77 and the 27th value is 78.

### Step 3: Interpolate Between the Two Scores
Since the 26.65th position is closer to the 27th value, we use linear interpolation:
\[
P_{65} = 77 + 0.65 \times (78 - 77) = 77 + 0.65 = 77.65
\]

Thus, \(P_{65}\) is approximately **77.65**. Since this is closer to **78**, the correct answer is:

\[
\boxed{78}
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the difference between percentiles and quartiles?
2. How would the method change if we had an even larger dataset?
3. Can percentiles be used to find outliers in a dataset?
4. How can you interpret the 65th percentile in terms of student performance?
5. What is the formula for finding the median, and how does it relate to percentiles?

**Tip:** Percentiles help understand how individual data points compare to the entire dataset. They are especially useful in grading systems.

The test scores of 40 students are listed below. Find P65.

Test Scores
30, 35, 43, 44, 47, 48, 54, 55, 56, 57, 59, 62, 63, 65, 66, 68, 69, 69, 71, 72, 72, 73, 74, 76, 77, 77, 78, 79, 80, 81, 81, 82, 83, 85, 89, 92, 93, 94, 97, 98

Learn how to calculate the 65th percentile (P65) of a dataset with 40 students' test scores, using the percentile formula and interpolation. This guide covers core statistical concepts and is suitable for students in Grades 9-12.

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