Math worksheet focused on calculating percentiles and quartiles, featuring problems with data sets, range, median, and interquartile range.

I'll work through each problem step by step, checking my calculations carefully.

Problem 1: Finding the Range

The range is the difference between the largest and smallest values in a data set.

a) Data: 6, 8, 11, 15, 24, 38
- Largest value: 38
- Smallest value: 6
- Range = 38 - 6 = 32

b) Data: 11, -6, -2, 16, 9, -8, 17, 19
- Largest value: 19
- Smallest value: -8
- Range = 19 - (-8) = 19 + 8 = 27

c) Data: 6.4, 3.8, 5.9, 4.7, 5.3, 7.1, 3.2
- Largest value: 7.1
- Smallest value: 3.2
- Range = 7.1 - 3.2 = 3.9

Problem 2: Finding Median and Quartiles

Data: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36

First, I need to order the data from smallest to largest:
6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

There are 11 data points (odd number).

Median (Q2): The middle value is the 6th value = 41

For quartiles with n=11:
- Q1 position: (n+1)/4 = 12/4 = 3rd value = 15
- Q3 position: 3(n+1)/4 = 3(12)/4 = 9th value = 43

Let me verify this using another method. With 11 values, the median splits it into two halves of 5 values each (excluding the median).

Lower half: 6, 7, 15, 36, 39
Upper half: 41, 43, 43, 47, 49

Q1 = median of lower half = 15 (middle of 5 values)
Q3 = median of upper half = 43 (middle of 5 values)

This confirms: Q1 = 15, Median = 41, Q3 = 43

Problem 3: Angela's Sales Data

Data: 34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37

First, order the data:
1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57

There are 12 data points (even number).

a) Median: For even n, median is average of the two middle values
Positions 6 and 7: 24 and 28
Median = (24 + 28)/2 = 52/2 = 26

b) Range = Largest - Smallest = 57 - 1 = 56

c) Upper and Lower Quartiles:
With n=12, split into two halves of 6 values each.

Lower half: 1, 11, 15, 19, 20, 24
Upper half: 28, 34, 37, 47, 50, 57

Q1 = median of lower half = (15 + 19)/2 = 34/2 = 17
Q3 = median of upper half = (37 + 47)/2 = 84/2 = 42

d) Interquartile Range (IQR) = Q3 - Q1 = 42 - 17 = 25

Problem 4: Heights of Students

Data: 65, 63, 68, 59, 74, 59, 68, 61, 64, 60, 69, 72, 55, 64

Let me count how many students fall into each interval:

Interval 55-58: Looking for heights 55, 56, 57, 58
- 55 appears once
Frequency = 1

Interval 59-62: Looking for heights 59, 60, 61, 62
- 59 appears twice
- 60 appears once
- 61 appears once
Frequency = 2 + 1 + 1 = 4

Interval 63-66: Looking for heights 63, 64, 65, 66
- 63 appears once
- 64 appears twice
- 65 appears once
Frequency = 1 + 2 + 1 = 4

Interval 67-70: Looking for heights 67, 68, 69, 70
- 68 appears twice
- 69 appears once
Frequency = 2 + 1 = 3

Interval 71-74: Looking for heights 71, 72, 73, 74
- 72 appears once
- 74 appears once
Frequency = 1 + 1 = 2

Total: 1 + 4 + 4 + 3 + 2 = 14 ✓ (matches the given 14 students)

Now let me find which intervals contain the median and upper quartile.

First, order all the data:
55, 59, 59, 60, 61, 63, 64, 64, 65, 68, 68, 69, 72, 74

With n=14 (even), median is average of positions 7 and 8:
Position 7: 64
Position 8: 64
Median = (64 + 64)/2 = 64

The median (64) falls in interval 63-66.

For Q3 (upper quartile):
Upper half: 64, 65, 68, 68, 69, 72, 74 (7 values)
Q3 = median of upper half = 4th value = 68

The upper quartile (68) falls in interval 67-70.

For part d: What percent of students are shorter than 5 feet 7 inches?
5 feet 7 inches = 5 × 12 + 7 = 60 + 7 = 67 inches

Students shorter than 67 inches means heights < 67:
From ordered data: 55, 59, 59, 60, 61, 63, 64, 64, 65
That's 9 students out of 14.

Percent = (9/14) × 100 = 64.29% ≈ 64.3%

Let me double-check by counting from the frequency table:
- 55-58: 1 student (all < 67)
- 59-62: 4 students (all < 67)
- 63-66: 4 students (all < 67)
- 67-70: This includes 67, 68, 69, 70. We need those < 67, so none from this interval
- 71-74: All ≥ 67, so none

Total < 67: 1 + 4 + 4 = 9 students
Percent = 9/14 × 100 = 64.2857...% ≈ 64.3%

Final Answer:

Problem 1:
a) Range = 32
b) Range = 27
c) Range = 3.9

Problem 2:
Ordered data: 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49
Median = 41
Q3 = 43
Q1 = 15

Problem 3:
a) Median = 26
b) Range = 56
c) Q1 = 17, Q3 = 42
d) Interquartile Range = 25

Problem 4:
a) Frequency table:
- 55-58: 1
- 59-62: 4
- 63-66: 4
- 67-70: 3
- 71-74: 2

b) Interval containing median: 63-66
c) Interval containing upper quartile: 67-70
d) Percent shorter than 5 feet 7 inches: 64.3%