Question

q9. create a frequency distribution table with 5 classes for the 45 raw data values below. 25 6 44 61 56 100 84 93 19 32 77 21 72 89 8 9 35 70 36 91 83 59 62 46 24 17 90 29 17 49 88 97 75 82 66 59 21 36 11 8 87 18 51 50 52

Explanation:

Step1: Find the range

First, find the minimum and maximum values in the data set. The minimum value ($\min$) is 6, and the maximum value ($\max$) is 100. The range ($R$) is calculated as $R=\max - \min = 100 - 6 = 94$.

Step2: Determine the class width

We need 5 classes, so the class width ($w$) is calculated by dividing the range by the number of classes and rounding up to a convenient number. $w=\lceil\frac{R}{n}
ceil=\lceil\frac{94}{5}
ceil=\lceil18.8
ceil = 19$. But to make the classes more intuitive, we can adjust slightly. Alternatively, we can use a class width of 20 (since 94/5 = 18.8, and 20 is a nice round number). Let's check: starting from 5 (a bit below the minimum 6) with width 20, the classes would be 5 - 24, 25 - 44, 45 - 64, 65 - 84, 85 - 104. Wait, but let's do it properly.

Wait, the minimum is 6, maximum is 100. Let's use class width 19. Then first class: 6 - 24 (6 + 19 - 1 = 24), second: 25 - 43 (25 + 19 - 1 = 43), third: 44 - 62 (44 + 19 - 1 = 62), fourth: 63 - 81 (63 + 19 - 1 = 81), fifth: 82 - 100 (82 + 19 - 1 = 100). Wait, but 100 is included in 82 - 100 (100 ≤ 100). Let's verify the number of classes.

Alternatively, let's list the data:

Data values: 25, 6, 44, 61, 56, 100, 84, 93, 19, 32, 77, 21, 72, 89, 8, 9, 35, 70, 36, 91, 83, 59, 62, 46, 24, 17, 90, 29, 17, 49, 88, 97, 75, 82, 66, 59, 21, 36, 11, 8, 87, 18, 51, 50, 52.

Let's sort the data: 6, 8, 8, 9, 11, 17, 17, 18, 19, 21, 21, 24, 25, 29, 32, 35, 36, 36, 44, 46, 49, 50, 51, 52, 56, 59, 59, 61, 62, 66, 70, 72, 75, 77, 82, 83, 84, 87, 88, 89, 90, 91, 93, 97, 100.

Now, let's define the classes. Let's use a class width of 20, starting at 0 (but our minimum is 6, so maybe 5 - 24, 25 - 44, 45 - 64, 65 - 84, 85 - 104. Wait, 100 is in 85 - 104. Let's count the frequency in each class:

1. 5 - 24: Values between 5 (inclusive) and 24 (inclusive). Let's list them: 6, 8, 8, 9, 11, 17, 17, 18, 19, 21, 21, 24. Wait, 24 is included? 5 - 24: 6 (yes), 8 (yes), 8 (yes), 9 (yes), 11 (yes), 17 (yes), 17 (yes), 18 (yes), 19 (yes), 21 (yes), 21 (yes), 24 (yes). Wait, 24 is 24, which is ≤24. So that's 12 values? Wait, let's count: 6,8,8,9,11,17,17,18,19,21,21,24. That's 12. Wait, but let's check the sorted data: positions 1 - 12: 6 (1), 8 (2), 8 (3), 9 (4), 11 (5), 17 (6), 17 (7), 18 (8), 19 (9), 21 (10), 21 (11), 24 (12). Then 25 is next (13).

2. 25 - 44: Values from 25 to 44 (inclusive). Sorted data: 25 (13), 29 (14), 32 (15), 35 (16), 36 (17), 36 (18), 44 (19). Wait, 25,29,32,35,36,36,44. Wait, 25 is 25, 29, 32, 35, 36, 36, 44. Let's count: 25 (1), 29 (2), 32 (3), 35 (4), 36 (5), 36 (6), 44 (7). So 7 values? Wait, no, in the sorted data, after 24 (12th), the next is 25 (13th), then 29 (14th), 32 (15th), 35 (16th), 36 (17th), 36 (18th), 44 (19th). So that's 7 values (13 - 19: 7 numbers).

3. 45 - 64: Values from 45 to 64 (inclusive). Sorted data: 46 (20), 49 (21), 50 (22), 51 (23), 52 (24), 56 (25), 59 (26), 59 (27), 61 (28), 62 (29). Wait, 45 - 64: 46 (yes, 46 ≥45), 49 (yes), 50 (yes), 51 (yes), 52 (yes), 56 (yes), 59 (yes), 59 (yes), 61 (yes), 62 (yes). What about 44? 44 is in the previous class. So 46 to 62: let's count the positions. 20th (46) to 29th (62): that's 10 values (29 - 20 + 1 = 10).

4. 65 - 84: Values from 65 to 84 (inclusive). Sorted data: 66 (30), 70 (31), 72 (32), 75 (33), 77 (34), 82 (35), 83 (36), 84 (37). Wait, 65 - 84: 66 (yes, 66 ≥65), 70 (yes), 72 (yes), 75 (yes), 77 (yes), 82 (yes), 83 (yes), 84 (yes). Let's check the positions: 30th (66) to 37th (84): that's 8 values (37 - 30 + 1 = 8).

5. 85 - 104: Values from 85 to 104 (inclusive). Sorted data: 87 (38), 88 (39), 89 (40), 90 (41), 91 (42), 93 (43), 97 (44), 100 (45). Let's count: 87 (yes, 87 ≥85), 88 (yes), 89 (yes), 90 (yes), 91 (yes), 93 (yes), 97 (yes), 100 (yes). Positions 38 - 45: 8 values (45 - 38 + 1 = 8). Wait, but 12 + 7 + 10 + 8 + 8 = 45, which matches the number of data points. Perfect.

Wait, but let's verify each class:

- 5 - 24: 6,8,8,9,11,17,17,18,19,21,21,24. Count: 12. Correct.
- 25 - 44: 25,29,32,35,36,36,44. Count: 7. Correct (25 to 44: 25,29,32,35,36,36,44 – 7 numbers).
- 45 - 64: 46,49,50,51,52,56,59,59,61,62. Count: 10. Correct (46 to 62: 10 numbers).
- 65 - 84: 66,70,72,75,77,82,83,84. Count: 8. Correct (66 to 84: 8 numbers).
- 85 - 104: 87,88,89,90,91,93,97,100. Count: 8. Correct (87 to 100: 8 numbers).

Now, let's create the frequency distribution table:

| Class Interval | Frequency |
|----------------|-----------|
| 5 - 24 | 12 |
| 25 - 44 | 7 |
| 45 - 64 | 10 |
| 65 - 84 | 8 |
| 85 - 104 | 8 |

Wait, but let's check the class boundaries to avoid overlaps. The first class: 5 ≤ x < 25, second: 25 ≤ x < 45, third: 45 ≤ x < 65, fourth: 65 ≤ x < 85, fifth: 85 ≤ x < 105. Then:

- 5 - 24 (boundaries 5 - 25): values 6,8,8,9,11,17,17,18,19,21,21,24 (all <25) – 12.
- 25 - 44 (25 - 45): 25,29,32,35,36,36,44 (all <45) – 7.
- 45 - 64 (45 - 65): 46,49,50,51,52,56,59,59,61,62 (all <65) – 10.
- 65 - 84 (65 - 85): 66,70,72,75,77,82,83,84 (all <85) – 8.
- 85 - 104 (85 - 105): 87,88,89,90,91,93,97,100 (all <105) – 8.

Yes, that works. So the frequency distribution table with 5 classes is as above.

Answer:

Class Interval	Frequency
25 – 44	7
45 – 64	10
65 – 84	8
85 – 104	8