Question: QUESTION 4
The data below represents the weights of students in Grade 12 in Kilograms;
35; 53; 54; 56; 58; 58; 59.5; 60; 61; 63.5; 65; 68; 68.3; 70; 70.2; 72.4
4.1 In what order is the data sorted?
(2)
4.2 Calculate the mean weight.
(3)
4.3 What is the median weight?
(2)
4.4 Complete the table below:
Weight Range Frequency Tally
30-39
40-49
50-59
60-69
70-79
4.5 Determine the five number summary of the dataset.
(7)
4.6 Draw a labelled box-and-whisker plot to show the distribution of the weights.
(5)
1. **State the problem:** We have a dataset of student weights in kilograms: $$35, 53, 54, 56, 58, 58, 59.5, 60, 61, 63.5, 65, 68, 68.3, 70, 70.2, 72.4$$ and we need to answer several questions about it.
2. **4.1 In what order is the data sorted?**
The data is sorted in **ascending order** (from smallest to largest).
3. **4.2 Calculate the mean weight.**
The mean is the sum of all weights divided by the number of weights.
Number of data points, $$n = 16$$.
Sum of weights:
$$35 + 53 + 54 + 56 + 58 + 58 + 59.5 + 60 + 61 + 63.5 + 65 + 68 + 68.3 + 70 + 70.2 + 72.4 = 991.9$$
Mean:
$$\text{Mean} = \frac{991.9}{16} = 61.99375 \approx 61.99$$
4. **4.3 What is the median weight?**
The median is the middle value when data is ordered.
Since $$n=16$$ (even), median is the average of the 8th and 9th values.
8th value = $$60$$, 9th value = $$61$$
Median:
$$\frac{60 + 61}{2} = \frac{121}{2} = 60.5$$
5. **4.4 Complete the frequency table:**
| Weight Range | Frequency | Tally |
|--------------|-----------|-------------|
| 30-39 | 1 | | |
| 40-49 | 0 | |
| 50-59 | 5 | ||||| |
| 60-69 | 6 | |||||| |
| 70-79 | 4 | |||| |
Explanation:
- 30-39: 35 (1)
- 40-49: none (0)
- 50-59: 53,54,56,58,58 (5)
- 60-69: 59.5,60,61,63.5,65,68,68.3 (7) but 59.5 is in 50-59 range? No, 59.5 is less than 60, so it belongs to 50-59 range.
Recount 50-59: 53,54,56,58,58,59.5 = 6
60-69: 60,61,63.5,65,68,68.3 = 6
Update table:
| Weight Range | Frequency | Tally |
|--------------|-----------|-------------|
| 30-39 | 1 | | |
| 40-49 | 0 | |
| 50-59 | 6 | |||||| |
| 60-69 | 6 | |||||| |
| 70-79 | 3 | ||| |
70-79: 70,70.2,72.4 (3)
6. **4.5 Determine the five number summary:**
The five number summary consists of:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
Minimum = $$35$$
Maximum = $$72.4$$
To find Q1 and Q3, split data into lower and upper halves (excluding median since even number of data):
Lower half (first 8 values): $$35, 53, 54, 56, 58, 58, 59.5, 60$$
Upper half (last 8 values): $$61, 63.5, 65, 68, 68.3, 70, 70.2, 72.4$$
Q1 is median of lower half:
8 values, median is average of 4th and 5th values:
4th = 56, 5th = 58
$$Q1 = \frac{56 + 58}{2} = 57$$
Q3 is median of upper half:
4th and 5th values in upper half:
4th = 68, 5th = 68.3
$$Q3 = \frac{68 + 68.3}{2} = 68.15$$
Five number summary:
$$\boxed{35, 57, 60.5, 68.15, 72.4}$$
7. **4.6 Draw a labelled box-and-whisker plot:**
- Draw a number line covering from minimum (35) to maximum (72.4).
- Draw a box from Q1 (57) to Q3 (68.15).
- Draw a line inside the box at median (60.5).
- Draw whiskers from minimum (35) to Q1 (57) and from Q3 (68.15) to maximum (72.4).
This visually shows the distribution of weights.
**Final answers:**
- 4.1: Ascending order
- 4.2: Mean = $$61.99$$
- 4.3: Median = $$60.5$$
- 4.4: Frequency table as above
- 4.5: Five number summary = $$35, 57, 60.5, 68.15, 72.4$$
- 4.6: Box-and-whisker plot described above