Histograms (Copy)
1.
The table shows masses of 40 students. Calculate the frequency density for each class.
| Mass (kg) | 40–50 | 50–60 | 60–80 | 80–100 |
|———–|——-|——-|——–|
| Frequency | 8 | 12 | 16 | 4 |
2.
Draw a histogram for Q1 using class widths and frequency densities.
3.
From Q1, which class interval has the greatest frequency density?
4.
The table shows times (minutes) taken by 50 runners. Complete the frequency density column.
| Time | 0–10 | 10–20 | 20–40 | 40–60 |
|——|——-|——–|——–|
| Freq | 5 | 10 | 20 | 15 |
5.
Draw the histogram for Q4.
6.
From Q4, which interval contains the median (25th runner)?
7.
A histogram has bars with frequencies:
0–5: 8, 5–10: 12, 10–20: 20.
Find frequency density for each.
8.
Explain why histograms use frequency density on the y-axis, not frequency.
9.
The table shows test scores of 60 students. Complete the frequency density column.
| Score | 0–20 | 20–40 | 40–60 | 60–100 |
|——-|——-|——-|——–|
| Freq | 6 | 18 | 24 | 12 |
10.
Draw a histogram for Q9.
11.
From Q9, estimate the modal class.
12.
A histogram has one tall narrow bar and one short wide bar, both with equal area. What does this mean?
13.
The heights (cm) of 80 plants are grouped:
| Height | 0–10 | 10–20 | 20–30 | 30–50 |
|——–|——-|——-|——–|
| Freq | 5 | 15 | 40 | 20 |
Calculate frequency densities.
14.
Draw the histogram for Q13.
15.
From Q13, which class interval is modal?
16.
The histogram of test scores shows the tallest bar in the 40–60 interval. What does this mean?
17.
Explain how you would use the histogram to estimate the median.
18.
The histogram shows more spread out bars at higher values. What does this suggest about data distribution?
19.
From a histogram with equal bar widths, what can be said about frequency density and frequency?
20.
The weights of 100 people are grouped:
| Weight | 40–50 | 50–70 | 70–90 | 90–110 |
|——–|——–|——–|——–|
| Freq | 20 | 40 | 30 | 10 |
Calculate frequency densities.
21.
Draw the histogram for Q20.
22.
From Q20, find the modal class.
23.
Explain why histograms are more suitable than bar charts for continuous data.
24.
The histogram shows a symmetric distribution. What does this imply about mean and median?
25.
The histogram shows a right-skewed distribution. Which is larger: mean or median?
26.
A histogram for exam marks is left-skewed. Which average would best represent the data?
27.
A histogram has bars: 0–5 (density 6), 5–15 (density 2), 15–20 (density 4). Find the total frequency.
28.
The time taken by buses to arrive is shown by a histogram. Most bars are concentrated around 10–20 minutes. What does this suggest?
29.
When drawing histograms, why must class intervals be continuous?
30.
Explain one situation where histograms are more useful than cumulative frequency diagrams.
1.
Frequency density = Frequency ÷ Class width.
40–50: 8 ÷ 10 = 0.8
50–60: 12 ÷ 10 = 1.2
60–80: 16 ÷ 20 = 0.8
80–100: 4 ÷ 20 = 0.2
2.
Histogram drawn with class widths proportional, heights as frequency densities from Q1.
3.
Greatest frequency density = 50–60 (1.2).
4.
Widths: 10, 10, 20, 20.
Densities: 5/10=0.5, 10/10=1, 20/20=1, 15/20=0.75.
5.
Histogram with bars of heights 0.5, 1, 1, 0.75.
6.
Cumulative frequencies: 5, 15, 35, 50.
Median = 25th lies in 20–40 interval.
7.
0–5: 8 ÷ 5=1.6
5–10: 12 ÷ 5=2.4
10–20: 20 ÷ 10=2
8.
Because class widths may differ, frequency density ensures area ∝ frequency.
9.
Widths: 20, 20, 20, 40.
Densities: 6/20=0.3, 18/20=0.9, 24/20=1.2, 12/40=0.3.
10.
Histogram drawn with these densities.
11.
Modal class = 40–60 (highest density 1.2).
12.
Equal areas mean equal frequencies, even though bar shapes differ.
13.
Widths: 10,10,10,20.
Densities: 5/10=0.5, 15/10=1.5, 40/10=4, 20/20=1.
14.
Histogram drawn with densities from Q13.
15.
Modal class = 20–30 (highest density=4).
16.
It means most data is concentrated in the 40–60 interval.
17.
Find cumulative frequencies, locate 50% mark, then read median from histogram using areas.
18.
Spread out at higher values = positive skew (right skewed).
19.
If widths equal, frequency density = frequency ÷ constant. So frequency density ∝ frequency.
20.
Widths: 10,20,20,20.
Densities: 20/10=2, 40/20=2, 30/20=1.5, 10/20=0.5.
21.
Histogram drawn with densities from Q20.
22.
Modal class = 50–70 (density=2, frequency=40).
23.
Histograms handle continuous intervals, bar charts only categorical data.
24.
Symmetric distribution → mean ≈ median.
25.
Right-skewed → mean > median.
26.
Left-skewed → median better than mean (not distorted by outliers).
27.
Areas = frequency.
0–5: 6×5=30
5–15: 2×10=20
15–20: 4×5=20
Total = 70.
28.
Most buses arrive between 10–20 mins → waiting time is consistent around that interval.
29.
Class intervals must be continuous so no gaps, ensuring correct area representation.
30.
Histograms show distribution shape (skew, spread) better than cumulative frequency diagrams, which mainly show medians and quartiles.