Histograms

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レッスンノート

Frequency Density

Frequency density = frequency ÷ class width.
Used for grouped data, especially when class intervals are unequal.
Class width = upper bound − lower bound of an interval.
Frequency density measures how dense data is within its interval.
Always calculate frequency densities before drawing a histogram.

Drawing Histograms

Histograms are for continuous data grouped into class intervals.
The area of each bar represents the frequency, not the height.
Bars have no gaps between them (continuous scale).
Horizontal axis: class intervals; vertical axis: frequency density.
To draw a bar, calculate frequency density = frequency ÷ class width.
Height of bar = frequency density; width = class width.
Often exam questions ask to complete a partially drawn histogram.

Interpreting Histograms

Frequency = frequency density × class width (area of bar).
The vertical axis may be labelled 'frequency density' or just 'frequency' if widths are equal.
To estimate frequency for part of an interval, find area of that part.
Estimates assume data is evenly distributed within the interval.
Histograms can compare distributions only if scales and intervals match.

Common Calculations

Given frequency and class width, find frequency density: fd = f ÷ w.
Given frequency density and class width, find frequency: f = fd × w.
Given frequency and frequency density, find class width: w = f ÷ fd.
Always show working for method marks in exams.

Worked Example: Frequency Density Table

Add columns for class width and frequency density.
Example: interval 20 ≤ s < 40, frequency 5, width 20 → fd = 0.25.
Example: interval 40 ≤ s < 50, frequency 15, width 10 → fd = 1.5.
Check calculations carefully to avoid errors.

Average speed of trains

Worked Example: Completing a Histogram

Missing bars: compute frequency density for each missing interval.
Draw bar from start to end of interval at correct height.
Example: interval 60 ≤ x < 70, frequency 8, width 10 → fd = 0.8.
Example: interval 80 ≤ x < 100, frequency 2, width 20 → fd = 0.1.

Javelin throw distances

Worked Example: Interpreting to Find Frequency

Use histogram bar height (fd) and width to find frequency.
Example: bar height 0.6, width 15 → frequency = 0.6 × 15 = 9.
To estimate frequency for part of an interval, use linear interpolation.
Example: for 13 ≤ w < 15, fd = 4, width 2 → estimated frequency = 8.

Weights of newborn bottlenose dolphins

Exam Tips

Always write down frequency densities before drawing.
The frequency density axis scale may not be 1 unit per square.
Method marks are awarded for showing use of frequency density.
Answers from histograms are estimates because data distribution within intervals is unknown.

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練習問題

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1.What is the formula for frequency density?
Easy
- AFrequency density = frequency / class width
- BFrequency density = class width / frequency
- CFrequency density = frequency × class width
- DFrequency density = total frequency / number of classes
2.In a histogram, what does the area of a bar represent?
Easy
- AFrequency
- BFrequency density
- CClass width
- DTotal frequency
3.A frequency table has a class interval 20 < t ≤ 50 with frequency 30. What is the frequency density?
Easy
- A1
- B0.5
- C2
- D30
4.The frequency density for a class interval 10 < h ≤ 20 is 3. The class width is 10. What is the frequency?
Easy
- A30
- B3
- C0.3
- D13
5.A histogram has a bar for the interval 0 < t ≤ 10 with height 2.4 cm. The frequency for this interval is 12. What is the scale for frequency density?
Medium
- A1 cm = 0.5 frequency density
- B1 cm = 2 frequency density
- C1 cm = 5 frequency density
- D1 cm = 12 frequency density
6.The class widths in a histogram are 5, 10, 20, and 5. Which class interval will have the tallest bar if all frequencies are equal?
Medium
- AThe interval with width 5
- BThe interval with width 10
- CThe interval with width 20
- DAll bars will have the same height
7.In a histogram, the bars are drawn with no gaps because:
Easy
- AThe data is continuous
- BThe data is discrete
- CThe data is categorical
- DThe data is grouped
8.The frequency density for a class interval is 5 and the class width is 4. What is the frequency?
Easy
- A20
- B1.25
- C9
- D5

もっと問題

9.A histogram shows a bar of height 3 cm for the interval 10 < x ≤ 20. The frequency is 15. What is the height of the bar for the interval 20 < x ≤ 30 if the frequency is 30?
Medium
- A6 cm
- B3 cm
- C9 cm
- D1.5 cm
10.In a histogram, the y-axis is labelled 'Frequency density'. The bar for the interval 0 < t ≤ 20 has height 4. The class width is 20. What is the frequency?
Easy
- A80
- B20
- C4
- D5
11.Which of the following is NOT a difference between a bar chart and a histogram?
Medium
- ABoth have bars of equal width
- BBar charts have gaps between bars, histograms do not
- CBar charts are for discrete data, histograms for continuous data
- DIn a histogram the area represents frequency, in a bar chart the height represents frequency
12.The histogram shows the time taken by 100 students to complete a task. The bar for 0 < t ≤ 10 has height 3.6 cm. The frequency is 9. What is the height of the bar for 10 < t ≤ 15 if the frequency is 18?
Medium
- A14.4 cm
- B7.2 cm
- C3.6 cm
- D18 cm
13.The table shows the speeds of cars. | Speed (v km/h) | Frequency | |---|---| | 0 < v ≤ 40 | 50 | | 40 < v ≤ 50 | 120 | | 50 < v ≤ 80 | 30 | What is the frequency density for the interval 40 < v ≤ 50?
Medium
- A12
- B8
- C10
- D15
14.A histogram has a bar for the interval 20 < t ≤ 30 with height (frequency density) 3. The class width is 10. What is the frequency?
Easy
- A30
- B3
- C10
- D0.3
15.The frequency density for a class interval 0 < h ≤ 20 is 6. The frequency is 120. What is the class width?
Medium
- A20
- B6
- C120
- D10
16.A histogram shows the masses of 80 bags. The bar for 0 < m ≤ 5 has height 4 cm (scale: 1 cm = 1 unit of frequency density). The frequency is 20. What is the height of the bar for 5 < m ≤ 10 if the frequency is 30?
Medium
- A6 cm
- B3 cm
- C2 cm
- D4 cm
17.The histogram shows the time taken by 80 people. The bar for 0 < t ≤ 10 has area 40. The bar for 10 < t ≤ 20 has height 5 and width 10. What is the total frequency for 0 < t ≤ 20?
Hard
- A90
- B50
- C40
- D80
18.Which of the following is a difference between a bar chart and a histogram?
Easy
- AHistograms have no gaps between bars
- BBar charts have no gaps between bars
- CHistograms show discrete data
- DBar charts show continuous data
19.The class widths in a histogram are 5, 10, 20, and 5. The frequencies are 15, 20, 40, and 10 respectively. Which bar has the greatest frequency density?
Medium
- AThe bar with class width 5 and frequency 15
- BThe bar with class width 10 and frequency 20
- CThe bar with class width 20 and frequency 40
- DThe bar with class width 5 and frequency 10
20.A histogram has a bar for the interval 10 < x ≤ 20 with height 3 cm. The frequency is 15. What is the scale for frequency density (i.e., how many units of frequency density per cm)?
Medium
- A0.5 units per cm
- B2 units per cm
- C3 units per cm
- D1 unit per cm
21.The table shows the heights of plants. | Height (h cm) | Frequency | |---|---| | 0 < h ≤ 10 | 5 | | 10 < h ≤ 20 | 15 | | 20 < h ≤ 30 | 10 | What is the frequency density for the interval 10 < h ≤ 20?
Medium
- A1.5
- B15
- C0.5
- D10
22.In a histogram, the bar for 0 < t ≤ 20 has height 4 cm. The frequency is 40. The bar for 20 < t ≤ 30 has height 8 cm. What is the frequency for the second bar?
Medium
- A80
- B40
- C20
- D60
23.The histogram shows the distances, in km, that 65 people travel to work. The bar for the interval 0 < d ≤ 5 has height 6 cm. The bar for the interval 5 < d ≤ 15 has height 2 cm. The bar for the interval 15 < d ≤ 30 has height 1 cm. The height is measured in cm and the scale is 1 cm = 1 unit of frequency density. How many people travel more than 15 km?
Hard
- A15
- B30
- C45
- D20
24.A histogram is drawn to show the masses, m kg, of 50 bags of flour. The table shows the class intervals and frequencies. | Mass (m kg) | Frequency | |---|---| | 0 < m ≤ 5 | 10 | | 5 < m ≤ 10 | 20 | | 10 < m ≤ 20 | 20 | What is the frequency density for the interval 10 < m ≤ 20?
Easy
- A2
- B1
- C4
- D0.5
25.In a histogram, what does the height of each bar represent?
Easy
- AFrequency density
- BFrequency
- CClass width
- DCumulative frequency
26.The table shows the times, t minutes, spent by 80 people in a gym. | Time (t min) | Frequency | |---|---| | 0 < t ≤ 20 | 16 | | 20 < t ≤ 30 | 24 | | 30 < t ≤ 50 | 40 | A histogram is drawn. What is the frequency density for the interval 30 < t ≤ 50?
Easy
- A2
- B0.5
- C1.33
- D0.8
27.The histogram shows the distances, d km, walked by 36 people in a week. The bar for 0 < d ≤ 4 has height 3 cm. The bar for 4 < d ≤ 10 has height 2 cm. The bar for 10 < d ≤ 20 has height 0.8 cm. The bar for 20 < d ≤ 30 has height 0.4 cm. The height is measured in cm and the scale is 1 cm = 1 unit of frequency density. How many people walked more than 10 km?
Medium
- A12
- B8
- C16
- D20
28.A histogram is drawn for the ages, a years, of 100 people. The class intervals are: 0 < a ≤ 10, 10 < a ≤ 20, 20 < a ≤ 40, 40 < a ≤ 70. The frequencies are 20, 30, 40, 10 respectively. Which bar has the greatest frequency density?
Medium
- A0 < a ≤ 10
- B10 < a ≤ 20
- C20 < a ≤ 40
- D40 < a ≤ 70
29.The histogram shows the speeds, v km/h, of 200 cars. The bar for 0 < v ≤ 40 has height 1.25. The bar for 40 < v ≤ 50 has height 12. The bar for 50 < v ≤ 80 has height 1. The heights are frequency densities. How many cars have speed greater than 50 km/h?
Medium
- A30
- B50
- C20
- D60
30.The frequency density for a class interval is 4 and the class width is 5. What is the frequency?
Easy
- A20
- B1.25
- C9
- D0.8
31.The histogram shows the masses, m kg, of some bags of sand. The bar for 0 < m ≤ 10 has area 40. The bar for 10 < m ≤ 20 has height 5 and width 10. The bar for 20 < m ≤ 40 has height 1. What is the total frequency?
Medium
- A110
- B90
- C120
- D100
32.The frequency density for a class interval is 2.5 and the class width is 8. What is the frequency?
Easy
- A20
- B10
- C3.2
- D0.3125
33.In a histogram, the area of each bar represents the frequency of that class interval.
Easy
True or false?
34.The frequency density for a class interval is 3.2 and the class width is 5. What is the frequency?
Easy
- A16
- B1.56
- C0.64
- D8.2
35.A histogram has a bar for the interval 0 < t ≤ 20 with height (frequency density) 2. The class width is 20. What is the frequency?
Easy
- A40
- B10
- C0.1
- D22
36.In a histogram, the bar for the interval 10 < x ≤ 20 has height 3 cm. The frequency is 15. What is the scale for frequency density (i.e. how many units of frequency density does 1 cm represent)?
Medium
- A0.5
- B2
- C5
- D1
37.Which of the following statements about histograms are true? (Select all that apply)
Medium
- AThe bars are drawn with no gaps.
- BThe height of each bar represents the frequency.
- CThe area of each bar represents the frequency.
- DHistograms are used for discrete data.
38.Match each term with its correct definition.
Easy
- Frequency density
- Class width
- Frequency
- Histogram
- Area of bar
- Height of bar
- Difference between upper and lower bound
- Number of data values in an interval
39.Put the following steps for drawing a histogram in the correct order.
Medium
- Calculate frequency densities
- Draw bars with heights equal to frequency densities
- Determine class widths
- Label axes with appropriate scales
40.A histogram shows the heights of 68 plants. The bar for the interval 0 < h ≤ 20 has height 0.5 cm. The bar for 20 < h ≤ 40 has height 1.5 cm. The bar for 40 < h ≤ 60 has height 0.8 cm. The bar for 60 < h ≤ 100 has height 0.3 cm. The height is measured in cm and the scale is 1 cm = 1 unit of frequency density. What is the total number of plants in the interval 60 < h ≤ 100?
Medium
- A12
- B30
- C40
- D24
41.In a histogram, bars are drawn with no gaps between them.
Easy
True or false?
42.A class interval has width 8 and frequency 20. Calculate the frequency density.
Easy
43.The frequency for a class interval is 45 and the frequency density is 9. What is the class width?
Easy
44.What is the y-axis of a histogram usually labelled?
Easy
45.The table shows the speeds of cars. | Speed (v km/h) | Frequency | |---|---| | 0 < v ≤ 40 | 50 | | 40 < v ≤ 50 | 20 | | 50 < v ≤ 80 | 90 | Calculate the frequency density for the interval 40 < v ≤ 50.
Medium
46.A class interval has width 12 and frequency 36. Calculate the frequency density.
Easy
47.Complete the sentence.
Easy
In a histogram, the ____ of each bar represents the frequency.
48.Explain why the area of a bar in a histogram represents the frequency, not the height.
Hard
49.A class interval has width 8 and frequency 20. Calculate the frequency density.
Easy
- A4.5
- B3.5
- C1.5
- D2.5
50.The frequency for a class interval is 45 and the frequency density is 9. What is the class width?
Easy
- A5
- B6
- C4
- D7
51.The table shows the speeds of cars. | Speed (v km/h) | Frequency | |---|---| | 0 < v ≤ 40 | 50 | | 40 < v ≤ 50 | 20 | | 50 < v ≤ 80 | 90 | Calculate the frequency density for the interval 40 < v ≤ 50.
Medium
- A2
- B4
- C1
- D3
52.A class interval has width 12 and frequency 36. Calculate the frequency density.
Easy
- A2
- B5
- C3
- D4

復習ノート

## Histograms (IGCSE Maths Extended) ### Key Formula - **Frequency density** = frequency / class width - **Frequency** = frequency density × class width ### Drawing Histograms 1. Calculate class width (upper bound − lower bound). 2. Compute frequency density for each class. 3. On the graph: horizontal axis = class intervals (continuous, no gaps); vertical axis = frequency density. 4. Draw bars with height = frequency density, width = class width. ### Interpreting Histograms - Area of bar = frequency. - To find frequency from histogram: frequency = frequency density × class width. - To estimate frequency for part of an interval: find area of that part. - Estimates are needed because data within an interval may not be evenly distributed. ### Common Exam Tips - Always write down frequency densities before drawing. - Check vertical scale carefully (may not be 1 unit per square). - For incomplete histograms, use given bar heights to find missing frequencies or densities. ### Example | Class | Frequency | Class Width | Frequency Density | |-------|-----------|-------------|-------------------| | 0–10 | 20 | 10 | 2 | | 10–20 | 30 | 10 | 3 | | 20–30 | 10 | 10 | 1 | Bar heights are 2, 3, 1 respectively.

Frequency density: Frequency divided by class width; used as the height of bars in a histogram.
Class width: The difference between the upper and lower boundaries of a class interval.
Histogram: A graphical representation of continuous grouped data where the area of each bar is proportional to the frequency.
Continuous data: Data that can take any value within a range; used in histograms with no gaps between bars.

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