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Statistics - Histograms, Bar Charts, and Pie Charts

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Bar charts are used to compare discrete categories. The height of each bar represents the frequency of that category, and all bars must have equal width with gaps between them.

A standard bar chart showing categories A, B, and C with discrete bars.
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Pie charts represent parts of a whole. Each sector's angle is proportional to the frequency it represents relative to the total, calculated as FrequencyTotal×360∘\frac{\text{Frequency}}{\text{Total}} \times 360^\circ.

A pie chart divided into three sectors.
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Histograms are for continuous data. Unlike bar charts, the area (not just height) represents the frequency. This is critical when class widths are unequal.

A histogram with unequal bar widths showing frequency density on the y-axis.
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Frequency Density is calculated as Frequency Density=FrequencyClass Width\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}. This value is plotted on the vertical axis of a histogram.

📐Formulae

Angle of Sector=FrequencyTotal Frequency×360∘\text{Angle of Sector} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ

Frequency Density=FrequencyClass Width\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}

Frequency=Frequency Density×Class Width\text{Frequency} = \text{Frequency Density} \times \text{Class Width}

Class Width=Upper Boundary−Lower Boundary\text{Class Width} = \text{Upper Boundary} - \text{Lower Boundary}

💡Examples

Problem 1:

In a survey of 60 students, 15 students chose 'Basketball' as their favorite sport. Calculate the angle for the 'Basketball' sector in a pie chart.

Solution:

90∘90^\circ

Explanation:

To find the angle, divide the frequency of the category by the total frequency and multiply by 360. 1560×360=14×360=90∘\frac{15}{60} \times 360 = \frac{1}{4} \times 360 = 90^\circ.

Problem 2:

A histogram is drawn to represent the weights of packages. For the class interval 20<w≤3020 < w \leq 30, the frequency is 45. Calculate the frequency density for this bar.

Solution:

4.54.5

Explanation:

First, find the class width: 30−20=1030 - 20 = 10. Then apply the formula: Frequency Density=FrequencyClass Width=4510=4.5\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} = \frac{45}{10} = 4.5.

Problem 3:

On a histogram, a bar has a width of 5 units (from x=10x=10 to x=15x=15) and a frequency density of 1.2. What is the frequency represented by this bar?

Solution:

66

Explanation:

Frequency is calculated as the area of the bar in a histogram. Frequency=Frequency Density×Class Width=1.2×5=6\text{Frequency} = \text{Frequency Density} \times \text{Class Width} = 1.2 \times 5 = 6.

Problem 4:

A group of 120 people were asked about their favorite fruit. 40 chose Apple, 50 chose Banana, and 30 chose Orange. Construct the sector angles for a pie chart.

Pie chart showing Apple, Banana, and Orange sectors with calculated angles.

Solution:

Total Frequency=40+50+30=120\text{Total Frequency} = 40 + 50 + 30 = 120 Apple=40120×360∘=120∘\text{Apple} = \frac{40}{120} \times 360^\circ = 120^\circ Banana=50120×360∘=150∘\text{Banana} = \frac{50}{120} \times 360^\circ = 150^\circ Orange=30120×360∘=90∘\text{Orange} = \frac{30}{120} \times 360^\circ = 90^\circ

Explanation:

To find the angle for each fruit, divide the frequency of that fruit by the total number of people surveyed and then multiply by the total degrees in a circle (360∘360^\circ).

Problem 5:

The table below shows the distribution of heights (hh cm) of students. For the interval 150<h≤165150 < h \leq 165, the frequency is 45. Find the height of the bar on a histogram if the vertical axis represents frequency density.

A histogram bar showing the interval 150 to 165 with a height of 3 units representing frequency density.

Solution:

Class Width=165−150=15\text{Class Width} = 165 - 150 = 15 Frequency Density=FrequencyClass Width\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} Frequency Density=4515=3\text{Frequency Density} = \frac{45}{15} = 3

Explanation:

The class width is the range of the interval. The frequency density, which is the height of the bar on a histogram, is the frequency divided by this width.