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Statistics and Probability - Representing Data (Bar Charts, Pie Charts, Line Graphs, Stem-and-Leaf Diagrams)

Grade 7IB

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

🔑Concepts

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Bar charts are used to compare categorical data. The height of each bar represents the frequency of that category, and gaps are kept between bars to show categories are distinct.

A bar chart showing three categories A, B, and C with different frequencies.
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Pie charts represent parts of a whole using sectors of a circle. The angle of each sector is proportional to the frequency of the category, calculated as Angle=ValueTotal×360∘\text{Angle} = \frac{\text{Value}}{\text{Total}} \times 360^{\circ}.

A pie chart with a 90 degree sector representing 25 percent.
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Line graphs are typically used to show trends or changes over a continuous interval of time. Data points are plotted and connected by straight line segments.

A line graph showing an upward trend over time.
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Stem-and-leaf diagrams organize numerical data while maintaining individual values. The 'stem' represents the tens (or higher digits) and 'leaves' represent the units digits. It must always include a key.

A stem and leaf plot showing values in the 10s and 20s.

📐Formulae

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

Percentage=Frequency of CategoryTotal Frequency×100%\text{Percentage} = \frac{\text{Frequency of Category}}{\text{Total Frequency}} \times 100\%

Mean(xˉ)=∑xn\text{Mean} (\bar{x}) = \frac{\sum x}{n}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

💡Examples

Problem 1:

A class of 2020 students was surveyed about their favorite fruit. 55 chose Apple, 1212 chose Banana, and 33 chose Orange. Calculate the angle for each sector to draw a pie chart.

Solution:

Step 1: Identify total frequency (n=20n = 20). Step 2: Calculate angle for Apple: 520×360∘=90∘\frac{5}{20} \times 360^{\circ} = 90^{\circ}. Step 3: Calculate angle for Banana: 1220×360∘=216∘\frac{12}{20} \times 360^{\circ} = 216^{\circ}. Step 4: Calculate angle for Orange: 320×360∘=54∘\frac{3}{20} \times 360^{\circ} = 54^{\circ}. Step 5: Check total: 90∘+216∘+54∘=360∘90^{\circ} + 216^{\circ} + 54^{\circ} = 360^{\circ}.

Explanation:

To represent data on a pie chart, we convert the frequency of each category into a proportional part of a full circle (360∘360^{\circ}).

Problem 2:

Represent the following test scores in a Stem-and-Leaf diagram and find the median: 21,25,33,33,38,40,4221, 25, 33, 33, 38, 40, 42.

Solution:

Step 1: Create the stems (tens place) and leaves (units place). Stem 22: 1,51, 5 Stem 33: 3,3,83, 3, 8 Stem 44: 0,20, 2 Step 2: Add a Key: 2∣1=212 | 1 = 21. Step 3: Find the median. Since there are n=7n = 7 values, the median is the 7+12=4th\frac{7+1}{2} = 4^{th} value. Step 4: Counting through the leaves: 21,25,33,33…21, 25, 33, 33 \dots the 4th4^{th} value is 3333.

Explanation:

The stem-and-leaf plot organizes the data in order. The median is the middle value of the ordered data set.

Problem 3:

A group of 4040 students were asked how they travel to school. 1010 walk, 2020 take the bus, and 1010 cycle. Represent this information as a pie chart and find the angle for the 'Bus' sector.

Pie chart split into one semicircle and two quarter circles.

Solution:

Total students=40\text{Total students} = 40 Bus sector angle=2040×360∘=180∘\text{Bus sector angle} = \frac{20}{40} \times 360^{\circ} = 180^{\circ} Walk sector angle=1040×360∘=90∘\text{Walk sector angle} = \frac{10}{40} \times 360^{\circ} = 90^{\circ} Cycle sector angle=1040×360∘=90∘\text{Cycle sector angle} = \frac{10}{40} \times 360^{\circ} = 90^{\circ}

Explanation:

To find the angle for each sector, divide the frequency by the total frequency and multiply by 360∘360^{\circ} (the total degrees in a circle). Since the bus represents half the total students (20/4020/40), it occupies exactly half the circle (180∘180^{\circ}).

Problem 4:

The following numbers represent the number of goals scored by a team in 12 matches: 1,0,2,1,3,0,1,1,2,4,1,01, 0, 2, 1, 3, 0, 1, 1, 2, 4, 1, 0. Create a frequency bar chart for this data.

Bar chart showing goal frequencies: 0:3, 1:5, 2:2, 3:1, 4:1.

Solution:

First, count frequencies:

  • 00 goals: 33 times
  • 11 goal: 55 times
  • 22 goals: 22 times
  • 33 goals: 11 time
  • 44 goals: 11 time

Explanation:

Identify the distinct values (0, 1, 2, 3, 4) to put on the horizontal axis and count how many times each value occurs (frequency) for the vertical axis heights.