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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: These are used to represent discrete or categorical data using rectangular bars. The height or length of each bar is proportional to the frequency (ff) of the category. Visually, bars should have equal widths, and there should be consistent gaps between them to indicate that the data categories are distinct rather than continuous.

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Pie Charts: A circular representation of data where the entire circle represents the total (100%100\% or 360∘360^{\circ}). Each 'slice' or sector represents a specific category's portion of the total. The size of the sector's angle is calculated based on the category's frequency relative to the total frequency.

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Line Graphs: Primarily used to track changes or trends over periods of time. Data points are plotted on a grid and connected by straight lines. The horizontal axis (xx-axis) usually represents the time interval, while the vertical axis (yy-axis) represents the value being measured.

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Stem-and-Leaf Diagrams: A way to display the distribution of numerical data while keeping all original data values visible. The 'stem' represents the leading digits (e.g., tens) and the 'leaf' represents the trailing digit (units). A key, such as 2∣1=212 | 1 = 21, is essential to define the scale of the values.

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Frequency Tables: A table used to organize raw data by listing categories or intervals alongside a tally and a final count called 'frequency'. This serves as the foundation for constructing bar charts and pie charts.

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Interpreting Data: This involves reading specific values from a graph, identifying the mode (the most frequent category, shown as the highest bar or largest sector), and identifying trends (upward or downward slopes in a line graph).

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Discrete vs. Continuous Data: Discrete data can only take specific values (like the number of students), often represented by bar charts. Continuous data can take any value within a range (like height or time), often represented by line graphs or histograms.

📐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.