Solved Exercises on Dot Plots in Grade 7

Master dot plots: creating, reading, and analyzing data displays through these 5 detailed exercises with step-by-step solutions.

Solution: Exercises 1 to 3
1 Creating a Dot Plot
Exercise 1
Create a dot plot for the following data showing the number of books read by students in a month: 3, 5, 2, 4, 5, 3, 6, 4, 3, 2, 5, 4, 3, 6, 5.
Definition:

Dot Plot: A statistical chart that displays data points on a simple scale using dots.

Frequency: The number of times a value appears in the data set.

Scale: The horizontal axis showing all possible values in the data set.

Dot Plot Creation Method:
  1. Identify the range of data (minimum to maximum values)
  2. Create a horizontal scale with all possible values
  3. Count the frequency of each value
  4. Place one dot above each value for each occurrence
  5. Stack dots vertically above each value
  6. Label the plot appropriately
Data Sorted
2,2,3,3,3,3,4,4,4,5,5,5,5,6,6
Values
2,3,4,5,6
Frequencies
2,4,3,4,2
Step 1: Organize the data

Original data: 3, 5, 2, 4, 5, 3, 6, 4, 3, 2, 5, 4, 3, 6, 5

Sorted data: 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6

Step 2: Identify the range

Minimum value: 2

Maximum value: 6

Range: 2, 3, 4, 5, 6

Step 3: Count frequencies

Books = 2: appears 2 times

Books = 3: appears 4 times

Books = 4: appears 3 times

Books = 5: appears 4 times

Books = 6: appears 2 times

Step 4: Create the dot plot

Draw horizontal scale from 2 to 6

Place 2 dots above 2

Place 4 dots above 3

Place 3 dots above 4

Place 4 dots above 5

Place 2 dots above 6

Step 5: Add labels and title

Title: "Number of Books Read by Students"

X-axis: "Number of Books"

Dot plot created with values 2,3,4,5,6 and frequencies 2,4,3,4,2
Final Answer:

The dot plot has been created with values 2, 3, 4, 5, 6 on the horizontal axis and the corresponding frequencies represented by stacked dots above each value.

Applied Rules:

• One-to-One Correspondence: Each data point is represented by one dot

• Vertical Stacking: Dots are stacked vertically above each value

• Consistent Scale: Values are evenly spaced on the horizontal axis

2 Reading a Dot Plot
Exercise 2
The dot plot below shows the number of hours spent on homework by 7th graders in one week. Use the dot plot to answer the following questions:
Hours: 0,1,2,3,4,5,6
Dots: 2,3,5,4,3,2,1
a) How many students were surveyed?
b) What is the most common number of hours spent on homework?
c) How many students spent 4 or more hours on homework?
Definition:

Reading Dot Plots: Interpreting data by counting dots and understanding their positions.

Mode: The value with the highest frequency (most dots).

Total Count: The sum of all frequencies.

Total Students
20
Mode
2 hours
4+ Hours
6 students
Step 1: Count total number of students

Hours: 0, 1, 2, 3, 4, 5, 6

Dots: 2, 3, 5, 4, 3, 2, 1

Total = 2 + 3 + 5 + 4 + 3 + 2 + 1 = 20 students

Step 2: Identify the mode (most frequent value)

Compare the number of dots for each value:

0 hours: 2 dots

1 hour: 3 dots

2 hours: 5 dots (highest)

3 hours: 4 dots

4 hours: 3 dots

5 hours: 2 dots

6 hours: 1 dot

The mode is 2 hours (5 dots)

Step 3: Count students spending 4 or more hours

4 hours: 3 students

5 hours: 2 students

6 hours: 1 student

Total = 3 + 2 + 1 = 6 students

Step 4: Verify calculations

Check that the total adds up correctly: 2+3+5+4+3+2+1 = 20 ✓

Confirm mode: 2 hours has the most dots ✓

Confirm 4+ hours: 3+2+1 = 6 ✓

a) 20 students, b) 2 hours, c) 6 students
Final Answer:

a) 20 students were surveyed. b) The most common number of hours is 2 hours. c) 6 students spent 4 or more hours on homework.

Applied Rules:

• Counting Dots: Each dot represents one data point

• Mode Identification: Value with most dots is the mode

• Range Analysis: Count dots for values meeting specific criteria

3 Analyzing Center and Spread
Exercise 3
Analyze the following dot plot showing quiz scores: 75, 80, 80, 85, 85, 85, 90, 90, 90, 90, 95, 95, 100. Calculate the mean, median, and range of the data.
Definition:

Measures of Center: Mean, median, and mode that represent typical values.

Measures of Spread: Range, IQR, and standard deviation that measure variability.

Data Set
75,80,80,85,85,85,90,90,90,90,95,95,100
Mean
87.7
Median
90
Range
25
Step 1: Identify the data set

From the dot plot: 75, 80, 80, 85, 85, 85, 90, 90, 90, 90, 95, 95, 100

Number of data points: 13

Step 2: Calculate the mean

Sum of all values: 75 + 80 + 80 + 85 + 85 + 85 + 90 + 90 + 90 + 90 + 95 + 95 + 100 = 1140

Mean = 1140 ÷ 13 = 87.69... ≈ 87.7

Step 3: Calculate the median

Since there are 13 values (odd), median is the middle value

Position of median: (13 + 1) ÷ 2 = 7th value

Ordered data: 75, 80, 80, 85, 85, 85, 90, 90, 90, 90, 95, 95, 100

7th value: 90

Median = 90

Step 4: Calculate the range

Maximum value: 100

Minimum value: 75

Range = 100 - 75 = 25

Step 5: Interpret the results

Mean (87.7) is slightly lower than median (90), suggesting a slight left skew

Range of 25 indicates moderate spread in scores

Mode is 90 (appears 4 times)

Mean = 87.7, Median = 90, Range = 25
Final Answer:

Mean = 87.7, Median = 90, Range = 25. The data is slightly skewed left as the mean is less than the median.

Applied Rules:

• Mean Formula: Sum of values ÷ Number of values

• Median Position: (n+1)÷2 for odd count, average of middle two for even

• Range Calculation: Maximum - Minimum

Rules and methods, laws,...
\(\text{Mean} = \frac{\sum x}{n}, \quad \text{Median} = \text{middle value}, \quad \text{Range} = \max - \min\)
Key Formulas
Dot Plot
One dot per data point
Vertical stacking
Measures of Center
Mean, Median, Mode
Represent typical values
Measures of Spread
Range, IQR
Measure variability
Key Definitions:

Dot Plot: A graphical representation of data using dots to show frequency of each value

Frequency: How often each value occurs in the data set

Scale: The horizontal axis showing all possible values

Stacking: Placing dots vertically above each value

Mode: The value that appears most frequently

Range: The difference between maximum and minimum values

Center: A measure representing the middle of the data set

Spread: How spread out the values are in the data set

Complete Methodology:
  1. Data Organization: Sort and organize the data values
  2. Scale Creation: Create a horizontal scale with all possible values
  3. Frequency Counting: Count how many times each value appears
  4. Dot Placement: Place one dot above each value for each occurrence
  5. Analysis: Calculate measures of center and spread
  6. Step 6: Interpretation: Draw conclusions from the data
Tip 1: Always start by organizing your data in ascending order.
Tip 2: Each dot represents exactly one data point.
Tip 3: Dots should be stacked vertically above each value.
Tip 4: The mode is the value with the tallest stack of dots.
Tip 5: Use dot plots to easily identify clusters, gaps, and outliers.

Common Errors: Miscounting dots, confusing frequency with value, not aligning dots properly, forgetting to label axes.
Exam Preparation: Practice creating dot plots, memorize formulas for measures of center, understand how to read dot plots effectively.
Solution: Exercises 4 to 5
4 Comparing Data Sets
Exercise 4
Two classes took the same test. Class A scores: 70, 75, 75, 80, 80, 80, 85, 85, 90. Class B scores: 65, 70, 75, 80, 85, 90, 95, 100. Create dot plots for both classes and compare their centers and spreads.
Definition:

Data Comparison: Analyzing and contrasting different data sets using statistical measures.

Center Comparison: Comparing means, medians, and modes.

Spread Comparison: Comparing ranges, IQR, and variability.

Class A
Mean=80, Med=80, Range=20
Class B
Mean=82.5, Med=82.5, Range=35
Comparison
B has higher center, greater spread
Step 1: Create dot plot for Class A

Scores: 70, 75, 75, 80, 80, 80, 85, 85, 90

70: 1 dot, 75: 2 dots, 80: 3 dots, 85: 2 dots, 90: 1 dot

Step 2: Create dot plot for Class B

Scores: 65, 70, 75, 80, 85, 90, 95, 100

Each score appears once, so 1 dot each

Step 3: Calculate Class A statistics

Mean = (70+75+75+80+80+80+85+85+90)÷9 = 720÷9 = 80

Median = 5th value = 80

Range = 90-70 = 20

Step 4: Calculate Class B statistics

Mean = (65+70+75+80+85+90+95+100)÷8 = 660÷8 = 82.5

Median = (80+85)÷2 = 82.5

Range = 100-65 = 35

Step 5: Compare the results

Class B has a higher mean (82.5 vs 80) and median (82.5 vs 80)

Class B has a greater range (35 vs 20), indicating more spread

Class A has more clustering around the center

Class A: Mean=80, Median=80, Range=20
Class B: Mean=82.5, Median=82.5, Range=35
Final Answer:

Class B has a higher center (mean=82.5 vs 80) but greater spread (range=35 vs 20) compared to Class A.

Applied Rules:

• Comparative Analysis: Calculate same measures for both data sets

• Center Measures: Compare means and medians

• Spread Measures: Compare ranges and variability

5 Real-World Application
Exercise 5
A store recorded the number of customers per hour over a 24-hour period: 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 25, 28, 30, 35, 40, 45, 50, 55, 60, 65. Create a dot plot and describe the distribution. What does this tell the store manager?
Definition:

Real-World Application: Using dot plots to analyze practical situations.

Distribution Shape: The pattern of how data values are spread out.

Data Set
24 values from 2 to 65
Mean
~27.3
Median
19
Range
63
Step 1: Organize the data

Sorted: 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 25, 28, 30, 35, 40, 45, 50, 55, 60, 65

Step 2: Create the dot plot

Since each value appears once, place one dot above each value from 2 to 65

Step 3: Calculate statistics

Mean = Sum ÷ Count = 656 ÷ 24 ≈ 27.3

Median = Average of 12th and 13th values = (18+20)÷2 = 19

Range = 65 - 2 = 63

Step 4: Analyze the distribution

Mean (27.3) > Median (19), indicating right skew

Large range suggests high variability in customer traffic

Higher values are more spread out than lower values

Step 5: Interpret for management

Peak hours likely have much higher customer traffic

Staffing should be adjusted based on expected traffic patterns

Consider investigating what causes the high-traffic periods

Distribution is right-skewed with mean > median
Final Answer:

The distribution is right-skewed with mean ≈ 27.3 and median = 19. The store has highly variable customer traffic, with some peak hours having significantly more customers than others.

Applied Rules:

• Distribution Analysis: Compare mean and median to identify skewness

• Real-World Context: Interpret statistical measures in practical terms

• Business Application: Use data to inform operational decisions

Detailed Summary: Dot Plot Fundamentals
\(\text{Dot Plot} = \text{Value} \times \text{Frequency}\)
Dot Plot Concept
Key definitions:

Dot Plot: A simple graphical display of data using dots to represent the frequency of each value

Frequency: The number of times a specific value appears in the data set

Scale: The horizontal axis that shows all possible values in the data set

Stacking: The vertical arrangement of dots above each value

Mode: The value that appears most frequently (tallest stack of dots)

Range: The difference between the maximum and minimum values in the data set

Center: A measure that represents the middle of the data set (mean, median, mode)

Spread: How spread out the values are in the data set (range, IQR)

Complete methodology:
  1. Data Collection: Gather all data points for analysis
  2. Data Organization: Sort values in ascending order
  3. Scale Creation: Draw horizontal axis with all possible values
  4. Dot Placement: Place one dot above each value for each occurrence
  5. Analysis: Calculate measures of center and spread
  6. Interpretation: Draw conclusions from the visual representation
Tip 1: Always ensure dots are aligned vertically above their respective values.
Tip 2: Each dot represents exactly one data point.
Tip 3: Use dot plots for small to medium-sized data sets (typically less than 50 values).
Tip 4: Dot plots are excellent for identifying clusters, gaps, and outliers.
Tip 5: The mode is easily identified as the value with the tallest stack of dots.

Common Errors: Miscounting dots, confusing value with frequency, not maintaining consistent scale, overlapping dots, forgetting to label axes.
Exam Preparation: Practice creating dot plots from raw data, memorize formulas for center and spread measures, understand how to read dot plots efficiently.
Formulas to know by heart:

• Dot Plot: Each value gets one dot per occurrence

• Mean Formula: Mean = (Sum of all values) ÷ (Number of values)

• Median Rule: Middle value when data is sorted (average of two middle values if even count)

• Range Formula: Range = Maximum - Minimum

• Mode: Value with the most dots (highest frequency)

Exercise with Visualization: Distribution Analysis
Exercise 6: Comparing Distributions
Compare the following distributions using dot plots:
Distribution A: 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9
Distribution B: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Analysis: The visualization shows how different distributions have different characteristics.

  • Distribution A: Symmetric with mode at 6 and 8
  • Distribution B: Uniform distribution (all values appear once)
  • Both have same range but different shapes

Questions & Answers

Question: How do I know if my dot plot is correct? What should I check?

Answer: Here are verification checks for your dot plot:

  1. Count Check: Total number of dots should equal total number of data points
  2. Value Alignment: Each dot should be directly above its corresponding value
  3. Frequency Accuracy: Number of dots above each value should match the frequency
  4. Scale Consistency: Values should be evenly spaced on the horizontal axis
  5. Completeness: Include all values in the range, even those with zero frequency

Example: If your data is [2, 3, 3, 4], you should have:

  • 1 dot above 2
  • 2 dots above 3
  • 1 dot above 4
  • Total of 4 dots

Always double-check that your visual representation matches the original data!

Question: What's the difference between a dot plot and a histogram?

Answer: Key differences:

  • Dot Plot: Individual dots represent each data point
  • Histogram: Bars represent ranges of values
  • Dot Plot: Best for small to medium data sets with discrete values
  • Histogram: Better for large data sets or continuous data
  • Dot Plot: Shows exact values and frequency clearly
  • Histogram: Shows overall distribution shape

Example: For data [2, 2, 3, 3, 3, 4], a dot plot shows 2 dots above 2, 3 dots above 3, 1 dot above 4.

A histogram might group values into ranges like 2-2.99, 3-3.99, 4-4.99.

Choose dot plots for smaller, discrete data sets; histograms for larger or continuous data!

Question: How can I use a dot plot to identify outliers in my data?

Answer: Dot plots are excellent for identifying outliers:

  • Visual Gaps: Look for large spaces between groups of dots
  • Extreme Values: Dots that are far from the main cluster
  • Isolated Dots: Single dots separated from the rest
  • End Behavior: Values at the far ends that seem distant

Example: If most dots are clustered between 10-20 but there's a dot at 50, that's likely an outlier.

Outliers can significantly affect mean calculations, so it's important to identify them when analyzing data.

Always consider whether outliers are valid data points or errors in collection!

Question: Can I use dot plots for categorical data like favorite colors or types of pets?

Answer: Dot plots are primarily designed for numerical data, but they can work for categorical data with modifications:

For categorical data, you would:

  • Place category names on the horizontal axis instead of numbers
  • Stack dots above each category to show frequency
  • Count occurrences of each category

However, bar graphs are generally better for categorical data as they're more conventional and easier to read.

Dot plots work best with numerical data where you can see patterns and distributions more clearly.

For example, dot plots work great for test scores (numerical) but bar graphs are better for favorite colors (categorical).

Question: How do I know when to use mean, median, or mode with dot plots?

Answer: Choose based on your data characteristics:

  • Mean: Use for symmetric distributions without outliers
  • Median: Use when data has outliers or is skewed
  • Mode: Use for categorical data or when looking for most common value

Look at your dot plot:

  • Is it symmetric? → Mean is appropriate
  • Is it skewed? → Median is better
  • Are there extreme values? → Use median
  • Do you see repeated values? → Mode is visible as tallest stack

Dot plots make it easy to see the distribution shape and identify the mode visually!