Solved Exercises on Analyzing Data Sets in Grade 8

Master data analysis: mean, median, mode, range, and data interpretation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Mean
Exercise 1
Find the mean of the following data set:
12, 15, 18, 20, 22, 25, 28
Definition:

Mean: The average value calculated by adding all values and dividing by the number of values

Mean calculation method:
  1. Add all the values in the data set
  2. Count the total number of values
  3. Divide the sum by the count
Data Set
12, 15, 18, 20, 22, 25, 28
Sum
130
Count
7
Mean
18.57
Step 1: Add all values

12 + 15 + 18 + 20 + 22 + 25 + 28 = 130

Step 2: Count the number of values

There are 7 values in the data set

Step 3: Divide sum by count

Mean = 130 ÷ 7 = 18.57 (rounded to 2 decimal places)

Mean = 18.57
Final answer:

The mean of the data set is 18.57

Applied rules:

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

• Precision: Round to appropriate decimal places

• Verification: Check addition accuracy

2 Finding Median
Exercise 2
Find the median of the following data set:
5, 12, 8, 15, 20, 3, 18, 10
Definition:

Median: The middle value when data is arranged in order from least to greatest

Original Data
5, 12, 8, 15, 20, 3, 18, 10
Ordered Data
3, 5, 8, 10, 12, 15, 18, 20
Median
11
Step 1: Arrange data in ascending order

3, 5, 8, 10, 12, 15, 18, 20

Step 2: Determine if count is odd or even

There are 8 values (even number), so median is average of 4th and 5th values

Step 3: Calculate median

Median = (10 + 12) ÷ 2 = 11

Median = 11
Final answer:

The median of the data set is 11

Applied rules:

• Ordering: Always arrange data from least to greatest

• Odd count: Median is the middle value

• Even count: Median is average of two middle values

3 Finding Mode
Exercise 3
Find the mode of the following data set:
4, 7, 3, 4, 9, 7, 4, 6, 8, 7
Definition:

Mode: The value that appears most frequently in a data set

Data Set
4, 7, 3, 4, 9, 7, 4, 6, 8, 7
Frequency Table
4:3, 7:3, others:1
Mode
4 and 7
Step 1: Count frequency of each value

3:1, 4:3, 6:1, 7:3, 8:1, 9:1

Step 2: Identify most frequent value(s)

Both 4 and 7 appear 3 times (most frequent)

Step 3: Determine the mode

This is a bimodal distribution (two modes: 4 and 7)

Modes = 4 and 7
Final answer:

The modes of the data set are 4 and 7

Applied rules:

• Frequency: Count occurrences of each value

• Unimodal: One mode

• Bimodal: Two modes

• No mode: All values appear equally

Rules and methods, laws,...
Mean = \(\frac{Sum\ of\ all\ values}{Number\ of\ values}\)
Mean Formula
Median Rule
Odd n: Middle value
Even n: Average of 2 middle values
Mode Rule
Most frequent value
Can have multiple modes
Range Rule
Max - Min
Measures spread of data
Key definitions:

Mean: Average value of a data set

Median: Middle value when data is ordered

Mode: Most frequently occurring value

Range: Difference between maximum and minimum values

Data Analysis Methodology:
  1. Organize data: Order from least to greatest
  2. Calculate measures: Mean, median, mode, range
  3. Interpret results: Understand what measures tell us
  4. Compare findings: Look for patterns and outliers
Tip 1: Always order data before finding median.
Tip 2: A data set can have one, multiple, or no mode.
Tip 3: Mean is affected by outliers, median is not.
Tip 4: Range shows the spread of the data.
Common errors: Forgetting to order data, miscalculating sums, confusing mean/median/mode.
Exam preparation: Practice calculations, understand when to use each measure, interpret results.
Solution: Exercises 4 to 5
4 Finding Range
Exercise 4
Find the range of the following data set:
15, 22, 8, 30, 18, 25, 12, 28, 5, 35
Definition:

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

Data Set
15, 22, 8, 30, 18, 25, 12, 28, 5, 35
Ordered
5, 8, 12, 15, 18, 22, 25, 28, 30, 35
Range
30
Step 1: Order the data from least to greatest

5, 8, 12, 15, 18, 22, 25, 28, 30, 35

Step 2: Identify the minimum and maximum values

Minimum = 5, Maximum = 35

Step 3: Calculate the range

Range = Maximum - Minimum = 35 - 5 = 30

Range = 30
Final answer:

The range of the data set is 30

Applied rules:

• Formula: Range = Maximum - Minimum

• Ordering: Organize data to identify extremes

• Interpretation: Range measures data spread

5 Comprehensive Analysis
Exercise 5
Analyze the following data set completely:
14, 16, 18, 15, 17, 14, 19, 15, 16, 14, 18
Definition:

Comprehensive analysis: Finding all measures of central tendency and spread

Data Set
14, 16, 18, 15, 17, 14, 19, 15, 16, 14, 18
Ordered
14, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19
Mean/Median/Mode/Range
16.1/16/14/5
Step 1: Order the data

14, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19

Step 2: Calculate the mean

Sum = 177, Count = 11, Mean = 177 ÷ 11 = 16.1

Step 3: Find the median

With 11 values, median is the 6th value = 16

Step 4: Find the mode

14 appears 3 times (most frequent) = Mode

Step 5: Calculate the range

Maximum = 19, Minimum = 14, Range = 19 - 14 = 5

Mean = 16.1, Median = 16, Mode = 14, Range = 5
Final answer:

Mean = 16.1, Median = 16, Mode = 14, Range = 5

Applied rules:

• Systematic approach: Calculate each measure in order

• Verification: Check that results make sense

• Interpretation: Mean > Median suggests slight right skew

Key Concepts for Data Analysis
Mean = \(\frac{\sum x_i}{n}\)
Mean Formula
Key definitions:

Mean: The arithmetic average of all data values

Median: The middle value when data is arranged in order

Mode: The value that occurs most frequently

Range: The difference between highest and lowest values

Data Set: A collection of numerical observations or measurements

Complete Analysis Methodology:
  1. Organize the data: Sort values from least to greatest
  2. Calculate central tendency: Mean, median, mode
  3. Measure variability: Calculate range
  4. Identify patterns: Look for clusters, gaps, outliers
  5. Draw conclusions: Interpret what the data tells us
Tip 1: Mean is sensitive to outliers, while median is robust.
Tip 2: In a symmetric distribution, mean ≈ median ≈ mode.
Tip 3: Range only considers extreme values, not the whole distribution.
Tip 4: Always check your work by verifying calculations.
When to use mean: When data is normally distributed and no significant outliers exist.
When to use median: When data has outliers or is skewed.
Essential Formulas:

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

• Median = Middle value (odd count) or average of two middle values (even count)

• Mode = Most frequently occurring value

• Range = Maximum value - Minimum value

Questions & Answers

Question: I don't understand when to use mean versus median. Can you explain the difference?

Answer: Great question! The choice depends on your data characteristics:

  • Mean: Use when data is symmetric and doesn't have extreme outliers. It uses all values in the calculation.
  • Median: Use when data has outliers or is skewed. It represents the middle value regardless of extreme scores.

Example: In the data set [10, 12, 13, 14, 100], the mean is 29.8, but the median is 13. The outlier (100) dramatically affects the mean but not the median.

Use mean when you want the average performance, median when you want the typical value unaffected by extremes.

Question: What happens when a data set has no mode? And can it have more than one?

Answer: Yes, both scenarios are possible:

  • No mode: When all values appear with equal frequency. Example: [2, 4, 6, 8] - each appears once, so no mode exists.
  • One mode (unimodal): When one value appears most frequently. Example: [3, 3, 5, 7, 7, 3] - mode is 3.
  • Two modes (bimodal): When two values tie for most frequent. Example: [4, 4, 6, 6, 8] - modes are 4 and 6.
  • Multiple modes: When more than two values appear with the same highest frequency.

The mode is useful for categorical data and shows the most common occurrence, unlike mean and median which work only with numerical data.

Question: Why do we need to calculate the range? Isn't mean enough to describe a data set?

Answer: No, mean alone is insufficient because it only tells us about the center of the data, not its spread!

Consider two data sets with the same mean of 10:
Set A: [9, 10, 10, 10, 11] - Range = 2
Set B: [1, 5, 10, 15, 19] - Range = 18

Both have the same mean (10), but Set A values are clustered closely together while Set B values are widely spread out. The range tells us about this variability.

Range is our simplest measure of dispersion - it shows how spread out the data is from the lowest to highest values. Together, central tendency (mean/median/mode) and spread (range) give us a complete picture.

Question: How can I quickly find the median without writing out all the numbers in order?

Answer: While ordering is essential for finding the median, here are some strategies to organize efficiently:

  • Scan method: Quickly identify the smallest value, then the next smallest, etc.
  • Grouping: Group numbers by tens or fives to see patterns
  • Count positions: For large data sets, knowing you need the middle value helps target where to focus

For example, with [23, 15, 30, 12, 25, 18]: Start with 12, then 15, then 18, then 23, then 25, then 30. Since there are 6 values (even), average the 3rd and 4th values: (18+23)/2 = 20.5.

The median is always the middle position: (n+1)/2 for odd counts, or average of n/2 and (n/2)+1 positions for even counts.

Question: What is the relationship between mean, median, and mode in different types of distributions?

Answer: The relationship between mean, median, and mode reveals the shape of the distribution:

  1. Symmetric distribution: Mean = Median = Mode (bell-shaped curve)
  2. Right-skewed (positive skew): Mode < Median < Mean (tail extends right)
  3. Left-skewed (negative skew): Mean < Median < Mode (tail extends left)

In Exercise 5, we found Mean (16.1) > Median (16) > Mode (14), indicating a slight right skew - there are higher values pulling the mean up while the mode remains at the lower end where values cluster most.

This relationship helps identify outliers and understand data distribution patterns!