Solved Exercises on Mean in Grade 7

Master mean calculations: finding averages, weighted means, and interpreting results through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Mean Calculation
Exercise 1
Find the mean of the following test scores: 85, 92, 78, 96, 88, 91, 83.
Definition:

Mean (Average): The sum of all values divided by the number of values.

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

Mean Calculation Method:
  1. Add all the values together
  2. Count the number of values
  3. Divide the sum by the number of values
  4. Round to appropriate decimal places if necessary
Sum
613
Count
7
Mean
87.6
Step 1: Add all values

85 + 92 + 78 + 96 + 88 + 91 + 83 = 613

Step 2: Count number of values

There are 7 test scores

Step 3: Divide sum by count

Mean = 613 ÷ 7 = 87.571428...

Step 4: Round appropriately

Mean ≈ 87.6 (rounded to one decimal place)

Mean = 87.6
Final Answer:

The mean of the test scores is 87.6.

Applied Rules:

• Mean Formula: Sum of values divided by count

• Arithmetic: Perform addition before division

• Rounding: Round to one more decimal than original data

2 Weighted Mean
Exercise 2
A student's final grade is calculated with the following weights: Homework (30%), Quizzes (20%), Midterm Exam (25%), Final Exam (25%). If the student scored 85 on homework, 90 on quizzes, 78 on the midterm, and 88 on the final, what is their weighted average?
Definition:

Weighted Mean: A mean where each value contributes differently to the final average based on its assigned weight.

Formula: Weighted Mean = Σ(value × weight) ÷ Σ(weights)

Homework
85 × 0.30 = 25.5
Quizzes
90 × 0.20 = 18.0
Midterm
78 × 0.25 = 19.5
Final
88 × 0.25 = 22.0
Step 1: Convert percentages to decimals

Homework: 30% = 0.30

Quizzes: 20% = 0.20

Midterm: 25% = 0.25

Final: 25% = 0.25

Step 2: Multiply each score by its weight

Homework: 85 × 0.30 = 25.5

Quizzes: 90 × 0.20 = 18.0

Midterm: 78 × 0.25 = 19.5

Final: 88 × 0.25 = 22.0

Step 3: Add all weighted values

25.5 + 18.0 + 19.5 + 22.0 = 85.0

Step 4: Verify weight total

0.30 + 0.20 + 0.25 + 0.25 = 1.00 (100%)

Weighted Mean = 85.0
Final Answer:

The weighted average grade is 85.0.

Applied Rules:

• Weighted Mean Formula: Σ(value × weight) ÷ Σ(weights)

• Percentage Conversion: Divide by 100 to convert percentages to decimals

• Verification: Sum of weights should equal 1.00 (100%)

3 Effect of Outliers
Exercise 3
Calculate the mean with and without the outlier for the following data set: 15, 18, 20, 17, 16, 19, 18, 100. Explain the effect of the outlier.
Definition:

Outlier: A value that is significantly different from other values in the data set.

Effect on Mean: Outliers strongly affect the mean because it uses all data values.

With Outlier
Mean = 27.9
Without Outlier
Mean = 17.6
Effect
Increase = 10.3
Step 1: Identify the outlier

Value 100 is significantly larger than other values (15-20 range)

This is the outlier.

Step 2: Calculate mean with outlier

Sum with outlier: 15 + 18 + 20 + 17 + 16 + 19 + 18 + 100 = 213

Count: 8

Mean = 213 ÷ 8 = 26.625 ≈ 26.6

Step 3: Calculate mean without outlier

Sum without outlier: 15 + 18 + 20 + 17 + 16 + 19 + 18 = 123

Count: 7

Mean = 123 ÷ 7 = 17.571... ≈ 17.6

Step 4: Compare the results

Mean with outlier: 26.6

Mean without outlier: 17.6

Difference: 26.6 - 17.6 = 9.0

Step 5: Explain the effect

The outlier (100) significantly increased the mean from 17.6 to 26.6.

This shows that the mean is sensitive to extreme values.

With outlier: Mean = 26.6
Without outlier: Mean = 17.6
Final Answer:

With outlier: Mean = 26.6. Without outlier: Mean = 17.6. The outlier increased the mean by 9.0 points.

Applied Rules:

• Outlier Sensitivity: Mean is sensitive to extreme values

• Data Analysis: Always consider outliers when interpreting means

• Robust Measures: Median is more robust than mean for skewed data

Rules and methods, laws,...
\(\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}\)
Basic Mean Formula
Weighted Mean
\(\text{WM} = \frac{\sum (x_i \times w_i)}{\sum w_i}\)
Each value has different weight
Outlier Effect
Mean sensitive to extremes
Median more robust
Properties
Uses all data points
Sum of deviations = 0
Key Definitions:

Mean: The arithmetic average of all values in a data set, calculated by adding all values and dividing by the count

Outlier: A value that is significantly different from other values in the data set

Weighted Mean: A mean where each value contributes differently based on its assigned importance

Deviation: The difference between each value and the mean

Central Tendency: A measure that represents the center of a data set

Complete Methodology:
  1. Basic Mean: Add all values, divide by count
  2. Weighted Mean: Multiply each value by its weight, sum results, divide by sum of weights
  3. Outlier Analysis: Identify extreme values and assess their impact
  4. Verification: Check that sum of deviations from mean equals zero
  5. Interpretation: Understand what the mean represents in context
Tip 1: Always verify that the sum of all deviations from the mean equals zero.
Tip 2: Mean is affected by every value in the data set, including outliers.
Tip 3: When weights sum to 1, the weighted mean formula simplifies.
Tip 4: The mean represents the "balance point" of the data set.
Tip 5: Use mean for symmetric distributions without significant outliers.
Common Errors: Forgetting to divide by count, miscounting values, not converting percentages to decimals, ignoring outliers.
Exam Preparation: Practice with various data sets, memorize formulas, understand when mean is most appropriate, know how to handle weighted data.
Solution: Exercises 4 to 5
4 Missing Value Problem
Exercise 4
The mean of five numbers is 24. Four of the numbers are 20, 25, 22, and 28. What is the fifth number?
Definition:

Missing Value: A problem where the mean and some values are known, and an unknown value must be calculated.

Algebraic Approach: Use the mean formula to solve for the unknown value.

Known Sum
95
Total Sum
120
Missing Value
25
Step 1: Use the mean formula to find total sum

Mean = Sum ÷ Count

24 = Total Sum ÷ 5

Total Sum = 24 × 5 = 120

Step 2: Calculate sum of known values

Known values: 20 + 25 + 22 + 28 = 95

Step 3: Find the missing value

Missing Value = Total Sum - Known Sum

Missing Value = 120 - 95 = 25

Step 4: Verify the answer

New data set: 20, 25, 22, 28, 25

Sum: 20 + 25 + 22 + 28 + 25 = 120

Mean: 120 ÷ 5 = 24 ✓

Fifth number = 25
Final Answer:

The fifth number is 25.

Applied Rules:

• Rearrangement: Mean formula can be rearranged to find missing values

• Algebraic Thinking: Use inverse operations to solve for unknowns

• Verification: Always check your answer by recalculating

5 Real-World Application
Exercise 5
A store recorded daily sales for a week: Monday $240, Tuesday $180, Wednesday $320, Thursday $260, Friday $200, Saturday $450, Sunday $380. What is the mean daily sales? If the store wants to increase average daily sales by 15%, what should the new weekly total be?
Definition:

Real-World Application: Applying mathematical concepts to practical situations.

Percent Increase: A change expressed as a percentage of the original value.

Current Weekly Total
$2,030
Current Daily Mean
$290
New Weekly Total
$2,334.50
Step 1: Calculate current weekly total

Monday: $240, Tuesday: $180, Wednesday: $320, Thursday: $260, Friday: $200, Saturday: $450, Sunday: $380

Weekly Total = $240 + $180 + $320 + $260 + $200 + $450 + $380 = $2,030

Step 2: Calculate current mean daily sales

Mean = Weekly Total ÷ Number of Days

Mean = $2,030 ÷ 7 = $290

Step 3: Calculate desired mean with 15% increase

New Mean = Current Mean × (1 + 0.15)

New Mean = $290 × 1.15 = $333.50

Step 4: Calculate new weekly total needed

New Weekly Total = New Mean × Number of Days

New Weekly Total = $333.50 × 7 = $2,334.50

Step 5: Verify the result

New Mean = $2,334.50 ÷ 7 = $333.50

Percent Increase = ($333.50 - $290) ÷ $290 × 100% = 15% ✓

Current Mean = $290
New Weekly Total = $2,334.50
Final Answer:

The current mean daily sales is $290. To increase average daily sales by 15%, the new weekly total should be $2,334.50.

Applied Rules:

• Real-World Context: Apply mathematical concepts to practical situations

• Percent Increase: New value = Original × (1 + percent increase)

• Verification: Always check that your answer makes sense in context

Detailed Summary: Mean Fundamentals
\(\text{Mean} = \frac{\sum x}{n}\)
Mean Formula
Key definitions:

Mean: The arithmetic average of all values in a data set, calculated by adding all values and dividing by the count

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

Deviation: The difference between each value and the mean (xi - mean)

Weighted Mean: A mean where each value contributes differently based on its assigned importance

Outlier: A value that is significantly different from other values in the data set

Central Tendency: A measure that represents the center of a data set

Complete methodology:
  1. Basic Mean Calculation: Add all values and divide by the number of values
  2. Weighted Mean Calculation: Multiply each value by its weight, sum results, divide by sum of weights
  3. Missing Value Problems: Use the mean formula to solve for unknown values
  4. Outlier Analysis: Identify extreme values and assess their impact on the mean
  5. Verification: Check that sum of deviations from mean equals zero
  6. Real-World Applications: Apply mean concepts to practical situations
Tip 1: Always verify that the sum of all deviations from the mean equals zero.
Tip 2: The mean represents the "balance point" of the data set.
Tip 3: Mean is sensitive to outliers, while median is more robust.
Tip 4: When solving for missing values, use the mean formula as an equation.
Tip 5: In weighted means, ensure weights sum to 1 (or 100%) when appropriate.
Common errors: Forgetting to divide by the count, miscounting values, not converting percentages to decimals for weighted means, ignoring the effect of outliers on interpretation.
Exam preparation: Practice with various data sets, memorize formulas, understand when mean is most appropriate, know how to handle weighted data and missing value problems.
Formulas to know by heart:

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

• Weighted Mean: WM = Σ(xi × wi) ÷ Σ(wi)

• Missing Value: Missing = (Mean × Total Count) - Known Sum

• Deviation Property: Σ(xi - Mean) = 0

• Percent Change: New Value = Original × (1 + Percent Change)

Exercise with Visualization: Mean Comparison
Exercise 6: Mean vs Other Measures
Compare the mean with other measures of center for the following data sets:
Set A: 10, 12, 14, 15, 16, 18, 20 (symmetric)
Set B: 5, 7, 9, 11, 13, 15, 50 (with outlier)
Set C: 2, 2, 3, 4, 5, 5, 5 (multimodal)

Analysis: The visualization shows how outliers affect the mean compared to other measures.

  • Set A: Mean ≈ Median (symmetric distribution)
  • Set B: Mean > Median (outlier pulls mean upward)
  • Set C: Mode differs from mean and median

Questions & Answers

Question: How do I know when to use the regular mean versus the weighted mean? They seem similar.

Answer: The key difference is in the importance of each value:

  • Regular Mean: Used when all values have equal importance
  • Weighted Mean: Used when values have different levels of importance

Examples:

  • Regular Mean: Average temperature over 7 days (each day equally important)
  • Weighted Mean: Course grade where exams count more than homework

Look for phrases like "weighted," "importance," "significance," or different percentages when deciding which to use. If values contribute differently to the final result, use weighted mean.

In the example with grades, homework (30%) is less important than the final exam (25%), so each contributes differently to the final grade.

Question: Why does the mean get affected by outliers? What makes it so sensitive?

Answer: The mean is sensitive to outliers because of how it's calculated:

  • Formula Includes All Values: Mean = (sum of all values) ÷ (count)
  • Every Value Counts: Each value contributes equally to the numerator
  • Extreme Values Have Large Impact: Very high or low values significantly change the sum

Example: For [10, 12, 14, 16, 18], mean = 14

Adding outlier 100: [10, 12, 14, 16, 18, 100], mean = 28.3

The outlier (100) increased the mean by 14.3 points! This happens because 100 is added directly to the sum.

In contrast, the median (which depends on position, not value) would only change slightly. This is why median is considered more "robust" than mean.

Question: In the missing value problem, how do I know whether to add or subtract when solving for the unknown?

Answer: Use algebraic thinking to determine the operation:

Starting with the mean formula: Mean = Sum ÷ Count

If you know Mean and Count, multiply both sides by Count:

Mean × Count = Sum (this gives you the total sum needed)

If you know the total sum and some values, subtract the known values from the total:

Missing Value = Total Sum - Known Sum

Think of it this way:

  • To find total sum: MULTIPLY mean by count
  • To find missing value: SUBTRACT known values from total

Always check your answer: Calculate the mean of the complete data set to ensure it matches the given mean!

This algebraic approach works for any missing value problem involving the mean.

Question: How can I check if my mean calculation is correct? Are there any quick verification methods?

Answer: Yes, there are several verification methods:

  1. Multiplication Check: Multiply your calculated mean by the count; it should equal the sum of original values
  2. Deviation Check: Calculate the deviation of each value from the mean; the sum of all deviations should equal zero
  3. Reasonableness Check: Does your mean fall within the range of your data?
  4. Estimation Check: Round values and estimate to see if your answer is reasonable

Example for [10, 12, 14]: Mean = (10+12+14)÷3 = 12

  • Multiplication: 12 × 3 = 36, and 10+12+14 = 36 ✓
  • Deviations: (10-12) + (12-12) + (14-12) = -2 + 0 + 2 = 0 ✓

These checks help catch arithmetic errors and ensure your calculation is correct!

Question: When would I encounter mean calculations in real life? How is this useful?

Answer: Mean calculations are incredibly common in everyday life:

  • School: GPA calculation, test averages, course grades
  • Business: Average sales, profit margins, customer ratings
  • Sports: Batting averages, scoring averages, performance metrics
  • Finance: Average returns, expense tracking, budget planning
  • Science: Experimental results, temperature averages, measurement data
  • Weather: Average temperatures, rainfall, humidity levels

The mean helps summarize data and provides a single representative value. For example, knowing the average daily temperature helps you plan what clothes to wear.

Understanding how to calculate and interpret means is crucial for making informed decisions based on data in both academic and real-world contexts!