Solved Exercises on Median in Grade 7

Master median calculations: finding middle values, comparing with mean, and interpreting results through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Median with Odd Count
Exercise 1
Find the median of the following data set: 15, 22, 18, 25, 12, 20, 17.
Definition:

Median: The middle value when data is arranged in ascending order.

For odd number of values: The median is the middle value.

Formula: Position = (n + 1) ÷ 2, where n is the number of values.

Median Calculation Method:
  1. Arrange all values in ascending order (smallest to largest)
  2. Count the number of values
  3. If count is odd: median is the value at position (n+1)÷2
  4. If count is even: median is average of two middle values
Sorted Data
12, 15, 17, 18, 20, 22, 25
Count
7 values
Median
18
Step 1: Arrange data in ascending order

Original: 15, 22, 18, 25, 12, 20, 17

Sorted: 12, 15, 17, 18, 20, 22, 25

Step 2: Count the number of values

There are 7 values (odd number)

Step 3: Find the middle position

Position = (n + 1) ÷ 2 = (7 + 1) ÷ 2 = 8 ÷ 2 = 4

Step 4: Identify the median

The 4th value in the sorted list is 18

Therefore, the median is 18

Median = 18
Final Answer:

The median of the data set is 18.

Applied Rules:

• Sorting Required: Always arrange data in order first

• Odd Count: Median is the middle value at position (n+1)÷2

• Position Formula: Use (n+1)÷2 to find middle position

2 Basic Median with Even Count
Exercise 2
Find the median of the following data set: 12, 18, 15, 21, 10, 16, 19, 14.
Definition:

Median: The middle value when data is arranged in ascending order.

For even number of values: The median is the average of the two middle values.

Formula: Median = (value at n÷2 + value at (n÷2)+1) ÷ 2

Sorted Data
10, 12, 14, 15, 16, 18, 19, 21
Middle Values
15, 16
Median
15.5
Step 1: Arrange data in ascending order

Original: 12, 18, 15, 21, 10, 16, 19, 14

Sorted: 10, 12, 14, 15, 16, 18, 19, 21

Step 2: Count the number of values

There are 8 values (even number)

Step 3: Find the two middle positions

Position 1 = n ÷ 2 = 8 ÷ 2 = 4

Position 2 = (n ÷ 2) + 1 = (8 ÷ 2) + 1 = 5

Step 4: Identify the two middle values

4th value: 15

5th value: 16

Step 5: Calculate the median

Median = (15 + 16) ÷ 2 = 31 ÷ 2 = 15.5

Median = 15.5
Final Answer:

The median of the data set is 15.5.

Applied Rules:

• Sorting Required: Always arrange data in order first

• Even Count: Median is average of two middle values

• Average Formula: Add the two middle values and divide by 2

3 Comparing Median and Mean
Exercise 3
Calculate both the median and mean of the following data set: 8, 12, 10, 15, 7, 11, 100. Explain the difference and which measure is more representative.
Definition:

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

Resistance: A measure that is not significantly affected by outliers.

Mean Sensitivity: Mean is affected by every value in the data set.

Sorted Data
7, 8, 10, 11, 12, 15, 100
Median
11
Mean
23.3
Step 1: Calculate the median

Sorted data: 7, 8, 10, 11, 12, 15, 100

Count: 7 (odd)

Median position: (7+1)÷2 = 4

Median: 11

Step 2: Calculate the mean

Sum: 7 + 8 + 10 + 11 + 12 + 15 + 100 = 163

Count: 7

Mean: 163 ÷ 7 = 23.285... ≈ 23.3

Step 3: Identify the outlier

Value 100 is significantly larger than other values (7-15 range)

This is the outlier.

Step 4: Compare the measures

Mean (23.3) is much higher than median (11)

The outlier pulled the mean upward significantly.

Step 5: Determine which is more representative

The median (11) better represents the typical value

Most values are between 7 and 15

Median = 11, Mean = 23.3
Median is more representative
Final Answer:

Median = 11, Mean = 23.3. The median is more representative because the outlier (100) significantly increased the mean.

Applied Rules:

• Outlier Sensitivity: Mean is sensitive to outliers, median is resistant

• Data Representation: Choose the measure that best represents the data

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

Rules and methods, laws,...
\(\text{Median Position (odd n)} = \frac{n + 1}{2}\)
Odd Count Formula
Median Position (odd)
\(\frac{n + 1}{2}\)
Find middle position
Median (even)
\(\frac{x_{n/2} + x_{(n/2)+1}}{2}\)
Average of middle values
Properties
Resistant to outliers
Based on position
Key Definitions:

Median: The middle value when data is arranged in ascending order

Ascending Order: Arranging values from smallest to largest

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

Resistance: A measure that is not significantly affected by outliers

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

Complete Methodology:
  1. Data Sorting: Arrange all values in ascending order
  2. Count Values: Determine the number of values (n)
  3. Odd Count: If n is odd, median is at position (n+1)÷2
  4. Even Count: If n is even, median is average of values at positions n÷2 and (n÷2)+1
  5. Verification: Ensure data is properly sorted before calculation
  6. Interpretation: Understand what the median represents in context
Tip 1: Always sort the data first before finding the median.
Tip 2: The median is resistant to outliers, unlike the mean.
Tip 3: For odd count: median is one specific value; for even count: median is average of two values.
Tip 4: The median represents the 50th percentile of the data.
Tip 5: Use median for skewed data or when outliers are present.
Common Errors: Forgetting to sort data, miscounting values, applying wrong formula for odd/even counts, not considering outliers.
Exam Preparation: Practice with various data sets, memorize formulas, understand when median is most appropriate, know how to handle different data arrangements.
Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A company has 11 employees with the following salaries (in thousands): 35, 40, 38, 42, 37, 45, 41, 39, 43, 44, 120. Find the median salary and explain why it might be a better measure than the mean for representing typical employee compensation.
Definition:

Salary Distribution: The arrangement of employee compensation values.

Executive Compensation: Typically much higher than regular employee salaries.

Sorted Data
35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 120
Median
$41,000
Mean
$49,100
Step 1: Arrange salaries in ascending order

Sorted: 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 120

Step 2: Count the number of values

There are 11 employees (odd number)

Step 3: Find the median position

Position = (n + 1) ÷ 2 = (11 + 1) ÷ 2 = 6

Step 4: Identify the median salary

The 6th value in the sorted list is 41

Median salary = $41,000

Step 5: Calculate the mean for comparison

Sum = 35 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 120 = 544

Mean = 544 ÷ 11 = 49.45... ≈ $49,455

Step 6: Explain why median is better

The outlier ($120,000) significantly increases the mean

Most employees earn between $35,000-$45,000

Median ($41,000) better represents typical compensation

Median = $41,000
Mean = $49,455
Final Answer:

The median salary is $41,000. The median is a better measure than the mean because the executive salary ($120,000) significantly increases the mean, making it less representative of typical employee compensation.

Applied Rules:

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

• Outlier Impact: Understand how extreme values affect different measures

• Measure Selection: Choose the most appropriate measure for the context

5 Missing Value Problem
Exercise 5
The median of the following data set is 15: 10, 12, 14, ?, 16, 18, 20. Find the missing value and explain your reasoning.
Definition:

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

Position Logic: Using the median position to determine where the missing value fits.

Known Values
10, 12, 14, 16, 18, 20
Count
7 total values
Missing Value
15
Step 1: Count the total number of values

Given: 6 known values + 1 missing = 7 total values

Step 2: Determine the median position

Since n = 7 (odd), median position = (7+1)÷2 = 4

Step 3: Understand the requirement

The median is 15, so the 4th value in the sorted list must be 15

Step 4: Determine where the missing value fits

Known values: 10, 12, 14, 16, 18, 20

For the 4th position to be 15, the missing value must be 15

Sorted with missing value: 10, 12, 14, 15, 16, 18, 20

Step 5: Verify the answer

Complete data set: 10, 12, 14, 15, 16, 18, 20

Sorted: 10, 12, 14, 15, 16, 18, 20

Median (4th value): 15 ✓

Missing value = 15
Final Answer:

The missing value is 15. When arranged in order (10, 12, 14, 15, 16, 18, 20), the median (4th value) is 15.

Applied Rules:

• Position Logic: Use median position to determine value placement

• Logical Reasoning: Work backwards from the known median

• Verification: Always check that your answer produces the correct median

Detailed Summary: Median Fundamentals
\(\text{Median} = \begin{cases} x_{(n+1)/2} & \text{if n is odd} \\ \frac{x_{n/2} + x_{(n/2)+1}}{2} & \text{if n is even} \end{cases}\)
Median Formula
Key definitions:

Median: The middle value when data is arranged in ascending order, calculated differently for odd and even counts

Ascending Order: Arranging values from smallest to largest (e.g., 2, 5, 8, 11)

Odd Count: When the number of values is odd, median is the single middle value

Even Count: When the number of values is even, median is the average of the two middle values

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

Resistance: The property of a measure to not be significantly affected by outliers

Complete methodology:
  1. Data Sorting: Arrange all values in ascending order (smallest to largest)
  2. Count Values: Determine the total number of values (n)
  3. Odd Count Calculation: If n is odd, median is the value at position (n+1)÷2
  4. Even Count Calculation: If n is even, median is average of values at positions n÷2 and (n÷2)+1
  5. Verification: Ensure data is properly sorted and median is correctly identified
  6. Interpretation: Understand what the median represents in the context
Tip 1: Always sort the data first before attempting to find the median.
Tip 2: The median is resistant to outliers, making it useful for skewed data.
Tip 3: For odd counts, median is a single value; for even counts, it's an average of two values.
Tip 4: The median represents the 50th percentile - half the data is above, half below.
Tip 5: Use median when data has outliers or is skewed; use mean for symmetric data.
Common errors: Forgetting to sort data, miscounting values, applying wrong formula for odd/even counts, not verifying the result, confusing median with mean or mode.
Exam preparation: Practice with various data sets of different sizes, memorize formulas for odd and even counts, understand when median is most appropriate, know how to handle missing value problems.
Formulas to know by heart:

• Odd Count: Median position = (n + 1) ÷ 2

• Even Count: Median = (value at n÷2 + value at (n÷2)+1) ÷ 2

• Sorting Requirement: Always arrange data in ascending order first

• Resistance Property: Median is not affected by extreme values

• Percentile: Median represents the 50th percentile of the data

Exercise with Visualization: Median vs Mean
Exercise 6: Comparing Central Tendency Measures
Compare median and mean 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, 6, 7 (skewed left)

Analysis: The visualization shows how outliers affect mean more than median.

  • Set A: Mean ≈ Median (symmetric distribution)
  • Set B: Mean > Median (outlier pulls mean upward)
  • Set C: Mean < Median (skewness affects mean more)

Questions & Answers

Question: Why do I need to sort the data before finding the median? Can't I just pick the middle number?

Answer: Sorting is absolutely essential for finding the median because:

  • Definition Requirement: The median is specifically the middle value when data is arranged in order
  • Position Matters: We need to know the exact position of the middle value based on magnitude
  • Accuracy: Without sorting, you might pick a value that isn't actually the middle in terms of value

Example: In the set [15, 12, 18, 10, 16], if you don't sort, you might think 18 is the middle (position 3), but actually it's 15. When sorted [10, 12, 15, 16, 18], the middle value is clearly 15.

Sorting ensures you're finding the true middle value based on the data's magnitude, not its original position in the list.

Always sort first, then identify the middle position!

Question: How do I remember whether to add 1 or not when finding the median position?

Answer: The formula is always (n + 1) ÷ 2 for the median position, where n is the number of values:

  • For odd counts: (n + 1) ÷ 2 gives you the exact position of the median
  • For even counts: (n + 1) ÷ 2 gives you a decimal that tells you which values to average

Memory trick: Think "n plus one, divided by two" for the median position.

Examples:

  • n = 5: (5 + 1) ÷ 2 = 3, so median is the 3rd value
  • n = 6: (6 + 1) ÷ 2 = 3.5, so median is average of 3rd and 4th values
  • n = 7: (7 + 1) ÷ 2 = 4, so median is the 4th value

The "+1" accounts for the fact that we're looking for a position, not just dividing the count.

Question: When is the median better than the mean? They both seem to measure the center.

Answer: The median is better than the mean in these situations:

  • When outliers are present: Median is resistant to extreme values
  • When data is skewed: Median represents typical values better
  • When data is ordinal: For rankings or categories with order

The key difference is that the mean is affected by every value in the data set, while the median only depends on position.

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

But for [10, 12, 14, 16, 100], mean = 30.4 while median = 14

The outlier (100) dramatically changes the mean but doesn't affect the median. This makes median more robust for skewed data or data with outliers.

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

Answer: Yes, there are several verification methods:

  1. Count Check: Ensure you have the right number of values before and after the median
  2. Sort Verification: Confirm that your data is truly in ascending order
  3. Position Check: For n values, verify that median is at the correct position
  4. Logic Check: Does your median make sense in the context of the data?

For odd count (n values): There should be (n-1)÷2 values before median and (n-1)÷2 values after.

For even count (n values): There should be (n÷2)-1 values before the middle pair and (n÷2)-1 values after.

Example: In [5, 10, 15, 20, 25], median = 15. There are 2 values before (5, 10) and 2 after (20, 25). ✓

Always double-check your sorting and counting to ensure accuracy!

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

Answer: Median calculations are very common in everyday life:

  • Housing: Median home prices (not affected by luxury mansions)
  • Income: Median household income (more representative than mean)
  • Testing: SAT/ACT scores, medical test results
  • Business: Median sales figures, customer satisfaction ratings
  • Health: Growth percentiles, vital signs
  • Sports: Performance rankings, competition results

The median is especially useful when there are extreme values that would distort the mean. For example, reporting median income gives a better picture of typical earnings than mean income, which could be skewed by very high earners.

Understanding medians helps you interpret statistical reports and make better decisions based on data in both academic and real-world contexts!