Questions For Mean Median Mode And Range

Understanding mean, median, mode, and range is fundamental for anyone delving into statistics and data analysis. These measures provide insights into the central tendencies and variability within a dataset. This article explores various types of questions related to mean, median, mode, and range, offering detailed explanations and examples to help you master these concepts.

Understanding Mean, Median, Mode, and Range

Before diving into specific questions, let's define each term:

• Mean: The average of a dataset. It is calculated by summing all the values and dividing by the number of values.
• Median: The middle value in a dataset when it is ordered from least to greatest. If there is an even number of values, the median is the average of the two middle numbers.
• Mode: The value that appears most frequently in a dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).
• Range: The difference between the largest and smallest values in a dataset.

Basic Calculation Questions

These questions focus on directly computing the mean, median, mode, and range from a given dataset.

Question 1:

Find the mean, median, mode, and range of the following dataset: 4, 7, 2, 9, 5, 5, 8, 3.

Solution:

1. Mean:
   • Sum of values: 4 + 7 + 2 + 9 + 5 + 5 + 8 + 3 = 43
   • Number of values: 8
   • Mean = 43 / 8 = 5.375
2. Median:
   • First, order the dataset: 2, 3, 4, 5, 5, 7, 8, 9
   • Since there are 8 values (an even number), the median is the average of the two middle numbers (5 and 5).
   • Median = (5 + 5) / 2 = 5
3. Mode:
   • The number 5 appears twice, which is more frequent than any other number in the dataset.
   • Mode = 5
4. Range:
   • Largest value: 9
   • Smallest value: 2
   • Range = 9 - 2 = 7

Question 2:

Calculate the mean, median, mode, and range for the dataset: 12, 15, 18, 21, 15, 12, 24, 15.

Solution:

1. Mean:
   • Sum of values: 12 + 15 + 18 + 21 + 15 + 12 + 24 + 15 = 132
   • Number of values: 8
   • Mean = 132 / 8 = 16.5
2. Median:
   • Order the dataset: 12, 12, 15, 15, 15, 18, 21, 24
   • The two middle numbers are 15 and 15.
   • Median = (15 + 15) / 2 = 15
3. Mode:
   • The number 15 appears three times, which is more frequent than any other number.
   • Mode = 15
4. Range:
   • Largest value: 24
   • Smallest value: 12
   • Range = 24 - 12 = 12

Question 3:

What are the mean, median, mode, and range of the following numbers: 3, 6, 2, 7, 9, 2, 4, 8, 5?

Solution:

1. Mean:
   • Sum the numbers: 3 + 6 + 2 + 7 + 9 + 2 + 4 + 8 + 5 = 46
   • Count the numbers: There are 9 numbers.
   • Calculate the mean: 46 / 9 ≈ 5.11
2. Median:
   • Arrange the numbers in ascending order: 2, 2, 3, 4, 5, 6, 7, 8, 9
   • The median is the middle number: 5
3. Mode:
   • The number that appears most frequently is 2 (appears twice).
   • Mode = 2
4. Range:
   • The largest number is 9 and the smallest is 2.
   • Range = 9 - 2 = 7

Advanced Calculation Questions

These questions require a deeper understanding and application of the concepts.

Question 4:

The heights of five basketball players are 180 cm, 185 cm, 190 cm, 195 cm, and 200 cm. What is the mean height of the players?

Solution:

1. Mean:
   • Sum of heights: 180 + 185 + 190 + 195 + 200 = 950
   • Number of players: 5
   • Mean height = 950 / 5 = 190 cm

Question 5:

A class of 10 students took a test. The scores are 60, 70, 70, 80, 85, 90, 90, 90, 95, 100. Find the median and mode of the scores.

Solution:

1. Median:
   • The scores are already in order.
   • Since there are 10 scores (an even number), the median is the average of the two middle scores (85 and 90).
   • Median = (85 + 90) / 2 = 87.5
2. Mode:
   • The score 90 appears three times, which is more frequent than any other score.
   • Mode = 90

Question 6:

The monthly sales for a small business over a year are: $10,000, $12,000, $11,000, $13,000, $12,000, $14,000, $15,000, $13,000, $12,000, $16,000, $14,000, $12,000. Calculate the mean, median, mode, and range of the monthly sales.

Solution:

1. Mean:
   • Sum of sales: 10,000 + 12,000 + 11,000 + 13,000 + 12,000 + 14,000 + 15,000 + 13,000 + 12,000 + 16,000 + 14,000 + 12,000 = 154,000
   • Number of months: 12
   • Mean sales = 154,000 / 12 ≈ $12,833.33
2. Median:
   • Order the sales data: 10,000, 11,000, 12,000, 12,000, 12,000, 12,000, 13,000, 13,000, 14,000, 14,000, 15,000, 16,000
   • The two middle values are 12,000 and 13,000.
   • Median = (12,000 + 13,000) / 2 = $12,500
3. Mode:
   • The value 12,000 appears four times, which is more frequent than any other value.
   • Mode = $12,000
4. Range:
   • Largest value: $16,000
   • Smallest value: $10,000
   • Range = 16,000 - 10,000 = $6,000

Word Problems

These questions involve applying the concepts of mean, median, mode, and range to real-world scenarios.

Question 7:

A teacher recorded the scores of a quiz for 15 students: 5, 6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10. Find the mean, median, and mode of the quiz scores.

Solution:

1. Mean:
   • Sum of scores: 5 + 6 + (2*7) + (3*8) + (4*9) + (4*10) = 5 + 6 + 14 + 24 + 36 + 40 = 125
   • Number of students: 15
   • Mean score = 125 / 15 ≈ 8.33
2. Median:
   • The scores are already in order.
   • Since there are 15 scores (an odd number), the median is the middle score, which is the 8th score.
   • Median = 9
3. Mode:
   • The score 9 appears four times, and the score 10 also appears four times. Thus, this dataset is bimodal.
   • Modes = 9 and 10

Question 8:

The number of customers visiting a store each day for a week is: 20, 25, 30, 22, 28, 24, 26. Calculate the mean and range of the number of customers.

Solution:

1. Mean:
   • Sum of customers: 20 + 25 + 30 + 22 + 28 + 24 + 26 = 175
   • Number of days: 7
   • Mean number of customers = 175 / 7 = 25
2. Range:
   • Largest number of customers: 30
   • Smallest number of customers: 20
   • Range = 30 - 20 = 10

Question 9:

John recorded the temperature (in degrees Celsius) each day for a week: 22, 25, 23, 24, 26, 25, 24. Find the median and mode of the temperatures.

Solution:

1. Median:
   • Order the temperatures: 22, 23, 24, 24, 25, 25, 26
   • Since there are 7 temperatures (an odd number), the median is the middle temperature, which is the 4th temperature.
   • Median = 24
2. Mode:
   • The temperatures 24 and 25 both appear twice. Thus, this dataset is bimodal.
   • Modes = 24 and 25

Questions Involving Algebra

These questions require using algebraic equations to find missing values when given the mean, median, mode, or range.

Question 10:

The mean of five numbers is 10. If four of the numbers are 5, 8, 12, and 15, what is the fifth number?

Solution:

1. Let the fifth number be x.
2. The sum of the five numbers is 5 + 8 + 12 + 15 + x.
3. The mean is (5 + 8 + 12 + 15 + x) / 5 = 10.
4. Solve for x:
   • 5 + 8 + 12 + 15 + x = 50
   • 40 + x = 50
   • x = 50 - 40
   • x = 10
   • The fifth number is 10.

Question 11:

The median of the numbers 3, 5, x, 9, 11 is 7. Find the value of x.

Solution:

1. Since the median is 7, and the numbers are already in ascending order except for x, we can deduce that x must be 7 to maintain the median value.
2. The ordered list is 3, 5, x, 9, 11. If x is less than 7, the median would be x. If x is greater than 7, the median would be 9. Therefore, x must be 7 for the median to be 7.

Question 12:

The mode of the numbers 2, 4, 6, 2, x, 4, 2, 8 is 2. What is the smallest possible value of x?

Solution:

1. Since the mode is 2, the number 2 must appear more frequently than any other number.
2. Currently, 2 appears three times, and 4 appears twice.
3. To maintain 2 as the mode, x cannot be 4 (otherwise, 2 and 4 would both appear three times, resulting in a bimodal dataset).
4. x can be any number other than 4 that doesn't appear more than three times. The smallest possible value for x is any number less than 4 other than 2. If x = 0 or 1, 2 will still be the mode.

Question 13:

The range of a set of numbers is 15. If the largest number is 27, what is the smallest number?

Solution:

1. Range = Largest number - Smallest number
2. 15 = 27 - Smallest number
3. Smallest number = 27 - 15
4. Smallest number = 12

Questions Involving Data Interpretation

These questions require you to interpret data presented in tables, charts, or graphs to find the mean, median, mode, and range.

Question 14:

Given the following frequency table of test scores, find the mean, median, and mode:

Solution:

1. Mean:
   • Multiply each score by its frequency: (60*2) + (70*3) + (80*5) + (90*4) + (100*1) = 120 + 210 + 400 + 360 + 100 = 1190
   • Total number of scores: 2 + 3 + 5 + 4 + 1 = 15
   • Mean score = 1190 / 15 ≈ 79.33
2. Median:
   • Since there are 15 scores, the median is the 8th score.
   • The cumulative frequencies are: 2, 5, 10, 14, 15. The 8th score falls in the category of 80.
   • Median = 80
3. Mode:
   • The score with the highest frequency is 80 (frequency = 5).
   • Mode = 80

Question 15:

The following data represents the number of books read by students in a class during the summer:

2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8

Find the mean, median, mode, and range of the number of books read.

Solution:

1. Mean:
   • Sum of books read: 2 + (2*3) + (3*4) + (2*5) + (4*6) + (2*7) + 8 = 2 + 6 + 12 + 10 + 24 + 14 + 8 = 76
   • Number of students: 15
   • Mean number of books = 76 / 15 ≈ 5.07
2. Median:
   • Since there are 15 students, the median is the 8th value.
   • The 8th value is 5.
   • Median = 5
3. Mode:
   • The number 6 appears four times, which is more frequent than any other number.
   • Mode = 6
4. Range:
   • Largest number of books: 8
   • Smallest number of books: 2
   • Range = 8 - 2 = 6

Critical Thinking Questions

These questions require a deeper understanding of how changes in a dataset affect the mean, median, mode, and range.

Question 16:

If you add the same number to every value in a dataset, what happens to the mean, median, mode, and range?

Solution:

• Mean: The mean will increase by the same number.
• Median: The median will also increase by the same number.
• Mode: The mode will increase by the same number.
• Range: The range will remain unchanged because the difference between the largest and smallest values remains the same.

Question 17:

If you multiply every value in a dataset by a constant, what happens to the mean, median, mode, and range?

Solution:

• Mean: The mean will be multiplied by the same constant.
• Median: The median will also be multiplied by the same constant.
• Mode: The mode will be multiplied by the same constant.
• Range: The range will be multiplied by the same constant.

Question 18:

Consider a dataset with a mean of 50. If one value is changed from 60 to 80, how does the mean change?

Solution:

1. Let the number of values in the dataset be n.
2. The original sum of the values is 50*n.
3. When 60 is changed to 80, the sum increases by 20 (80 - 60 = 20).
4. The new sum is 50*n + 20.
5. The new mean is (50*n + 20) / n = 50 + (20 / n).
6. The mean increases by 20 / n.

Conclusion

Mastering the concepts of mean, median, mode, and range involves understanding their definitions and practicing various types of questions. This article has provided a comprehensive overview of the different types of questions you might encounter, from basic calculations to more complex algebraic and data interpretation problems. By working through these examples, you can build a solid foundation in descriptive statistics and improve your ability to analyze and interpret data effectively.