Mean (Arithmetic Average)
When your teacher announces the class test results, she might say, "The average score of the class is 72." What does this mean? The mean, also called the arithmetic average, is a single number that represents the central value of a group of numbers.
The mean gives you an idea of what a "typical" value in the data looks like. It is one of the most commonly used measures of central tendency in mathematics and real life.
In Class 7 NCERT Maths, you will learn how to calculate the mean of a data set, understand what it tells you, and use it to solve problems related to marks, temperatures, runs, and more.
What is Mean (Arithmetic Average) - Grade 7 Maths (Data Handling)?
Definition: The mean (or arithmetic average) of a set of observations is the sum of all observations divided by the total number of observations.
In simple words:
- Add up all the values.
- Divide the total by how many values there are.
- The result is the mean.
Mean (Arithmetic Average) Formula
Formula:
Mean = Sum of all observations / Number of observations
Where:
- Sum of all observations = Add every value in the data set
- Number of observations = The count of values in the data set
If the observations are x₁, x₂, x₃, ..., xₙ, then:
Mean = (x₁ + x₂ + x₃ + ... + xₙ) / n
Types and Properties
Key facts about the mean:
- The mean does not have to be one of the values in the data set. For example, the mean of 2, 4, and 6 is 4, which is in the set. But the mean of 3, 5, and 7 is 5, and the mean of 1, 2, and 6 is 3, which is not one of the original values.
- The mean is affected by very large or very small values (called outliers). For example, if five students scored 40, 42, 45, 43, and 95, the mean is 53 — higher than most of the scores because of the one high score.
- The mean is useful when the data values are spread fairly evenly without extreme values.
Mean of grouped data (frequency table):
When data is given in a frequency table, use this formula:
Mean = Sum of (observation x frequency) / Sum of frequencies
That is: Mean = Σ(f × x) / Σf
Solved Examples
Example 1: Finding the Mean of Test Scores
Problem: The marks obtained by 5 students in a maths test are: 80, 65, 72, 90, 78. Find the mean.
Solution:
Given: Observations = 80, 65, 72, 90, 78. Number of observations = 5.
Step 1: Find the sum: 80 + 65 + 72 + 90 + 78 = 385
Step 2: Divide by the number of observations: 385 / 5 = 77
Answer: The mean score is 77.
Example 2: Mean of Daily Temperatures
Problem: The temperatures recorded over a week (Mon to Sun) were: 32°C, 35°C, 33°C, 30°C, 31°C, 34°C, 33°C. Find the mean temperature.
Solution:
Given: 7 observations.
Step 1: Sum = 32 + 35 + 33 + 30 + 31 + 34 + 33 = 228
Step 2: Mean = 228 / 7 = 32.57°C (approx.)
Answer: The mean temperature is approximately 32.57°C.
Example 3: Finding a Missing Value Using the Mean
Problem: The mean of 4 numbers is 25. Three of the numbers are 20, 30, and 22. Find the fourth number.
Solution:
Given: Mean = 25, Number of observations = 4.
Step 1: Sum of all 4 numbers = Mean × Number of observations = 25 × 4 = 100
Step 2: Sum of known numbers = 20 + 30 + 22 = 72
Step 3: Fourth number = 100 − 72 = 28
Answer: The fourth number is 28.
Example 4: Mean of Cricket Runs
Problem: A batsman scored the following runs in 6 innings: 45, 52, 0, 33, 67, 23. Find his average (mean) score.
Solution:
Step 1: Sum = 45 + 52 + 0 + 33 + 67 + 23 = 220
Step 2: Mean = 220 / 6 = 36.67 (approx.)
Answer: The mean score is approximately 36.67 runs.
Example 5: Mean from a Frequency Table
Problem: The number of goals scored in 20 football matches is given below:
- Goals: 0, 1, 2, 3, 4
- Frequency: 4, 6, 5, 3, 2
Find the mean number of goals per match.
Solution:
Step 1: Calculate f × x for each:
- 0 × 4 = 0
- 1 × 6 = 6
- 2 × 5 = 10
- 3 × 3 = 9
- 4 × 2 = 8
Step 2: Σ(f × x) = 0 + 6 + 10 + 9 + 8 = 33
Step 3: Σf = 4 + 6 + 5 + 3 + 2 = 20
Step 4: Mean = 33 / 20 = 1.65
Answer: The mean number of goals per match is 1.65.
Example 6: Effect of Adding a New Value
Problem: The mean weight of 5 students is 42 kg. A new student weighing 54 kg joins. Find the new mean.
Solution:
Step 1: Total weight of 5 students = 42 × 5 = 210 kg
Step 2: New total = 210 + 54 = 264 kg
Step 3: New mean = 264 / 6 = 44 kg
Answer: The new mean weight is 44 kg.
Example 7: Mean of First 5 Natural Numbers
Problem: Find the mean of the first 5 natural numbers.
Solution:
Step 1: First 5 natural numbers: 1, 2, 3, 4, 5
Step 2: Sum = 1 + 2 + 3 + 4 + 5 = 15
Step 3: Mean = 15 / 5 = 3
Answer: The mean of the first 5 natural numbers is 3.
Example 8: Word Problem on Average Marks
Problem: Rohan scored 85 in English, 76 in Hindi, 92 in Maths, 68 in Science, and 79 in Social Studies. What is his average score?
Solution:
Step 1: Sum = 85 + 76 + 92 + 68 + 79 = 400
Step 2: Number of subjects = 5
Step 3: Mean = 400 / 5 = 80
Answer: Rohan's average score is 80.
Example 9: Finding Total Using Mean
Problem: The mean height of 8 plants is 15 cm. What is the total height of all 8 plants?
Solution:
Step 1: Mean = Sum / Number of observations
Step 2: So, Sum = Mean × Number of observations = 15 × 8 = 120 cm
Answer: The total height of all 8 plants is 120 cm.
Example 10: Mean with Decimal Values
Problem: Find the mean of: 3.5, 4.2, 5.8, 6.1, 2.4
Solution:
Step 1: Sum = 3.5 + 4.2 + 5.8 + 6.1 + 2.4 = 22.0
Step 2: Mean = 22.0 / 5 = 4.4
Answer: The mean is 4.4.
Real-World Applications
Real-life uses of the mean:
- School results: Teachers calculate the class average to understand overall performance.
- Cricket: A batsman's batting average is the mean of runs scored across innings.
- Weather: Meteorologists report average temperatures for a month or season.
- Business: Companies calculate average sales per month to track performance.
- Health: Doctors check average blood pressure readings over multiple visits.
- Economics: Per capita income is the mean income of a country's population.
Key Points to Remember
- Mean = Sum of all observations / Number of observations.
- The mean is also called the arithmetic average.
- The mean may or may not be one of the values in the data set.
- The mean is affected by extreme values (outliers).
- For grouped data: Mean = Σ(f × x) / Σf.
- If you know the mean and the number of observations, you can find the total sum.
- If you know the mean and all but one observation, you can find the missing value.
- Mean is one of the three measures of central tendency (mean, median, mode).
Practice Problems
- Find the mean of: 12, 18, 22, 15, 28.
- The mean of 6 numbers is 14. Five of the numbers are 10, 12, 16, 18, and 15. Find the sixth number.
- A student scored 75, 82, 68, 90, and 85 in five subjects. Find the average score.
- The runs scored by a batsman in 8 innings are: 56, 0, 34, 72, 45, 23, 67, 11. Find the mean.
- The mean monthly rainfall for 4 months is 120 mm. What is the total rainfall for these 4 months?
- In a class, 5 students weigh 38 kg, 42 kg, 35 kg, 40 kg, and 45 kg. If a student weighing 50 kg joins, what is the new mean?
Frequently Asked Questions
Q1. What is the mean in maths?
The mean is the sum of all observations divided by the number of observations. It is also called the arithmetic average. For example, the mean of 4, 6, and 8 is (4 + 6 + 8) / 3 = 6.
Q2. What is the difference between mean and average?
In Class 7 maths, mean and average refer to the same thing — the arithmetic average. The word 'average' is used more in everyday language, while 'mean' is the mathematical term.
Q3. Can the mean be a decimal?
Yes. The mean can be a decimal even when all the observations are whole numbers. For example, the mean of 3, 4, and 5 is 4, but the mean of 3, 4, and 6 is 4.33.
Q4. How do you find a missing number if the mean is given?
Multiply the mean by the number of observations to get the total sum. Then subtract the sum of the known values. The result is the missing number.
Q5. Does the mean have to be one of the values in the data?
No. The mean does not have to be one of the values. For example, the mean of 2 and 5 is 3.5, which is not in the data set.
Q6. What happens to the mean when a very large value is added?
The mean increases. Very large or very small values (called outliers) pull the mean towards them. This is why the mean is said to be sensitive to extreme values.
Q7. What is the mean of the first 10 natural numbers?
The first 10 natural numbers are 1, 2, 3, ..., 10. Their sum is 55. Mean = 55 / 10 = 5.5.
Q8. How is mean different from median and mode?
Mean is the sum divided by count. Median is the middle value when data is arranged in order. Mode is the value that occurs most often. All three are measures of central tendency but can give different results for the same data.
