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Average is a numeric value in Mathematics that is used to represent a large amount of data. It uses a single number to represent all the other numbers that you might find in a large data set. The calculation of average has numerous practical applications and understanding it helps us handle large sets of data more easily. Sometimes it is not possible to make decisions based on one single data or a large set of data, thus, the average value is taken as it helps to represent all the values in a single value. The average can be easily calculated by adding all the data values and dividing them by the number of items. In simpler terms, the average can be defined as the ratio of the sum of all given observations to the total number of observations.
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Key Takeaways: Average, Arithmetic Mean, Ratio, Observations, Data, Sum, Whole Numbers, Data Sets, Average Formula
Definition of Average
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Average refers to the arithmetic mean of a set of numbers which is equal to the ratio of the summation of all the numbers present in a particular set to the total number of numbers present in that set. Therefore, to put it simply, an average is the sum of all given values divided by the total number of values present.
Average
Average Formula
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Solving the average of a given set of numbers or observations requires a simple formula which is as follows: Sum of all the given values/Total number of values given.
Average Formula
For example, the age of all the students in a class is taken and an average is calculated to get a single value that represents the average age of the students of a class.
Read More: Difference between Average and Mean
Steps to Calculate Average
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Suppose you are given a set of observations that contain the following values: 12, 34, 45, 46, 267. 100.
- Step 1: As per the given formula, you will first need to find the sum of all the given values which is (12+34+45+46+267+100) 504
- Step 2: Next, you will have to count the total number of values given which is 6
- Step 3: Now, by using the formula mentioned above, you can calculate the average of the given set of numbers as follows: 504/6, which is equal to 84. Thus, the average for the given set of numbers here will be 84.
Symbol of Average
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\(\overline {X}\)
To use average in mathematics, it is important that we familiarize ourselves with the symbol that is widely used to represent the average value of a set of observations. We use the symbol ‘\(\overline {X}\)’, also known as X bar, to denote average. Other than the X bar, we also use the symbol ‘μ’ to denote average in certain cases.
Read More: Geometric Mean Formula
Applications or Uses of Average Calculation
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The calculation of Average has numerous applications in real life and helps us deal with many problems on a day-to-day basis. Some of them are as follows:
- We cannot study the exact amount of time every day but keeping an average time goal for a week helps us meet our study requirements for a particular week.
- Investing in shares and stocks is a risky business as the prices of shares keep changing every week. Therefore, it is important that we are able to calculate the average value of the shares of a certain company to decide whether it would be wise to invest in them or not.
- The time taken by a train to travel between two places does not remain the same every time. Therefore, by calculating the average time taken by the train to travel between two places, we can form a correct estimate regarding how long it will take us to reach our own destination.
- Average calculation is especially useful in sports. Coaches often calculate the different speeds and other such stats of sportspersons over a particular period of time to calculate their average. This helps them keep track of the player’s condition at any given time.
- Average is also useful for population surveys. By calculating the average number of males and females in a certain area we are able to learn the male to female ratio of that area and other such important details.
Read More: Difference between Mean, Median and Mode
Things to Remember
- Average is the ratio of the summation of given values to the total number of values.
- Average helps us with a single number that is used to represent all the other numbers in a large data set.
- We use the or μ to denote average in mathematics.
- The formula for calculating the average is (sum of given numbers)/ (total number of numbers)
- Average helps us to deal with large amounts of data in real life.
Sample Questions
Ques. Ratna practices singing for 4 hours, 6 hours and 8 hours respectively on 3 consecutive days. How much did Ratna practice daily on average? (3 Marks)
Ans. The total number of practice hours = 4+6+8
Total number of days for which she practised = 3
Therefore, the average practice time of Ratna would be = (4+6+8)/3 = 6 hours.
Thus, Ratna practices 6 hours on average.
Ques. What is the average of the first 7 prime numbers? (2 Marks)
Ans. The first 7 prime numbers are = 2,3,5,7,11,13,17
Therefore, the average of the first 7 prime numbers is = (2+3+5+7+11+13+17)/7 = 8.285
Thus, the average of the first 7 prime numbers is equal to 8.285.
Ques. A batsman scored 12, 30, 45, 32 and 25 runs in 5 separate innings. What is the average number of runs scored by him per inning? (3 Marks)
Ans. The total number of runs scored by the batsman = 12+30+45+32+25
Total number of innings played by the batsman = 5
Therefore, the average number of runs scored by the batsman per inning= (12+30+45+32+25)/5 = 28.8 runs
Thus, average runs scored by the batsman per inning is 28.8
Ques. Use the system of average calculation to find at least 2 numbers between 1/3 and 1/4. (3 Marks)
Ans. The average of 1/3 and 1/4 = (1/3+1/4)/2 = 7/24
The average of 1/3 and 7/24 = (1/3+7/24)/2 = 15/48
Therefore, 2 numbers between 1/3 and 1/4 are 7/24 and 15/48.
Ques. The average height of 5 family members is 160 cm. If the heights of 4 family members are 161 cm, 162 cm, 164 cm and 163 cm, what is the height of the 5th family member? (3 Marks)
Ans. The summation of the heights of the 4 family members = (161+162+164+163) 650 cm
Total number of family members = 5
Average height of 5 members, as given = 160 cm,
Therefore, the height of the 5th member = (160 x 5) – 650 = 150 cm
Thus, the height of the 5th family member is 150 cm.
Ques. Given below are the marks scored out of 100 by Rohan, Rita and Raj in three different subjects. Whose average is the highest and by how much? (5 Marks)
| Subjects | Rohan | Rita | Raj |
|---|---|---|---|
| English | 72 | 75 | 81 |
| Maths | 99 | 85 | 80 |
| History | 65 | 72 | 85 |
Ans. Total marks scored by Rohan in 3 subjects = (72+99+65) 236
Therefore, his average marks = 236/3 = 78.667
Total marks scored by Rita in 3 subjects = (75+85+72) 232
Therefore, her average marks = 232/3 = 77.333
Total marks scored by Raj in 3 subjects = (81+80+85) 246
Therefore, his average marks = 246/3 = 82
Thus, Raj has the highest average.
Difference between the average of Raj and Rohan = (82-78.667) = 3.333
Difference between the average of Raj and Rita = (82-77.333) = 4.667
Thus, Raj’s average marks is 3.333 marks higher than Rohan and 4.667 marks higher than Rita.
Ques. Over a period of 4 days, a train took 5 hours 16 minutes, 4 hours 22 minutes, 4 hours 45 minutes and 5 hours respectively to reach its destination. Find the average time taken by the train to reach its destination. (3 Marks)
Ans. Total time taken by the train on the first day = (5x60) + 16 = 316 minutes
Total time taken by the train on the second day = (4x60) + 22 = 262 minutes
Total time taken by the train on the third day = (4x60) + 45 = 285 minutes
Total time taken by the train on the fourth day = 4x60 = 240 minutes
Therefore, the average time taken by the train = (316+262+285+240)/4 = 275.75 minutes or 4.596 (275.75/60) hours.
Thus, the average time taken by the train to reach its destination is 275.75 minutes.
Ques. A café made a total sale of 2,30,000 rupees in a week. What is their average per day sale? (3 Marks)
Ans. Total sales in a week = 2,30,000 rupees
Total number of days in a week = 7
Therefore, average sales per day = 2,30,000/7 = 32,857.14 Rupees
Thus, the café made an average sale of 32,857.14 Rupees per day.
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