Average
Average, or the arithmetic mean, is the single value that best represents a group of values, and in the ADRE Grade IV examination it is one of the easiest chapters to score from. The average of a set of values is found by adding all the values and dividing the sum by the number of values. The idea is so simple that it looks almost too easy, yet questions on average are asked every year in some form, and the same idea also lies behind several questions on mixture, speed and data interpretation. The method followed in the Master Book: Maths Masti (D. Hurina) is to write the total and the count clearly before doing anything else, because almost every average problem becomes simple once the total is known. The chapter rewards careful reading more than hard calculation, and a candidate who masters the two or three standard results given below can solve most questions in a few seconds.
Basic Terms
- Average or mean: the sum of all the observations divided by the number of observations.
- Observation: each individual value in the set.
- Total: the sum of all the observations.
- Number of terms: how many observations there are.
- Weighted average: an average in which some values count for more than others.
- Average speed: the total distance divided by the total time.
Only two quantities matter in every average question, the total and the count, and the average is simply the total divided by the count. Rearranging this one relation gives the total when the average and the count are known, and the count when the average and the total are known.
The Basic Formula and Its Rearrangements
Average equals the sum of the observations divided by the number of observations. From this, the sum equals the average multiplied by the number, and the number equals the sum divided by the average. Two consequences are used again and again. If the average of a group of n values rises by x, the total rises by n multiplied by x. And if a person is replaced by another and the average rises by x, the new person is greater than the old one by n multiplied by x. These small results turn long questions into a single line of work.
Average of Consecutive Numbers
The average of consecutive numbers is the middle term. For an odd number of terms it is the middle number itself, and for an even number of terms it is the average of the two middle numbers. The average of the first n natural numbers is n plus one divided by two. The average of the first n even numbers is n plus one, and the average of the first n odd numbers is simply n. Recognising a run of consecutive numbers saves the whole addition, and questions on roll numbers, consecutive ages and page numbers are solved instantly with this idea.
Weighted Average
When two or more groups of different sizes are combined, the average of the whole is the total of all the values divided by the total number of values, and it is not the simple average of the separate averages. This is the single most tested trap of the chapter. If a class of ten students has an average of sixty and a class of twenty students has an average of thirty, the combined average is the total marks divided by thirty, which is forty, and not the mid-value of the two averages.
Finding Missing Values and Replacement
If a value is removed from a set, the new total is the old total minus that value, and the new average follows at once. If a new value is added, the new total is the old total plus the new value. In replacement questions, one person leaves and another joins, and the change in the total is the difference between the two values. These questions become easy as soon as the candidate keeps the two running figures, the sum and the count, in view.
Average Speed
Average speed is the total distance divided by the total time, and it is not the average of the several speeds. If a journey is covered in two equal distances at speeds x and y, the average speed is twice x y divided by x plus y. When the distances are not equal, the safest method is to find the total distance and the total time separately and then divide. Confusing average speed with the mean of the speeds is a frequent mistake.
Applications in Ages, Marks and Money
Average questions in the paper are usually set in familiar situations. The average age of a group changes when a member joins or leaves. The average of runs, marks or income is found from the total. Averages are also used to find a new member's age, a missing mark, or the average of the remaining items after some are removed. In each case the route is the same: find the relevant total, adjust it, and divide by the relevant count.
How This Chapter Appears in the Examination
Average questions in ADRE Grade IV, APSC and Assam Police papers usually ask the candidate to find the average of a given set, to find the missing value when the average is known, to find the effect of joining or leaving on the average, or to combine two groups of different sizes. Data interpretation sets use average to report a representative figure, and the concept lies behind many questions on speed and on mixture. Because the method is short and fixed, this is one of the surest scoring sections.
Shortcut Methods for Rapid Calculation
Use the assumed mean method when the numbers are large and close together. Take a round number near the values, find the average of the differences from it, and add it back to the assumed mean. For consecutive numbers use the middle term. In replacement questions use the change in the total directly instead of recalculating the whole average. These shortcuts save time and reduce the chance of an arithmetic slip.
Common Mistakes to Avoid
- Averaging the averages instead of the totals.
- Using the wrong number of terms after a value is added or removed.
- Taking the average of the speeds instead of total distance divided by total time.
- Assuming the average must be a whole number, when it may be a fraction or a decimal.
- Forgetting to change the count when the size of the group changes.
How to Prepare This Topic
- Learn the one basic formula and its two rearrangements.
- Memorise the results for consecutive, even and odd numbers.
- Practise weighted average with groups of different sizes.
- Work on replacement, missing value and average speed questions.
- Finish with a mixed set and revise the total method every week.
Frequently Asked Questions
What is the difference between average and average speed?
Average is the sum divided by the count. Average speed is the total distance divided by the total time. The two are different, and average speed is not the mean of the speeds used on a journey.
Why can I not average the two averages?
Because the two groups may have different numbers of items. The correct average of the combined group is the total of all the values divided by the total count, which is a weighted average and not the mid-value of the two averages.
What is the average of the first ten natural numbers?
The average of the first n natural numbers is n plus one divided by two, so for ten numbers it is eleven divided by two, which is five point five. The sum is fifty-five, and dividing it by ten gives the same answer.
How do I handle replacement questions?
Compare the old total with the new total. The change in the total is the difference between the value that left and the value that joined. Dividing this change by the number of members gives the change in the average, and the relation also works in the reverse direction.
Is the average always an integer?
No. The average may be a fraction or a decimal, and reporting it correctly matters. It is an integer only when the total is an exact multiple of the count.